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<a href="../derivatives/more_zeros.html" class="sidebar-item-text sidebar-link"><span class="chapter-number">31</span> <span class="chapter-title">Derivative-free alternatives to Newton’s method</span></a>
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<a href="../derivatives/lhospitals_rule.html" class="sidebar-item-text sidebar-link"><span class="chapter-number">32</span> <span class="chapter-title">L’Hospital’s Rule</span></a>
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<a href="../derivatives/implicit_differentiation.html" class="sidebar-item-text sidebar-link"><span class="chapter-number">33</span> <span class="chapter-title">Implicit Differentiation</span></a>
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<a href="../integrals/ftc.html" class="sidebar-item-text sidebar-link"><span class="chapter-number">37</span> <span class="chapter-title">Fundamental Theorem or Calculus</span></a>
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<a href="../ODEs/solve.html" class="sidebar-item-text sidebar-link"><span class="chapter-number">50</span> <span class="chapter-title">The problem-algorithm-solve interface</span></a>
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<a href="../ODEs/differential_equations.html" class="sidebar-item-text sidebar-link"><span class="chapter-number">51</span> <span class="chapter-title">The <code>DifferentialEquations</code> suite</span></a>
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<a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-6" aria-expanded="false">Differential vector calculus</a>
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<a href="../differentiable_vector_calculus/vector_valued_functions.html" class="sidebar-item-text sidebar-link"><span class="chapter-number">54</span> <span class="chapter-title">Vector-valued functions, <span class="math inline">\(f:R \rightarrow R^n\)</span></span></a>
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<a href="../differentiable_vector_calculus/scalar_functions.html" class="sidebar-item-text sidebar-link"><span class="chapter-number">55</span> <span class="chapter-title">Scalar functions</span></a>
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<a href="../differentiable_vector_calculus/scalar_functions_applications.html" class="sidebar-item-text sidebar-link"><span class="chapter-number">56</span> <span class="chapter-title">Applications with scalar functions</span></a>
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||
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<a href="../differentiable_vector_calculus/vector_fields.html" class="sidebar-item-text sidebar-link"><span class="chapter-number">57</span> <span class="chapter-title">Functions <span class="math inline">\(R^n \rightarrow R^m\)</span></span></a>
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<a href="../differentiable_vector_calculus/plots_plotting.html" class="sidebar-item-text sidebar-link"><span class="chapter-number">58</span> <span class="chapter-title">2D and 3D plots in Julia with Plots</span></a>
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<a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-7" aria-expanded="false">Integral vector calculus</a>
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<a href="../integral_vector_calculus/double_triple_integrals.html" class="sidebar-item-text sidebar-link"><span class="chapter-number">59</span> <span class="chapter-title">Multi-dimensional integrals</span></a>
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<a href="../integral_vector_calculus/line_integrals.html" class="sidebar-item-text sidebar-link"><span class="chapter-number">60</span> <span class="chapter-title">Line and Surface Integrals</span></a>
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<a href="../integral_vector_calculus/div_grad_curl.html" class="sidebar-item-text sidebar-link"><span class="chapter-number">61</span> <span class="chapter-title">The Gradient, Divergence, and Curl</span></a>
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<h2 id="toc-title">Table of contents</h2>
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<ul>
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<li><a href="#the-slope-of-the-secant-line" id="toc-the-slope-of-the-secant-line" class="nav-link active" data-scroll-target="#the-slope-of-the-secant-line"> <span class="header-section-number">22.1</span> The slope of the secant line</a></li>
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<li><a href="#the-slope-of-the-tangent-line" id="toc-the-slope-of-the-tangent-line" class="nav-link" data-scroll-target="#the-slope-of-the-tangent-line"> <span class="header-section-number">22.2</span> The slope of the tangent line</a></li>
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||
<li><a href="#the-derivative" id="toc-the-derivative" class="nav-link" data-scroll-target="#the-derivative"> <span class="header-section-number">22.3</span> The derivative</a>
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<ul class="collapse">
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<li><a href="#some-basic-derivatives" id="toc-some-basic-derivatives" class="nav-link" data-scroll-target="#some-basic-derivatives"> <span class="header-section-number">22.3.1</span> Some basic derivatives</a></li>
|
||
</ul></li>
|
||
<li><a href="#rules-of-derivatives" id="toc-rules-of-derivatives" class="nav-link" data-scroll-target="#rules-of-derivatives"> <span class="header-section-number">22.4</span> Rules of derivatives</a>
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<ul class="collapse">
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<li><a href="#power-rule" id="toc-power-rule" class="nav-link" data-scroll-target="#power-rule"> <span class="header-section-number">22.4.1</span> Power rule</a></li>
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<li><a href="#sum-rule" id="toc-sum-rule" class="nav-link" data-scroll-target="#sum-rule"> <span class="header-section-number">22.4.2</span> Sum rule</a></li>
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<li><a href="#product-rule" id="toc-product-rule" class="nav-link" data-scroll-target="#product-rule"> <span class="header-section-number">22.4.3</span> Product rule</a></li>
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||
<li><a href="#quotient-rule" id="toc-quotient-rule" class="nav-link" data-scroll-target="#quotient-rule"> <span class="header-section-number">22.4.4</span> Quotient rule</a></li>
|
||
<li><a href="#chain-rule" id="toc-chain-rule" class="nav-link" data-scroll-target="#chain-rule"> <span class="header-section-number">22.4.5</span> Chain rule</a></li>
|
||
</ul></li>
|
||
<li><a href="#higher-order-derivatives" id="toc-higher-order-derivatives" class="nav-link" data-scroll-target="#higher-order-derivatives"> <span class="header-section-number">22.5</span> Higher-order derivatives</a></li>
|
||
<li><a href="#questions" id="toc-questions" class="nav-link" data-scroll-target="#questions"> <span class="header-section-number">22.6</span> Questions</a></li>
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</ul>
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<div class="toc-actions"><div><i class="bi bi-github"></i></div><div class="action-links"><p><a href="https://github.com/jverzani/CalculusWithJuliaNotes.jl/edit/main/quarto/derivatives/derivatives.qmd" class="toc-action">Edit this page</a></p><p><a href="https://github.com/jverzani/CalculusWithJuliaNotes.jl/issues/new" class="toc-action">Report an issue</a></p></div></div></nav>
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<h1 class="title d-none d-lg-block"><span class="chapter-number">22</span> <span class="chapter-title">Derivatives</span></h1>
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</div>
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<p>This section uses these add-on packages:</p>
|
||
<div class="sourceCode cell-code" id="cb1"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="im">using</span> <span class="bu">CalculusWithJulia</span></span>
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<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a><span class="im">using</span> <span class="bu">Plots</span></span>
|
||
<span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a><span class="im">using</span> <span class="bu">SymPy</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<hr>
|
||
<p>Before defining the derivative of a function, let’s begin with two motivating examples.</p>
|
||
<section id="example-driving" class="level5">
|
||
<h5 class="anchored" data-anchor-id="example-driving">Example: Driving</h5>
|
||
<p>Imagine motoring along down highway <span class="math inline">\(61\)</span> leaving Minnesota on the way to New Orleans; though lost in listening to music, still mindful of the speedometer and odometer, both prominently placed on the dashboard of the car.</p>
|
||
<p>The speedometer reads <span class="math inline">\(60\)</span> miles per hour, what is the odometer doing? Besides recording total distance traveled, it is incrementing dutifully every hour by <span class="math inline">\(60\)</span> miles. Why? Well, the well-known formula relating distance, time and rate of travel is</p>
|
||
<p><span class="math display">\[
|
||
\text{distance} = \text{ rate } \times \text{ time.}
|
||
\]</span></p>
|
||
<p>If the rate is a constant <span class="math inline">\(60\)</span> miles/hour, then in one hour the distance traveled is <span class="math inline">\(60\)</span> miles.</p>
|
||
<p>Of course, the odometer isn’t just incrementing once per hour, it is incrementing once every <span class="math inline">\(1/10\)</span>th of a mile. How much time does that take? Well, we would need to solve <span class="math inline">\(1/10=60 \cdot t\)</span> which means <span class="math inline">\(t=1/600\)</span> hours, better known as once every <span class="math inline">\(6\)</span> seconds.</p>
|
||
<p>Using some mathematical notation, would give <span class="math inline">\(x(t) = v\cdot t\)</span>, where <span class="math inline">\(x\)</span> is position at time <span class="math inline">\(t\)</span>, <span class="math inline">\(v\)</span> is the <em>constant</em> velocity and <span class="math inline">\(t\)</span> the time traveled in hours. A simple graph of the first three hours of travel would show:</p>
|
||
<div class="cell" data-hold="true" data-execution_count="4">
|
||
<div class="sourceCode cell-code" id="cb2"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a><span class="fu">position</span>(t) <span class="op">=</span> <span class="fl">60</span> <span class="op">*</span> t</span>
|
||
<span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(position, <span class="fl">0</span>, <span class="fl">3</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="5">
|
||
<p><img src="derivatives_files/figure-html/cell-5-output-1.svg" class="img-fluid"></p>
|
||
</div>
|
||
</div>
|
||
<p>Oh no, we hit traffic. In the next <span class="math inline">\(30\)</span> minutes we only traveled <span class="math inline">\(15\)</span> miles. We were so busy looking out for traffic, the speedometer was not checked. What would the average speed have been? Though in the <span class="math inline">\(30\)</span> minutes of stop-and-go traffic, the displayed speed may have varied, the <em>average speed</em> would simply be the change in distance over the change in time, or <span class="math inline">\(\Delta x / \Delta t\)</span>. That is</p>
|
||
<div class="cell" data-execution_count="5">
|
||
<div class="sourceCode cell-code" id="cb3"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb3-1"><a href="#cb3-1" aria-hidden="true" tabindex="-1"></a><span class="fl">15</span><span class="op">/</span>(<span class="fl">1</span><span class="op">/</span><span class="fl">2</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="6">
|
||
<pre><code>30.0</code></pre>
|
||
</div>
|
||
</div>
|
||
<p>Now suppose that after <span class="math inline">\(6\)</span> hours of travel the GPS in the car gives us a readout of distance traveled as a function of time. The graph looks like this:</p>
|
||
<div class="cell" data-hold="true" data-execution_count="6">
|
||
<div class="cell-output cell-output-display" data-execution_count="7">
|
||
<p><img src="derivatives_files/figure-html/cell-7-output-1.svg" class="img-fluid"></p>
|
||
</div>
|
||
</div>
|
||
<p>We can see with some effort that the slope is steady for the first three hours, is slightly less between <span class="math inline">\(3\)</span> and <span class="math inline">\(3.5\)</span> hours, then is a bit steeper for the next half hour. After that, it is flat for the about half an hour, then the slope continues on with same value as in the first <span class="math inline">\(3\)</span> hours. What does that say about our speed during our trip?</p>
|
||
<p>Based on the graph, what was the average speed over the first three hours? Well, we traveled <span class="math inline">\(180\)</span> miles, and took <span class="math inline">\(3\)</span> hours:</p>
|
||
<div class="cell" data-execution_count="7">
|
||
<div class="sourceCode cell-code" id="cb5"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb5-1"><a href="#cb5-1" aria-hidden="true" tabindex="-1"></a><span class="fl">180</span><span class="op">/</span><span class="fl">3</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="8">
|
||
<pre><code>60.0</code></pre>
|
||
</div>
|
||
</div>
|
||
<p>What about the next half hour? Squinting shows the amount traveled was <span class="math inline">\(15\)</span> miles (<span class="math inline">\(195 - 180\)</span>) and it took <span class="math inline">\(1/2\)</span> an hour:</p>
|
||
<div class="cell" data-execution_count="8">
|
||
<div class="sourceCode cell-code" id="cb7"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb7-1"><a href="#cb7-1" aria-hidden="true" tabindex="-1"></a><span class="fl">15</span><span class="op">/</span>(<span class="fl">1</span><span class="op">/</span><span class="fl">2</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="9">
|
||
<pre><code>30.0</code></pre>
|
||
</div>
|
||
</div>
|
||
<p>And the half hour after that? The average speed is found from the distance traveled, <span class="math inline">\(37.5\)</span> miles, divided by the time, <span class="math inline">\(1/2\)</span> hour:</p>
|
||
<div class="cell" data-execution_count="9">
|
||
<div class="sourceCode cell-code" id="cb9"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb9-1"><a href="#cb9-1" aria-hidden="true" tabindex="-1"></a><span class="fl">37.5</span> <span class="op">/</span> (<span class="fl">1</span><span class="op">/</span><span class="fl">2</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="10">
|
||
<pre><code>75.0</code></pre>
|
||
</div>
|
||
</div>
|
||
<p>Okay, so there was some speeding involved.</p>
|
||
<p>The next half hour the car did not move. What was the average speed? Well the change in position was <span class="math inline">\(0\)</span>, but the time was <span class="math inline">\(1/2\)</span> hour, so the average was <span class="math inline">\(0\)</span>.</p>
|
||
<p>Perhaps a graph of the speed is a bit more clear. We can do this based on the above:</p>
|
||
<div class="cell" data-execution_count="10">
|
||
<div class="sourceCode cell-code" id="cb11"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb11-1"><a href="#cb11-1" aria-hidden="true" tabindex="-1"></a><span class="kw">function</span> <span class="fu">speed</span>(t)</span>
|
||
<span id="cb11-2"><a href="#cb11-2" aria-hidden="true" tabindex="-1"></a> <span class="fl">0</span> <span class="op"><</span> t <span class="op"><=</span> <span class="fl">3</span> ? <span class="fl">60</span> <span class="op">:</span></span>
|
||
<span id="cb11-3"><a href="#cb11-3" aria-hidden="true" tabindex="-1"></a> t <span class="op"><=</span> <span class="fl">3.5</span> ? <span class="fl">30</span> <span class="op">:</span></span>
|
||
<span id="cb11-4"><a href="#cb11-4" aria-hidden="true" tabindex="-1"></a> t <span class="op"><=</span> <span class="fl">4</span> ? <span class="fl">75</span> <span class="op">:</span></span>
|
||
<span id="cb11-5"><a href="#cb11-5" aria-hidden="true" tabindex="-1"></a> t <span class="op"><=</span> <span class="fl">4.5</span> ? <span class="fl">0</span> <span class="op">:</span> <span class="fl">60</span></span>
|
||
<span id="cb11-6"><a href="#cb11-6" aria-hidden="true" tabindex="-1"></a><span class="kw">end</span></span>
|
||
<span id="cb11-7"><a href="#cb11-7" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(speed, <span class="fl">0</span>, <span class="fl">6</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="11">
|
||
<p><img src="derivatives_files/figure-html/cell-11-output-1.svg" class="img-fluid"></p>
|
||
</div>
|
||
</div>
|
||
<p>The jumps, as discussed before, are artifacts of the graphing algorithm. What is interesting, is we could have derived the graph of <code>speed</code> from that of <code>x</code> by just finding the slopes of the line segments, and we could have derived the graph of <code>x</code> from that of <code>speed</code>, just using the simple formula relating distance, rate, and time.</p>
|
||
<div class="callout-note callout callout-style-default callout-captioned">
|
||
<div class="callout-header d-flex align-content-center">
|
||
<div class="callout-icon-container">
|
||
<i class="callout-icon"></i>
|
||
</div>
|
||
<div class="callout-caption-container flex-fill">
|
||
Note
|
||
</div>
|
||
</div>
|
||
<div class="callout-body-container callout-body">
|
||
<p>We were pretty loose with some key terms. There is a distinction between “speed” and “velocity”, this being the speed is the absolute value of velocity. Velocity incorporates a direction as well as a magnitude. Similarly, distance traveled and change in position are not the same thing when there is back tracking involved. The total distance traveled is computed with the speed, the change in position is computed with the velocity. When there is no change of sign, it is a bit more natural, perhaps, to use the language of speed and distance.</p>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="example-galileos-ball-and-ramp-experiment" class="level5">
|
||
<h5 class="anchored" data-anchor-id="example-galileos-ball-and-ramp-experiment">Example: Galileo’s ball and ramp experiment</h5>
|
||
<p>One of history’s most famous experiments was performed by <a href="http://en.wikipedia.org/wiki/History_of_experiments">Galileo</a> where he rolled balls down inclined ramps, making note of distance traveled with respect to time. As Galileo had no ultra-accurate measuring device, he needed to slow movement down by controlling the angle of the ramp. With this, he could measure units of distance per units of time. (Click through to <em>Galileo and Perspective</em> <a href="http://www.mcm.edu/academic/galileo/ars/arshtml/mathofmotion1.html">Dauben</a>.)</p>
|
||
<p>Suppose that no matter what the incline was, Galileo observed that in units of the distance traveled in the first second that the distance traveled between subsequent seconds was <span class="math inline">\(3\)</span> times, then <span class="math inline">\(5\)</span> times, then <span class="math inline">\(7\)</span> times, … This table summarizes.</p>
|
||
<div class="cell" data-hold="true" data-execution_count="11">
|
||
<div class="cell-output cell-output-display" data-execution_count="12">
|
||
|
||
|
||
<div class="table-responsive">
|
||
<table class="table table-hover">
|
||
<tbody><tr><th>t</th><th>delta</th><th>distance</th></tr>
|
||
|
||
<tr><td><div class="markdown"><p>0</p>
|
||
</div></td><td><div class="markdown"><p>0</p>
|
||
</div></td><td><div class="markdown"><p>0</p>
|
||
</div></td></tr>
|
||
<tr><td><div class="markdown"><p>1</p>
|
||
</div></td><td><div class="markdown"><p>1</p>
|
||
</div></td><td><div class="markdown"><p>1</p>
|
||
</div></td></tr>
|
||
<tr><td><div class="markdown"><p>2</p>
|
||
</div></td><td><div class="markdown"><p>3</p>
|
||
</div></td><td><div class="markdown"><p>4</p>
|
||
</div></td></tr>
|
||
<tr><td><div class="markdown"><p>3</p>
|
||
</div></td><td><div class="markdown"><p>5</p>
|
||
</div></td><td><div class="markdown"><p>9</p>
|
||
</div></td></tr>
|
||
<tr><td><div class="markdown"><p>4</p>
|
||
</div></td><td><div class="markdown"><p>7</p>
|
||
</div></td><td><div class="markdown"><p>16</p>
|
||
</div></td></tr>
|
||
<tr><td><div class="markdown"><p>5</p>
|
||
</div></td><td><div class="markdown"><p>9</p>
|
||
</div></td><td><div class="markdown"><p>25</p>
|
||
</div></td></tr>
|
||
|
||
</tbody></table>
|
||
</div>
|
||
|
||
</div>
|
||
</div>
|
||
<p>A graph of distance versus time could be found by interpolating between the measured points:</p>
|
||
<div class="cell" data-execution_count="12">
|
||
<div class="sourceCode cell-code" id="cb12"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb12-1"><a href="#cb12-1" aria-hidden="true" tabindex="-1"></a>ts <span class="op">=</span> [<span class="fl">0</span>,<span class="fl">1</span>,<span class="fl">2</span>,<span class="fl">3</span>,<span class="fl">4</span>, <span class="fl">5</span>]</span>
|
||
<span id="cb12-2"><a href="#cb12-2" aria-hidden="true" tabindex="-1"></a>xs <span class="op">=</span> [<span class="fl">0</span>,<span class="fl">1</span>,<span class="fl">4</span>,<span class="fl">9</span>,<span class="fl">16</span>,<span class="fl">25</span>]</span>
|
||
<span id="cb12-3"><a href="#cb12-3" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(ts, xs)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="13">
|
||
<p><img src="derivatives_files/figure-html/cell-13-output-1.svg" class="img-fluid"></p>
|
||
</div>
|
||
</div>
|
||
<p>The graph looks almost quadratic. What would the following questions have yielded?</p>
|
||
<ul>
|
||
<li>What is the average speed between <span class="math inline">\(0\)</span> and <span class="math inline">\(3\)</span>?</li>
|
||
</ul>
|
||
<div class="cell" data-execution_count="13">
|
||
<div class="sourceCode cell-code" id="cb13"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb13-1"><a href="#cb13-1" aria-hidden="true" tabindex="-1"></a>(<span class="fl">9</span><span class="op">-</span><span class="fl">0</span>) <span class="op">/</span> (<span class="fl">3</span><span class="op">-</span><span class="fl">0</span>) <span class="co"># (xs[4] - xs[1]) / (ts[4] - ts[1])</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="14">
|
||
<pre><code>3.0</code></pre>
|
||
</div>
|
||
</div>
|
||
<ul>
|
||
<li>What is the average speed between <span class="math inline">\(2\)</span> and <span class="math inline">\(3\)</span>?</li>
|
||
</ul>
|
||
<div class="cell" data-execution_count="14">
|
||
<div class="sourceCode cell-code" id="cb15"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb15-1"><a href="#cb15-1" aria-hidden="true" tabindex="-1"></a>(<span class="fl">9</span><span class="op">-</span><span class="fl">4</span>) <span class="op">/</span> (<span class="fl">3</span><span class="op">-</span><span class="fl">2</span>) <span class="co"># (xs[4] - xs[3]) / (ts[4] - ts[3])</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="15">
|
||
<pre><code>5.0</code></pre>
|
||
</div>
|
||
</div>
|
||
<p>From the graph, we can tell that the slope of the line connecting <span class="math inline">\((2,4)\)</span> and <span class="math inline">\((3,9)\)</span> will be greater than that connecting <span class="math inline">\((0,0)\)</span> and <span class="math inline">\((3,9)\)</span>. In fact, given the shape of the graph (concave up), the line connecting <span class="math inline">\((0,0)\)</span> with any point will have a slope less than or equal to any of the line segments.</p>
|
||
<p>The average speed between <span class="math inline">\(k\)</span> and <span class="math inline">\(k+1\)</span> for this graph is:</p>
|
||
<div class="cell" data-execution_count="15">
|
||
<div class="sourceCode cell-code" id="cb17"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb17-1"><a href="#cb17-1" aria-hidden="true" tabindex="-1"></a>xs[<span class="fl">2</span>]<span class="op">-</span>xs[<span class="fl">1</span>], xs[<span class="fl">3</span>] <span class="op">-</span> xs[<span class="fl">2</span>], xs[<span class="fl">4</span>] <span class="op">-</span> xs[<span class="fl">3</span>], xs[<span class="fl">5</span>] <span class="op">-</span> xs[<span class="fl">4</span>]</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="16">
|
||
<pre><code>(1, 3, 5, 7)</code></pre>
|
||
</div>
|
||
</div>
|
||
<p>We see it increments by <span class="math inline">\(2\)</span>. The acceleration is the rate of change of speed. We see the rate of change of speed is constant, as the speed increments by <span class="math inline">\(2\)</span> each time unit.</p>
|
||
<p>Based on this - and given Galileo’s insight - it appears the acceleration for a falling body subject to gravity will be <strong>constant</strong> and the position as a function of time will be quadratic.</p>
|
||
</section>
|
||
<section id="the-slope-of-the-secant-line" class="level2" data-number="22.1">
|
||
<h2 data-number="22.1" class="anchored" data-anchor-id="the-slope-of-the-secant-line"><span class="header-section-number">22.1</span> The slope of the secant line</h2>
|
||
<p>In the above examples, we see that the average speed is computed using the slope formula. This can be generalized for any univariate function <span class="math inline">\(f(x)\)</span>:</p>
|
||
<blockquote class="blockquote">
|
||
<p>The average rate of change between <span class="math inline">\(a\)</span> and <span class="math inline">\(b\)</span> is <span class="math inline">\((f(b) - f(a)) / (b - a)\)</span>. It is typical to express this as <span class="math inline">\(\Delta y/ \Delta x\)</span>, where <span class="math inline">\(\Delta\)</span> means “change”.</p>
|
||
</blockquote>
|
||
<p>Geometrically, this is the slope of the line connecting the points <span class="math inline">\((a, f(a))\)</span> and <span class="math inline">\((b, f(b))\)</span>. This line is called a <a href="http://en.wikipedia.org/wiki/Secant_line">secant</a> line, which is just a line intersecting two specified points on a curve.</p>
|
||
<p>Rather than parameterize this problem using <span class="math inline">\(a\)</span> and <span class="math inline">\(b\)</span>, we let <span class="math inline">\(c\)</span> and <span class="math inline">\(c+h\)</span> represent the two values for <span class="math inline">\(x\)</span>, then the secant-line-slope formula becomes</p>
|
||
<p><span class="math display">\[
|
||
m = \frac{f(c+h) - f(c)}{h}.
|
||
\]</span></p>
|
||
</section>
|
||
<section id="the-slope-of-the-tangent-line" class="level2" data-number="22.2">
|
||
<h2 data-number="22.2" class="anchored" data-anchor-id="the-slope-of-the-tangent-line"><span class="header-section-number">22.2</span> The slope of the tangent line</h2>
|
||
<p>The slope of the secant line represents the average rate of change over a given period, <span class="math inline">\(h\)</span>. What if this rate is so variable, that it makes sense to take smaller and smaller periods <span class="math inline">\(h\)</span>? In fact, what if <span class="math inline">\(h\)</span> goes to <span class="math inline">\(0\)</span>?</p>
|
||
<div class="cell" data-cache="true" data-hold="true" data-execution_count="16">
|
||
<div class="cell-output cell-output-display" data-execution_count="17">
|
||
<div class="d-flex justify-content-center"> <figure class="figure"> <img 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" class="card-img-top figure-img" alt="A Figure">
|
||
<figcaption class="figure-caption"><div class="markdown"><p>The slope of each secant line represents the <em>average</em> rate of change between \(c\) and \(c+h\). As \(h\) goes towards \(0\), we recover the slope of the tangent line, which represents the <em>instantatneous</em> rate of change.</p>
|
||
</div> </figcaption>
|
||
</figure>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>The graphic suggests that the slopes of the secant line converge to the slope of a “tangent” line. That is, for a given <span class="math inline">\(c\)</span>, this limit exists:</p>
|
||
<p><span class="math display">\[
|
||
\lim_{h \rightarrow 0} \frac{f(c+h) - f(c)}{h}.
|
||
\]</span></p>
|
||
<p>We will define the tangent line at <span class="math inline">\((c, f(c))\)</span> to be the line through the point with the slope from the limit above - provided that limit exists. Informally, the tangent line is the line through the point that best approximates the function.</p>
|
||
<div class="cell" data-cache="true" data-hold="true" data-execution_count="17">
|
||
<div class="cell-output cell-output-display" data-execution_count="18">
|
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<div class="d-flex justify-content-center"> <figure class="figure"> <img 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4UfxOswAftLxC4Tsvvc7DFBe5fVLpKJU3shcH+J3F1C95jYXSY/X3nQYzL0kHgb3JmmK85JLl2TP73niMc7TPQudL53JN16fkjgXTL4lhQeJoePSeKnvniYNH4jRT96REbfktKz5PQvST1MVn/21r/k9Ri5BzW0bg6K0J4ltl8J7l2i+5fwPu++dwnwK8L2WjADHrg7/kqSr5Llt6T5Lnn+/+aj35LpSwQezKjFM+SBEe2rhPsp8T5LwN8S8b+E84z3fEQeX+aMuD8l8IcS8rcS9McS9ucS+Od6+ucQoFdzsCd5S0d58GV5Y4d5qqd590d+LGF+CoEPxDUM2oB0HPF/KBGAJzGAKlGAK3GALZGAv7eACCF8t4AMxQcSJHgSJmgSKJgSKqgSLMgSLih9MFgQ1bcM2CcSN2gSOVgSO4gSPZgSP7gSQVh+QygQ8EBx7EcSSVgSS0gSTXgST4gSUagSU7iBQ8h/J7GFJNGFI/GFJhGGJzGGKVGGKzF9miZz1+CAaQiBXyeBOkeBdWeBu4eBCKiBdch33tUL4QWI88aHcf/nh03HiKgniM5HiC1oiCoxdDJIgy6hhiPBhiLhhiUBhyYhhyhBh5mYX0V4hJ3oiIIHiWEniblHieFniUCIiSnxWVdIZFkIE54oEqAYEqJIEqRYEqZ4EqiYiyyFhjPxiyERjCAxjCNRjCRxjCaRjCjhUQ2oXxZXa5QWHZZmHx2RiIsIWt04IbYGjrh2aRqxiTU4V+f4Ium4HusojhghD89gffAwUPG4JPN4IeG4IRexi83Qix/Vj2fyjzMSkDdSEcy4TAg5KAr5JAw5JRIBc8gwc3r4ShH5KRO5JhX5JhCRiMIADrLYfq5IerA4dyf5fbRYf7YohbiYjfyyierwPB3/uSsfeSghuSgMsYotqRHOCBLQ+BHSKBLUOBLWWBLYeBK2QpAGiZP64Y0Yp44ax47dtgzf1n9fk5PXspOj0pOnUhD01mkb2ZVTiY7fSI9XaY/8QJImeSteOS9g+StiOSz8YJPylpbyuJYAWY/HcQ7QgGz7eG586Y9+uZCAWRVQWWyHmZCJSZGLKRVhxgvScA66tiMNNzAPty45smlmyYEvl5K1t5KEF5QE+JIGGJNkOJMfAZeSIZosMZQfUZQecZQhkZQisZQk0ZQcoZcEIZsrQZseYZsdgZsgoZshwZsj4ZsZgY9EVpgFIZxER5rIZ5qmh5opqJoryJpz6JoY0Q7p/wcNURmcVTic1rl92Hl72smD3OmD3nmK4FkR5pAM6rYQ1JkSxNkRxskRyPkRygkSzCkSzjl7oGlvDJGfKLGfHNGfG/GfHhGgHzGgIVGgDgGbD6GgeziXD1OX23KXOQGcEKGhJsGgG+GgGgGhHSGhHkGhIGGhCbGKFEGiJWGiGoGiGaGiHMGiHeGiHwGjBnGFtrB+FkGjWpie77eeyteeTvieUBifyDifBwF661ak51mdHLoyHnovIPoSZYmgGGGkI2GjGYGjGKGjG8GjHOGjHmGh+FANAMd1GyGmSIikAKik3cekYOikYgil1zif9kANM3iTHkGnIUGmGGGmF4GmGv+hphvBph3RlDIKEoZqg3Zagngaf3r6hnwah37KlLgonltJUpX6EYh6EYpqEYyaEY6qEZDKEXRIpSVRqh5xqhaRqhWxqhjRqhnxqhsRhF8qgiRBqx1hq4OVqQK4qaPYqaX4qb3ZehhKk5l5JZv5MZ15MCoRqINqhtM6cBcXbBnnbCcRD/nYDNVGhd16b9/KbOGqbSXRDloJeUKYrqtSrTtzrSNDErKaf/R6LPZ6Nfj6M4d6oMKqgP3KbuuKbe1abh4RrWl3sCxhCEWAAmxwEMqQAw7wAp4gEPQwBREQAnjAJLhKEbp6EbyKEb6qEbqkrchAqPgFsSsxB2cgA1xwEDz/gATYwAYWABhsYALKUAkF8Aq0ppZVyZbi+pz9dX0rBbMs8QNcgB7TIAC7IBAR0FwmYAcCwQNhACMjOxElaxEnexEpmxFKBK9jFnIPxbQr4bTo4QkGMBAyUGoFAAp7EQZSdhC+EAGqsLdCi5hF+5dtORH7Ok8RMQ7RcLiIe7iu4ArHgLjKmnSXmhBsWxCN4AADoQNkoA8BwAoCcQY2QFkFEAGiKwI+VrosBmSmm7owdmKq27oq5mKuG7sjRmMrRlwzd2Syy2KzEAqFAAh0oAZioAVNQAQ4MAMuQAIkIAESkADM27zMq7zQG70M4LzN2wASQAIp4AI4gAM9QARaIAZq/0AHgGAJoWAMrou6OJZkGDG5BEEKBDAQMDAH+rAAmSAQXCAEQ9uXf6uYgdsQWMcLxLANNlEPtMAIeqAGwtsDMQACDZAADAACLjAD3XsF4AsHf8AIlsAJrmAKh7sO61APIJxp+lAPHlwP2RANG8wJjFAIevC7wUsEPTADKaAB05sAENwDTSAGFmwJrrAOKJkT7DsQ6IAAG+sOBkC3MlCx/JDE+eu34GqVR9sQ9iANyAZgLrEOplAIcCAGTSDDDawBLtADFGzBjMDBA4EP4aAParwP+0CB+sDGBKHG4+AKjPAHcJDALqABDkwCOJDDdMAIPSx6kWsQlRAGJQADYyBocf/wufyQBSoQCD4gA3shCA6AB2HgWk0Mmfsrmf0bbc9gC+aqEsdgCXQgBkTgAsurATPQBFoAB4XACdGALcRKENnACYWAwEQQA3rcAGEsBnQAywjhjIpQBcRcBRuLCW+wF/TABjzABa8lEJDMBKogssh6go87jcxqjM7anLQCr6NKEuPACXSgBThAAgkAAb0MB4xwDNQzywxxDOIsBj3gAg2cAj2gBb88Do2YpUezpRMjloPrEdHACGrQxctrz2WgB6Ogzx3hzhJRy3pgyimQABqAA1dAB6Pgw5qZsOO2sPDWdQTbEfUwCnRwBTgAAQkQA02gBoxAC7N6pSdRD67wB1r/0APmDAI9IAZ64ArXfKz/OjcBuzUFAZcbsQ6WoAZEcNM5XQgunYpbsQ/dIBDrQNLkrMckQARwwAk9/RBdKxFfWxFhaxFjixHgoa3J4LIVsQ6cAAdJnQAp0ARZXQ8GW2cHodZsbc4u0AR04Aq/xtHu5tH7Jm1/VxGuQAdNMNEkANdaXXUwOw5H3QN6PANa8AfH8EQ//ThBXTHmoJUgNxFGXQY4YL09oAacoNF/pbYDEQ2FIAYnLQE4UAaMYNovddmrk9lKcWe4EHoQEQ1/cAUu4NZXoAeVfU2ofRC0oAe+nQAucAV/EA1LRdvHY9vdc6Dh8BDjUAhXMNGvHdtLVNwK//HZq/zcfp1vgN0UvBZe/CAs9cAJYhADKa0FhSDb3U3XAwLd4yPdQTHFt3DWA+EmrqAGOJAAJHAFhcDQo+PdfU2VT2y0TSHYoUwQ9XEMdEAEECABREAHzn0xCP499v0/+M0TDwnh6DAOekAEGiABPUAHw42X9M1w411w5U0UAU0QtKAGM5AAOKAGpiCQLb7RCs6uUGwUw1Vc2oAQrlAGv53i2bBxjlnfL+5wMe4T/4sMAlwQ9cAIV3DiTcAI+gCklArTCzrIXFjNOrjVyZnN1bjNUjyYbicQ2fAHPaDcYjAKIh5x08LPY+PPL0MUDn6uAvHfNz4DKn4QXv4RDu1/Yv++hmTOhGYOoGiulGqOEA9pCmKQvE1QCHKdEIVeqGC+oY8pkZEJkj+xr65Q6RJwBZzgEJve0J1eoon+iYvuhY0eoY++m5FucyFNC0gOAVseEavOEYf+w5/ukaHOkzqRiAHHD8ew601gCYDz63Pa6jX66sAY62046yta68u5zQrXDuNABzPQAFue6RQB7RoR7BdhrBTR1RHx1RQR1hWxlLv4DPHACESQADOgBwZuEeaeEehuEeo+EewOEe4+EfBOEcfYcR8XClogAS5QBitOtlI6Ev9eEQEvEQP/EAUvEQc/EXKYbrCgBiCgAWLA12068SJR8cZH7c9o7aGI7Tuq7QL/aouxJwk90AA9UAgEivIhofITcfERkfEOsfER0fEScYBT3gbXqwb7/qI8/+Ubvj8dvkEffo/9FQlG0ABXsON/aufvgud/o+dLoxJ+twcfgAOY/p1evzBgvzlifzYowQ7FcApgQAIQf4trfzJtfztvPzgmYQ6o0AlQMAN/ULBqz+TQNuw6WexhiYTg8AmO8ARXwNPjl/dDs/fT0/efExL3AAupkAdGoAfkXvmIz22K/5WMb5ceIQ+icApksAWUr3iWj5ZO/uMKG+QYcQ6ioAlnkAbjoKz9HqbSfqSY/z6avzsbwQ2RoAlgoAdnCXSzvzfFv0DHfz0ZsQqQQAloEAr4/2D40F/6hln7RLvggLsQ1hAHXPAGz/yWkuAIjiAHtJAPjR78F+HzF8nyROnywgjzaSrzAMFP4ECCBQ0eHIjOHcKCihxQqEDBgSByiTrl6ZNNH0OOHT1+ZOgOHUiSJU2e7EhPHEqWLV0y1Ift5UyaJH1NqJlTJ0JsG3f+BCqOHlCiOhV6FLdgRAumIwxocvPonc+iQEVWxepSZVauJ2N2BcvxZliyPKmWRTtQaFq2Ao92FESB6VwKbOqdbXvyal60W/mW/fqX61jBXXsWDrsWcde3HOPIndui7mKTeylj9Xu5amDNOwl3/nkYNFHFo382ZjjnQWTJgvCaJmgZds7Ms/9rcrbd8nNul6J5zyz92yXqgvgkdSI0ocJcDhbmCR8oGzrK2tO9yrRecnd2kL65lwz+HSRxgfcuacI4zpeMhwteyMouXbzH6vM74rZ/cHv+g975Mwzvv4OIk+c8N/qoZyB9ZFEEPu7kE7Cg+iIkCD8K96OQH/8yVGsoDg1q7JxITnEDkAwh5HBCDi2MEEMKN/wwwA+PYicSTdgoZB8OUcxQxQxZFNDFCGHkUEYOyQEHEkrQsASf1/7jkUIfKQTyPyEFJDJDIyPE55qLvBglnycFjDLCKSOskr8r/8uSwi35swcWEr8I5UODyhTwTAHTzG9N/tqM8M355IGGxC1mIWf/ITtjG2nRgfT8j0/7/MwPUAEF5Q6eZk4xIwlTNiLvQzz/g5Q/Seej1D5L/8N0OnaWOQWMHupMSFFH+RmVv1LzO1W8VOdblb9WedOnHGRSUQIHTnQkKNQdG711V/t6/e5X8YLNb9jZuuzlEyCURcjZE6F1VNr5qOXO2u+wtU/b0e6h5hZUaJiBkwTDtdXRXPMzVzx0s1OXO3bnc1czQmsRpYYUGLmXIXEp3Ne+fr/717qAsxtYvIIX0xThJDTQo2GOHo4w4vkm5q7i6S62LuPvNhbMnGRugaUJCdQQuSOSySR3UZSzUxk6lqdzmTuY29IHHGF+ieUKBsr4xqSdoezZ/86frQtauKGhKzq7o9HiVhhwxGjgimhQmpo/k8W7erqsf9tauK6t+zoseG9BRh06JMDBlZbSzm/t79qG7m3e4v5t7unq5urgZeBhxIUYOHkJcPsE545w4QzPDXHeFIeO8ao6bkaeUXAAecySLJ8P8+w0/41z2zzPDXThRAfqVV6kseeYKxooI+e/813UdetgJxa7D2m3zfbfcM8p6aWvwWccMRjQIhujiLfT+OmQz0322ZifzXneoJ8pbHA22psIWk7jXtSqPwTfNvFhIx8283NDv6W78xYIJ1yQAksQhXXi8R506jeb+5kmf6bZn236dxJ5PKMWzICHQI5hMzhU5f+A30mgcBYImwaO5oGjieBsJkgS0slDIOsoQwO0MA6sfPBB80vRSm7FjxKC5oSgSSFsVugRme3OHgP5A9/elxUbxgeHPdLhrXrYmR92JoimGSJCpPcL6g2EEzEgIFiaaJ0Q/maEppmiZqqomSuOJovFuUYvxEaVaDQBAnRQHVDGOJ0y8uaMo0njZdZ4mTaC5o3liRcACZLEJpyNLHuETh9z80fQBJIyg6RMITvzRsdlkCCjiIELKIcWSApHkrahZGcs+ZJ0pEMnmFyMJjUzxBYWJBtagEAH2VLK35xyNqnUzCpZkg8kIAABTMDLCQywTAOcgR+CKAAzfYEQWCJGlpf/meA2kMELatzDIIx05C7j96wd8gOYlxEmSvDQAXFgowN8QIgzBkAKfvBBBh6pZmGuSRno6UMbS9MGswhiihmEkS+85I0vYXNOyqTzJDkgg0DIoAOEnOEFArGnM57DkHwKZp+LEZ36XjMOLQAvj2FBaG4UahqGLsahJhEBPPmBBxMgRARxwCgBJkAAJGzUIL5wgCGEqogPfRQxdfufOhCSxB4cAzEptc1KR9NSxLy0JB0QhEAC0YGDeKIA3hDINEYSjBA4Uz8GgMELYGADa7TVrW+Fa1zlOle61tWud5VrMPC6V7721a9+VcY0/tpWblgwGdwQbFxDEQMQMCKxg4Xs/12V4YzIVtayl7WGM5SBWc52Fq/T2KxnRTvat+p1tOR4yQvmIJA4wOAgSEDCQcZAUf3gxFFGLczGNGWLZ8hDHyPVwtPyIdDCQHU2UgUNVQtjVZJk4Qcb8QEXeKg8chgAEwfxAROoadtF4VYw7jLHNrvJEKY6VTPGhQ1yO6NcwTAXJLpAQBaqsIBpgiIAPplDCKgyhSiQgQcIUMV2b+Xdv7Rqi9cgbkFcUdACgga9plGvZtj7F/eCRBZkIIMuwopTgWBiEwQBxRm48AZncJS7diIwXwQlUobUI4Y4M82DRxPhy0yYLxVOS0f/kuK8bMke0rDFMtiRR06kYAZLjPE4x/9VThvnBcdo0TFfeNwWGcXDgs04R0dImsvcyBg0NKZMk5GmPA5FOS9TZkt42sEMW0DDHidlhAZ6EM7ZeLkzYF6MmNny5LKYuS1oTktpwstNb3YkGkSQQCGEY2fN4Bkxek4Ln8niZ7YAGi1COXCClyqBK9Bw0UqG2BOlFEVHSToslE6LpcvSjTjO8SO0wAEJRgkdRl/G0YWBNFpMDRZUo0XVYLEHNWyhyI+ooQFiEB5vak2ZWwsm14Ahc4Z6XZZfZ4VQtmAGN0BNEAHOwG/cWfZimv2XZ5Nl112ZNlmqXZR2LKPNb/aGhziyDjFIQJffCTdixs2Xcofl3IM5cVFPyh3/Y/GiGoXmh6DijAPziiffhdl3XvoNln9nJd1hWXf0lMbFBG/p0BpQdH4eLpiIt2XiXak4Vi4Oloynr9XgQIiRCsHpqPFn5H8pOVtOzpWUV2Xlhhl4boKNN6UCSN4DGYfNGBGhm/Ml52nZeVZ6XpSfc6XlKLGyLZoRD6QcnR9xJoL2mL5tnjGZ1IuaOlGqnpWrl6Td7w56acZxBQksPUNNz8vT+3J2O6UdKGvHSts/MujxkkQxYBf73clONbOX0+8/AXxVBA+TjSP4JEKZe92Dju/Fq03UZuL7hx7vmYBzaPIGuUc1eIGMcrREHIjfPOfLqfeyRB0ro39l6TN0+oHY/wMaQW6HS5IO8h3ivS20J4vtNxPtC+n+RbFvSzyaUYutvwTs0YC+wzsfuM/nKfQrYn6LnD+k7Jfl7W5+SeYLccjOGJ8tyA+L8ouC+5xEvihtJzzCW4L4hHvdTu5PC/gDC/kjCvqrCfsjiozLtJlYB7oLuf4rJwBECwHsCgIECgOkCQQEimrrEl4YBpijiVEAgR5IPAgsvu27nO4jle/7kfALkvHDkvIjih+7hWQoOpowNjowCPY7LxRsHRXUFRakEhe0EhhkExncCemjPq6riWOYgRhAMoLgwcuQwLKgQK6wwJ/AwJnQwNBAwpp4lXfTCT2AADEYkymkjCokiyvMiv8s3IktfIku3IkUyz+dODQQmLWDQMPFUMOwYEPMEEI0IUI1McI/+UKW2Adt+AVh0IafiLMmWIeO2MOn8kEEAkJ+CcQ9GcQ+KcRKOUST6MAP/Il60IK6+4hJLK5KBKFLlJhMjJRNnJROVJVPBAkaXIbg+4lRIAEcoDOOQEXB6EOw+MOqcEOdgEOXkEOd0CQlrD6gUIM7Ag//mxFVvKHG26Fj1A1ZBBZaZIgwRD+gcEIoNIlf/Itg7IphLIpijB5YRBVtvBZuNIg6JAoyNMPLk0YOMUeuQEeiUMfbYEdfccd1gUeBWECiyAYiwEOWIMeDokYnskYp+sdqCUiBgUduIQb/ECwKS9CAJvA0lFjIvMjHrNhHoOhHmsBGlkjGnAAd36uFW8QKUky0l/jItghJrBjJnyjJmThJlEjJmpibjumtrKAFF2A44LhHxZs9VjwZVzSViEyXicQYJPTGI8oKPQCemphJcUrKhyw1pwQYqGyZ8pNHrFiHJtCAPJTJo6SQmqyKm9yJnHyJnTyJnqQJbEmaX+C4rtBFEtSJrEwLtiwKt9QJuHQJuTQJupyJVUm91Ws9sICDBtDBnfBLUmpIMlJKtmFKXvFKiwFLojmpa3NJsMiGHiABUwgKtRy7rdwhwmwJw9SOzuQa1TmH6WtGsOCEjYzE04zAyuSjyxyczJyW/81cGdiUmyeZyrIoAwj4g6qYzLIATKIQTNoAznMRTqEhzsQ5i6Qhhl64Bv3rinCMQt08QdWMlun0l+rUmuv8HJ/Yh2tYxPVBC0aAAC0YSKPbTfIsF/OkGPSEG/WsHX1YTNZLC5i0u6xozkfizUjyzczRz5Thz8PxT9s4B2hoSVxEi6Esyq44UJRKUFNa0NdpUKB50M6JUNOYzSVsiz+4ysRATQF5Tqv40OMJUawZ0dkp0c7wxnNItq4oy7Msiw0Vow7tpRj9nhl1mxodnxuljDpMsb0sQbAAUsYQ0oQiUgUy0sJBUvxR0vYCB7y0PIEgMDiAxkBr0f940Z+IzppgTf+WcE2b2NK8gJfVMweDwK1xIILSpLIytbkpVakqFaEr3ZwsdaA3TYuDwaAnMSpXSIEe6Egyvc8dSlOaWFOUaFOQQEyEgAdmqIWgZIiPUtF7y9NHvZVInYlJvY5FudSCOM6OuCZS9NEC01OR49Oo8lMzAtTYEVQTIlSuIDx86I76RAgM7cVQHU9IrVU/utXkQdVdLQp8UMRhCKiS0CT5LAOQilX7ONOdINWXMFWTqNSPoMv/mdOTKKR6K9Dcutb5yFad2FatSNbwyVUfYlad6CQkvCInnIFhhVVR1ZdjnaR3tZ94paJ5pYla6g1gFQiNpM9ZSlftw0+fAVgGElg1Ilj/l1CHmaEGqnyJFHrGB8SmhpU9Y+VKtJtYQapYrwCHYeBOX/XJ+swGHHCB8OQnkAW3WT0uf0WliCWhkr2kkyUJbvlAYDWfUcBNLKLZ7FjXnGjXlujWkvhWfPJZjxDXBITHx9QDFTpa60jamlhalmhaknjajri4er0/Wkw6Evg2Icra6dhamuha6tBZNOLZxZg2g5W8TzQFEiCC3MRafi0enP2luAWkuUUMVFtVtjvEPxhT/llbWrPZ9ALchRLcSiLcwjCzYkGGXqgGlgW6lqi3Bjufxv20h7WayVWlyhUMHWMxdZPBaMCBfA0d0f2Ntp2Jtz2JrwWJsBWLk02qSis//6K9AoSFUtlVtseFsMhlKdMNJtT9C1iqoAvyJN81CTpogKulG+LtMuOdMeSdKuVFJ+bli0HK1FqABheSstirhyvQgFEwGuy1Ddp9Cds1Cdz9CN01saI43PMFiWOIAV58GfetM+39Mu5NLu9tKPDNiwfSTs3lXI8KOk6QgHr8X7/tHgJeLwN2KQQmiXnwhGkyiMxyBmdQnnwAhVfYXZ1Y3VgaODH12PalYPkZ2b7T4I9gBQtQAQfgArxAAQeYgAmYAoFQhhAwgQiILQGjid61ojxahztNW40BYNiAX5eQ35KgX4+YOh+oAn4IBgNQhbNAAUQoiCrwAX6YhgiYBCN2Cf+yBSLVoYVFbVQnfmFyWk0MrqoZ9ogC8ASB0IGIIggUiINKsIaBsABDEAgm+OHacgm7RaExkU81uJQnTjLSpR86Xi475ohpCIBpEIgp0C6CSKsXMIA6EIh5kqgdoCYHYINUfgN6YOVWduVXpgd1KIZ3mwdYtuVbxuVc1uVdnoZahmU1qLtdFuZhJuZi1mVrSAdjVuZlZmZj9gZyaOZoluZppgdyiDdqxuZs3uV0sAZt9uZvduV5mAZwzgeXcIYACGR+qIIifhSBMAQCsAZ9EIAu5gczyAFqWoAo0GcuEAdy8Od/Bmh/FgdtUNlqwIaARuiEVuiFZuiGZmhl6GeAPgb/mEUUh7boi8bojG5oEdbojvboj9ZowQLpkSbpkiYHa5gGk1bplbZobHAGlobpmAZocVAGma5MeaYnfvCBMBgTA8gEfoiARhCIKujkn9LGFF5PgnAFEGiCHSUYSB6NKG6JKSaJKr4PS+YIGzCDhHOA61IGehITgfAEAVAGfgiCLJ6HEAiEJ6EUJPbMgZBPUA0UqHYwAb4zC5YwSm4vrGYITECAIjAB2mIDFeAHVcAAH/ABA+BjUliAH1ArnyqINekkgRQIYAZdLaHr9rPrRsPrGtNrCuNrjpqDQ/CJYAAFfcgHT7ADPDDhgVCGOhAEyI7s8VPkqFziFJDZuY7jJZtj/8cLbV7TPd3hnUo5BhdgVEeJUq6QapagapCwao6wXzTmocpr4PlghAjeoeRmos22tc4Os8++sd9GN9vCh2rohYuMEOptYTvR7hribmbz7jwDbycTb4D7MVtIBnaIkPQFgVDIbIbd7VCLYdGrb4tbAuqL3v+IhhiYgWyI53Jqbw96b3GL70eb7zFDVRNgwggB3o3gvZ+I8KJYbpRo7o94bpgocJWbAOEtDMW1Xg1hcTf6byqccH2rcFy78D1LcZ+LWrIQg1cF0xg3pBlPwxqHuBt3thyPtB2nuh7vipeNgV78cMkkcj40cpJDcnJTcl1jcrVzcqHUW+GZ8r6sckqU5P8c8u1l5Y+4NgsIL/NUPHMoSnM7SVWGKAMJuOyCGPOcCHEDunKcy3J+23JoU3PxuO2G6w8h36Q3B8Y/d7pAl7hBN7cu/7sv/4kFx4E3plNFB/BiHVVINzlJ9zdKhzxL1wlT0AAt+Ig9x0pGL0dHzztQ1zlRpzhSJz3uKAQIeHFW5fSPDfCSkXWoo3WUs/Xcyw5gRkuOYHWa6HM9gvXjC/a9m/PlMfWXqIcmSAFE94hlN8pfL7vevsZir79qbwlM13Red3NvZzxwh8hC5w1UV3Vy7fWZVXfPG3Dwc3fbyPVdl9Z5t9Z65757b8F8hw1kbwluT0uAT0GBH0KCB41rz/b/g013T+9XhhdEh9eMaCBKvmUJhHeJZoefOB+1aS8zcv8Ioo33jfX3o3J1hhR50CN5aTP5jtj3lp34WxlxvYj22ht2nhP3A5x5O5eAZD/4lUdXhf9Bi9dEjBeMeiCCiFdJo/+ulgfJZ3+/nU++npe6n8/AoLelGcj0OZT6faX4v1X6V2T6vKAFEmgChPV416N6mrT6AMT6+NP62+N6LvT6hIUAR97AsVexuNdKkWX3rkz7tCDD9Y76m3eUnK+Muh/Au1++w0fO9b1bxl8Uxy+JEqcPyZ+/vI/DmXd6qC9bzP+/uZ9AyK9Azy9A0EdGk9d4HOB4qjX9aXx574v55uML/1dI9c7NbsH/S9S3QtXHQta/QNfPxryQz8i0OsDvMeCnzNtfwdwXv7agAwg4V8StfXwU/jUk/jY0fi1EfpS09B9nX4xzfmLF+e73w+8HROp/QbRYhx6IcmpL/zSDfudkf2F0f2IECHH8BhIsaPAgwoQKFzJs6FAfNocSJ1Ks2NDXBIsaDR5zQWTdxpADsekTafKkSHH0ULJs6RCdO5cyZxZ0h44mTpf0BObsaRKiz6AUMQotaEqDlqIISSpt2lCl06gJYUqtWvOmVas7s1oFylUpUaWFGtDJyvRrVKhonVJd29Sm26Zb4xb1ShdnWKFwIFjievauT7WAe7YdnBOu4f+ccxPTtMsYZV6fV0DQ+vr3sUvBmFsW3swSsWeWi0OfdEzaYmSc63DMyIb28umQmmNv7ExbI+jbGkfrrmi6N8PUM6N5BPm6JHCLs5O/jMmcYu7nDXlLX/i7ekHhLl0hjQsbe8Ll4A/aHn8wunmD1NMTvD5eO0tOEuDQ/c5+oPj75e/zQ89//X3ugQffSWMVgpxb9t2XH3v73efffQCyJyB2BIq0FyeAKcgeg+k5yB6E7EmYHoXVWbiRFpQNtmF6HZr3YXohpjeieSVKd2JF6/QQQzSGsWiei+PBaJ6M5tE4no3P4TgRcT3Uk9iP4wUJ3pDjFTnekeAlydySDtECwhX/j0UJ3pTYVQneleBlid2WyXXJkHxqYDYmdmVWdyZ2aWK3ZnVtAvemQmP9sRmd1dkpHZ7V6Vkdn9L52RugCGHoWaHSHfpcotItKl2jzz2qW6QGpehKaJU+dylzmT636XOdMvfpbaEOpCOPpJnKHKrJqcocq8y5mhystMmazQw9GFcqgvwRlCtwuybXa3K/AhdsbKHSQkITtN2aHLO9OQsctMBJ2xu1p0Ua523bAtetbt/2Fm5v4+pWLmmACqqbur2xe5u7usGrm7y30Rvam5Pim6yy/OxLW7+3/XtbwLQN7FmXYoBAam/56rZwbA3T9jBtEcc28WY41kOECz0Cp/Ft/xyf5nFsIMcm8mkkY3YisTgcmzHCyrpMGsynyXwazaTZ/JiF1zbRc7pML7hSwgQFTdrQpBUd2tGMEYiudCzT9nNoU4dWdWhXe5Z1YvAVAsGg1XkdG9ieie0Z2Z6ZvRnahmlnsNtOcwh11PzMvVndm92NWd6DCVeGBqaM9/ZpcW82OGaFY3b4Y4kDFtnJKRxjHuSkSY4Z5Y9Z/hjmjGl+V145j5NeMOkEPpAyPAXujDWz82ONM7qLo4zu6QSTcFjXhsmeDYLo7sMcfptXBRm6j5GF8+PF8YPugdhAfEajSCDnfckv3/zs0EtP/ezXZ7+9shhZAkFZ/Ik/O/PVg2f+7P/T24+d+rNrT7wCGACIaTijgAY8IAITqMAFMrCBDnzgAmWABwhSsIIWvKAFecAGDHKwgx6EIBPC8MERkvCDXGBCCVOoQgiuwQcrfCEMD2gHGcQwInexRgYuUAAD8LCHPvwhEIMoxCESsYhGHCIBdnjEJTKxiU5kYhKfKMUpUvGIUawiFrNIxStqsYteLGIBCPDFMZLRh2Esow8Asw961GN/unsjHOMoxznSsY52vCMe86jHPfKxj378IyADKchBErKQhjwkIhOpyEUyspGOfCQkI+mSTYwhDlgpiCHsoElMEIQUZGCDDW8jCzOcAXgGCYQmNUkKfrwilXZwTmzmsQn/PKiCaYEIQyAKQsk3XHJkpOADJw8iizeQQRDIIUUq8aCbfICCD5tAiCLwoMlGEIQVZthgb2QRiEMgRBCu9AQ/fOHKUEqSJYJYQBhkAIODmMAGQhBCHAaCCQNwgQchgOVpWIGAKiAhAr0rSBWKIIQgBIAP/JjDBN4pBHLcxgYdQAAbEMKFDowhBFwoiSEQEAYbwMCNgOFCBCZQhIPMAwFFmGga+WGGDrwTCbqpwgQiEIWeyUAG71zDQEiBgCz8YAK5o40dFtCBF/SMCwIVwgDewA9BOEChpiwnS16gVHpEgJoFMYEiDKKDMPBDHyWww22i4FJ+2MAMCUHEAmIyhxwk/2ce/JBBRA1ijQKsUhUFyJ1U+TGPCWSVNm7NwlgLog+e+CIAwDNDYHXj1ijM9CAyMGhBkDCFgcD1Nm6dA1EV4gkC5E4QL4DqTPIhAFkMxAdhSJYJkMAF5Q1kAcGsAhNuUwLInoGtCOFBFQYyhxBkAZu9qaxBGhEBglhAEaJ9RWlPqxvALkQVAoiIGVSQhTeQkzaMpakPuMAH5HSAtWTgQW8w6zcmYG+pGODCGYYHWpQEIwAM5QdsDVKFMIRhAkzQRz4CgFx+kMG2sXGAVe2AAoRMYwCgGIghpmCGHSxgF7+Na0HwUAKCoKAOyggAT2DrUcAwNyH0kEFu+YGHKpjBBv8T+GdYG2sQLoRhDB0ALz8QkImBxGGduhFvQtKBgEkMpBJMMIMPDHDg9ZrEGgGYxkCYEGJhBmB4A1glP8KQ0thYABE0trFB2KAChOQgCw8+SCBCgBwR8MHISOaHknvT4YPkwwc2wOdAXmBW3VxXIc4gwIEjwGN+sMEGG74LjhGCBwz0DAkjJbJJCvBMsp4hIfrAsz4wwNofeJk2MFAqP7gw5YKUANMG4emXDeIJA7g1Hwh4pgEWbYNGLzex7SmCDHpJECGgL8XOi0BWVTCHgfAzvJlFiAyid5ASI9okSHApKQrgC35kYtfWqCU/1rCAm3DBz75YACj+7BYtu8Makg7/p7D5sdmfinslspiAp2kDXH40Qpn86MCu7dCBkiBBCPxI9rJb3cmI6oMJMLCduPPBD1AsIJd0VjEo3qCPdAy5DgXo3RlgMA9n4NrXyHFGGKAmCwHke+CyC0YIwl1sjQQDBRFAgML57GddOMABCAiBVa0hA5dz1WE8cMACYssPTDiAIFm4L0GQUIAIGKAK2nZLFnxYiSintBITsICeBxIMFZwcwrHBgw8jGmZ+OOOH4NTB0DV6Yx/GEw8n2F0EFrCAbwsuBzhfcmxG3cMo8MOuWDEDjAeShaEXAAluHblIpiE7hFgD4AOxhqy1ZfiFuAPF5smHMwReEMHPDh1nTo81TLxxkMLPbuKA/zzoQy/60ZO+9KY/PepTr/rVs771rn897GMv+9nTvva2vz3uc6/73fO+977/PfCDL/zhE7/4xj8+8pOv/OUzv/mKDAgAIfkEBWQAAAAsCwALAG8CtQEACP8AAQgcSLCgwYKN2BT8x/CgQ4fq+D2cSLGiRYf/JvJTd7Gjx48WzRFsCLKkSYrr9pkkebIlRJcXWcJ8KHKmTYEyO8Kzd7NnR5Y1fbqc82NhRpfaJApdCiAnwX3gmEqdiG3k0alSwaks6RQrRX7avDa9KrWqWI9dK5qrd5YpS7NtLRI1CjNpXJdpAUC9OxXuWL43ta4kC7ggWLF5b/otbDAxTbaMYb6N7HCu1bpKKcck/DSqZpt+HX9+KJgr58iHvYqGufjzaoNrRw8e2Fqz5YGvJ9qVTTHvXt4tQ58GTnoryNxiU2NFbrI2ZeYCYxP/edV55Ns4h3vcPb2x9t/dPwr/D1+x9HHtgJVPhQ7SOmP20slPnAwc+1+kmeXfPwhef8Xx/h1k3kfsSaWeVAV65F5h8EEWoHe0EWdfggNxp59vnj1I1WUaEjQgWujxdaBbIc60IGANdkiXQCfyNWGJFVkoH4YqPgRgjR9Sx9uIS1F4UYt3pVhjdhHWVxSHLclIHo1DGnSjijlutmNYqsHoEpBxCTkkfby9iNmDTDZJ0JMdRmmRjz7xKBSa/xGnZY1cyuYlfmB+l6GYLCIJpXEEWnmWmj6xSRGWbb2pYpyjzZlkfjPaieeYepbJJ4hTIubnSYSeZWiHiMJUjydz8HHQOpi8ochEip6kZHhhPkqmhmZW/ySoTYD2NKuNbl46UHxwVucTHhiYoMJBYYTQwQ+MEpSqSat21yqerz4Ya2+6YlXrTbc6lKmlJfF6qK9C1TGsQ1z4gOqRuFVrULPTPStmtAFOO5+6BlK5HL0K5tqtg70W6ZO4D5V77oqL1vlQf9BGCuukOsp2rU3ZHrRtlfuK2elMAJNr7kNzTADDCzBkgc3IJJds8sknD1MNyiy37PLLMMdccjXDyGzzzTjHDEzOPPecs8o+By00yzQPbfTROx+ttM/MQLP00zknbTQ5A2V8kMAc54DJ1qrsp2qyrDr6aJ7pNinvQxG39PBMaRc08b0VN3kxTFYbhHVl6BJJZ4DuNv8Jr39nY4TvUmtLNviP+oLkLafg3mQPNG+YAE1U0FQBDwDrQJMFD9AEVdCyJbGbOH93Jlz2kIEf1LZJheN1uEVvr6fu4hrOfRIoE+Q+QRYAzBKBSILoPoFCB4EOkujA9T3k3/qlDqHD9sruVewIzs7vt/7KmbfXzILtrNhjMy+f8wR/1npLq2c/2qa1N659+d0b7BDC7yosLcNSQs8tVtSTGPeW7kvU9tCEPN4or0biIw/57MeY850kfWQzoPUsFsDPGO8jBZTNAVWUwPAs8HSjceBs+Dc6j9BOfgDo31IuuB3vtQt8rmIg4PB3ptf1RISmmV4JO3JCvlXQNgN8XQb/1wdD0+kNRzSUlQ1vgsPz6DB5E5TbDynDwo4M0TVFrB8I9zTCEEavek+U4P/6FcH3ydCKLtxhZ8ZWRghO5YNHNN8X/UdCKI4Re2UUIPxCl0Y7zq90WoyjpLoox/31RY0huR7j1GfBIH7JhwcDpN/OOL4kUqtSFKujGBWnyPYxEoh7PF4fN0k6NqaQkgq05LwwCTdNajCKAPwkFR25N/9ssEMd7A4cuUeZJvYpjK+84yLz2MhQYnCUwfyjKXM5nV260SO+pJQriShMTxITlKiMETKpqUw2MpM4zlwirebYI3Hiyo8f6aEtp3gdWhYMkt0MXzZ1qUq0mRMm0WzYIdFp/8JOolCFQqniRa6omVtq6JvACScrpTdNLFbznxJy59dQaBD6TXKLg8yh/jK5T1L2k4KybKcxW0jRglh0efNsZj0Ft1AwNrSgsCTjKY00UjSWdI3eTCk4V6q6e7okn/nraDLT6U94zrRLEo2fUSsqSZRidGGE1AxQawhMbhIVpNecZU0Huk2HxjOGT71fVHtJzjX51EmIVEtR1xlSxgjUIgR9ThYvKkioatSLhixLWimizgux061J5eNNB3JSBOo0oTx9Hl45qld+8nCtfm1rYd6qzcEKpLAcPCxvFLrRVgrVqh+VomQBQ1mKxDUyBn0QQjeb2K3eZapKrKpXryrarP+KVLPr6ipMI7lM3I6Gs4v1bGM9+lisHtWMYT2mZfXSVMMmd4ZjRU1ZA3VWt+11In1t1GhdFFhRLhezuPTtZ4BbSMYyBaDUfahR0TsPcdiktLrRrVx5m9PnNq+14iXcdG1VXUg59iLZXdJfL4KPZ+AiG+/trnKXatLmZta+lYxuA/eLrf5uN0gxxeNxQZIPatziGfi4CXwfctr3zNWpdRXrXcsr3PNe9zHGfZs/rpELZ8zDJyN2SIkZdGLnpjhe+IXwaykMMQvb1sTqZeuRHcIPbvxiGe9YoYJJymCcylPIHgzyjwsD20u+dL6cjPFH2FEMYnRDKjk+yI5R1OMHb/n/vhLmMpHZZuQNzza0sVwyQeSxjF9wQ74fSXNuv+vg8GKZnnFOz5wNJ9vdhrm2dj7IPJyRi2v4wyuCLsia+ZLaAK1WNuSV6qJd12gw0zbPkSYIPqKBC2rk4yyZJsimMUzfK785wisWdV5d/N9Eitkh+rDGLZxBj7jEukKARvJXjchLFTuxswz97J2LC+naNNkXy0gHd11b2SoTttAHzS9lQk3WXS8FvfxNcmSJeY5gGMMdk52yTb19WXCrVtyRIbd0zS0UdFdY3dotI5nNfNtDx5fQvTX4ThMtolGjr87+LjLABVwkPvs52T05tkBmnaU2G/rWqWT4kPntk4jTeeJh/6MNPSht6WLim+OF8ni4FY5Ykcely6v8MmozPEx8COMW0Xi1Hl+OcTbXGqwgz7LN24Jze5Z65yj/nj6ugYtjFBup3DZt0Tkt83vTnLVL/5PDHwjxFzskwOThBrblMXaxaBwAMNdU1z2Nb8boe8Ik74nJGf3oGrX73XreNtERXt+kIzrX5Tbvuc1+ELQDZ+DnGDBp5c1Vwtu62UAOe3Lajvh+Mx42kCWOPJTh5/3sfSiUh+vWab3sQGIeup3Hu+I932u1PmjluGi56SOa9YPTm7kJN7xKNe+VprNU58o+NXk6DHShH/H0LXl73PMO3pkLf+Gxl3PeFfP5gji+MFPHBf+xFQv9k0h/9R0/OrOfyZS7a3/2Je8+Qb7/WrUvQx7HTzXZu8KQ/men/xlxfpaHdK8HZ9mnaNsHGvK3K6FXGH8Hb04XeFQFgD1FgQDYEAL4e9XndddXcwfYcAloIgsYHQ14F+xgDMFwDrElgfbkf7fWEDmRgUpWSoVXgLj2bMEVbcM1VHj2GXzmC9mATLajRC5oX/53gQIhg+tGg5fHfkvhfggIf3o3ggBAf1JBD6xGDfqgTxODhIrlHUdxgQEYRGJYhmbYf0lxhmq4hmzYhm54gVDxhnI4h3R4hmNRh3iYh22oFXrYh354gWDxh4Loh3c4iIY4h2txiHXIfNGAD3r/WIiE+GNFeBlKGHBMSIBOKBRQCIJSyH21x1clyBThN344iCVeGIFGwX9j2Hskhn4xp36ul4k+sYkj14kK+InYFYppYn/410UtcoqCY4EwiIRiCACVSHGtR1c2GHIfWIsttni4CGN8IQ7uBoFk11bFSIRHGIbESIHGmHrdNoNMFXzLqHTNeHOch4M7CFrU1hYnmIInl1XA6HbgqHUDuH51lhVaVo5Ml46/hHw8FnUwMQ/N4AvX4Iq7l4qymIT16Hvi2GDkuJCBsY8S+RDG11NPl3w9KBWr1mpb+G9t5I2Tx4o6hpDUZ29093WgRpH52BEX+YW8RlwAposlwYghll5k/zOPgqeSmmaSUriBKdmBYHeO/RiCrEGFVtgRo3h1ZkUbOjmSg6eBKOkfn/ZbLNlSdCRtjqZ8TMSL0sMQ2DCJWhWVD2llmNiST3iV0OZSWmlqGzkT7UYM6FAl/Vd+JnGMKZeMKMaPw0eUYmeUV4KUNEkR70B63GAp/meXJYGX3wOLyliRM0GL6AiYwSGYNjFpBnlp91KEigkSjPlCjrmXkAkTklmUtiiC0Xh2g2kQHelqdCmSnRloDdmK9xiLaKmJapmDbLmO0zaTLZEP0nAL0nCT1eONQxhvJKlmPvmM3xaRtzmLucliOhiTPNiOHzF1NcaUJCKSCYl1ZLmE41iDo//pEqX5l6d5lKnZeIN5bcrQi8Wpihf2EdXgObKGCcwwMN9pieHZhM/ZE+W5eZSJKZbpEehADHK5mXlxnCARBg4QAGFwEIrgADCwAHHAMbNZkrX5mP05kcRnLf4oTW0JdX1XEe/QZ4cpO2JJfj7hCbDgA2GQLCIwBwCwCQZANcVzocqZoaK5oTbxn8X3ofrEm1v5lpLmDJmJoquhoCXhoowCCwLAEQDQAYGAN8k5aFLpnGYTnbp2noGZnqBHEa3pfOWUoqi4d0xaEJMQAQMhA8RjEHPwAmywBmwQCPBQp3Z6p3iap3kKDeugp376p4AaqII6qHeqDtVAqIiaqIo6qND/sKiO+qiLWg3qAKkAAKmW+qeZc6mEWqmaKqiN2qmgeqnYQA5/Og4eFg30EKqquqehyhMCcaYE0QhqKhA2wAZgMwclEAW6Ggfm0Ku++qvAGqzBygzkIKzGeqzImqzKuqy/Cg6dw6zQGq3SqqzMMK3Weq3TOjnYuq3caqzkUK3dGq7hCq7iWq7X6jTC+g3C1gzeoA7m+q7ASq7dejmv+qIFoQoD4KolICo3WqU9qaM+xpfYp45bypzx56Xep0ji4AvGEGXbSaZBZab2CgDMQAoAwA8WIAi9QwDQQKX5iYyXiI9Z2qFT8ZL+mi8IO3/8cg7EUAzs8J6vo6QeUQcw4AAW/wADkwAAZ/ACAmEHDlAEEzCxboqjVlqWzSmePBqZWpp4BjuFKcuAAuEOpOdeD7tEMtsRwLA1W9OxwwAKA0EKdoAJ+MmTsrac02mWIos6S7tvXFqZT0uCk1ZpmmlWEOuLvPexeRmytjmyfgmgbSugbwsA6ZCFYlph3ClxLNgWn7lXQEmVdVcYPuqhAdocC2iTY7o6V1sR95BgJ4tsAOtmAuuBBMu0Z0t7MmmR2YALy+ANq0kth4uT+kcR7TAKeiAGROACDUAEnIu3jamXATueLRG5JQukQSWkbjkR5/ALDVuFrRuMz3S1r/AHYtADIJAAINADWkAHljANIka0/3qlSP/Lt6PLtk3riadbEAN3ZiRYZHXblLY1DYygBk0QAwwgATNwBXDACWY7EIv7iY2rH1U5Xmsre+V7i+crEBb3ZyrrOq/rWXAxu3BwBTMgAQngAk2gBoxQDorrvWX7uR8XukM5vgRcugd7vnGrews8GA1MMffwCnQwvRqQACTQA2VQCMkAWJ27cfublft5luL7j7rJw9TJjgAQpqppGu07FbZQCGVwuwmgATiQvaOwuS5Htp4Lvvz5wyAqnbs5xHdmudJIHSvMFKMAB03gAgkgAVFMB6PQDrGLnLwLmr4LusB7EsJbL5NbErEpTdeWbba3GUl8E7bwB1cQA9ZLBGpgCdv/8Hx3a8U67MHWB8IrSbJ4/LeUa0cP6JvaaGTbwAhl0AMaAAEzoAV/wL0qSlNxzLhTCcCPCxh3zBQm28rnZEDpS6RWEcgmcQ9l3AQkkAAp0AR0cMNe9sZQ6chwt8PlFJq/m7SkOcDvV8CoKRsJLF8nlI09YQp00AQp4MvAPAq6lbmwxsFXbLT1hqVqS8mwTLxUBZBxsXJyK0xPeRL3wAllgAMNoAE9kMhubLeobMzT95OrLB8BrBmvrF953B6fsWrN9ybxDBLzXM8MoAFEEMyIS8w7KZRFC54QGb7n3Lc/etDiQRlLidHS0dAd0cliYMgkcAWl7L4WfRf9e8D/K9Cy/3wXBS0UsWzMbcJlXqmSiTjGFjENhXAFaBwDWlAIGsyWe9wRMV2daLu3HS3Cz0zCTssXmZyc//DT9JIMdEAEGtAAOFAGnEDFHLXUF9HURHy0WRzVQMzFQgyNcVHLGCnGiQgS2/AHTQACDdADcDAKRpe44ZzDxwzJHCjJVonOBm3JehwXP3ii+ee6DOF485zSCRADYqC/IvrSxibOj4zFPszWW1ywVG2+WLFyt6CFw9yC/pddr6AGOCDDV1AI+7yVZi0XnD3Ynp22SITYOK3OKxiiN8GIhfvYYCgTJ8TVRCABEjDRpkxKtV0RaN2bav3Zu+3RkqvYCC0VI128t+wUvP9SDnit13xtCyD73BQR3UO60Wtd3VIdhdCMnjjd00EKicMRG609AwkwA2XgzTNo3mOL0d9LzsDH0ezd1qLdxXDdE1cNxNZsEPdgCVpAAstNB4tsTZq9wYL9z0070+Qx0OPmzO492gZsE++ogrPR0MmgBz0gw1qA2TLl3xaa4cjs0j2s21xk3cML0ig7Ez8YhA+3wqOQ0gyAA3AwDUk5VDDusf4847Bb41Bd4KFNughuui1BD89w2h+5f05RDoXQBBCgAbFN1kduVUner6nsvwHd4TUdFzedJr6d2l78EWCs5YTxCmVgyDigBuT9pdXWyADewbn95Dfe3pz43l1aEtv/rcJeGOQkAAFNINu5+GveueSEHZSGLcC87eY63hETww/Z4AvN4J5I7IL/MAoRLgFNwAiajGplPrQyXumOq9OQC+KFLuLR3BHUCHh0LhD8wAlaoAEa0OJcOUwXHthnLtNpHh4enm+07oy2Dt8WUeI/3hD3wAheTgJaMAo8Z+Gt/jm3reFUzeHKvuZt0eY39OY5B9wEQZAGWXQXCN5E0AAuUAZ7zj4Q1c9/Ps4a/dQaqsXzLeVvTeUO0ZpZfh4MAd4rPgNwIMw+3byz1e3K8u1Mnm5z/MF1bBLmzkTojorGKxBz3icHTwev3QN0kNQwueov7ueXru/6qd7UPegGDvDJ//xliQ7y91AICQ8HFc7x1qlhEM+/Eg/rrCzrrtzsk4ndIW0Y8t0n/GDt8q4GOw/nKO/zKk9AEw+SetvvoP3v5Pvshy4Qf/eypmHqEkACZcDw3D31xP7zDPnqga71UM71IzzlJVyYpccVpg7sWvAKJDvmD1/1QnT1Fe3kcA/zUd71dD+F8lCQ3DC3HmELZZACEnAFfv1wAjk6bP+Nbi/g4t4dy253Rm+aht4S3mBg1uD4FzENcBADjq7q8Tjs3A74j8T5ye755H4WGT9Om14RH28R27D6DEAEjOCT9r5esl9L+z7dNp5RhO7sie8S2OkM3qBbjLDiPfAHZE3xsH/vk/+e751N++Yc92mP+AH/U2rXni9tC2JQ9mUwDf1V/P19/O8E/gRu+HI/1c9fEnE5l0tWDgBBJwaEJpz+HQSQUOFChg0dNkT4UCIAc/UmXsSYUePEiACwbQQZsuGcHww7ilSojR9Kli0VnmS4D5xLmjUbfnz5z+ZOkeD2sYTJkyU/bUIlBm35btkvbgxxOuREJAGOP/eQGgVw9WJFrF2P6kz41CtKkibBulQ5FqtWmWrd3lyo9W1Nn0DPzsVIdK7cjfOc+brmDy5DW2IkkCgz7eBdt3wfcsXbtaPYyBfLxmWMMm1ll2xnch4r1jHojXVRjo6s9y1qifii4aKW7+HTcgL/CRpczJn1QsikbU72PfFyzpqbg4P0fNyo6MzKQZoWufut6sbNNeajdisavokfGfVg0OPPYuury0/s7fz0WcrqhyeU7tC4+ovJ6ddkfr/nz/X6qauNDwB9rsnFGXoumkYMDVJQQ7HcggswPf00Am7C97I6T6P5JoSovLY4ZCk/EDOCLqQAx/pvrPjE8cWYdy6KKjzcHjwuQotGzKhC/S48caENccRQog+B3EhEIh8qEbkMOUvRq93OIaYYdiaaRg0QSFAjHGzIo8/GIzlij0Mel8ToRxzt+xIjI9NcKMmNeuyqScnIBEAppiaKaqpCFtuSTry8ZLNDhdpzbsziVkoT/81AZ8Ns0Tb5i87PueRca0m/cgnsoSpJ0EAMB/nkEFBH4QtTKC47fJA8hAylycwRFR11oTUXdZNCSacryryJXINNNqikwqEQDB8kVDlRR9Vxp1PN4lJVnVhFC9EvYY01rEZjrTXHW92i1KirsNOOu4Y27XSaYU8qtsZtKbox1mR5opFUsBCKaDFoW3IVRGqrnTXQbDGCE6tuTc3Mn2twWWaeX/WUN16PQl1XwkXfVRamjujFGIAxlw0pX4iF/KxaWa8d9d/61kUx1+oYOucXFxsqBw4SQGiwYevShfDWgypy1lkgKf7N4rNUJVXjkjDrOWmlVVK6aaefhjpqVWWSuv9qq69W+lyst+baaZ+6BjtsVYkS22p2iiGmG2dt0aKBGfa0Wuuy57aaZ7rvjpvunBi7+NRVjyauVWmP3DfWftk0GUz/VAZQJ3mWaoohTnpo4ApbbM4I50rxXkxipDkPOEhrN2dW7qLvZcnjCQsf9fA0E/9q8b0uzVShdmTGshzMNdLcrqv3Jhojz9e7q+s3S4WX6HqVJxp1zQYnknVHXf8S9odCF2rgnVy7xRpfE2KbgbeBv7V3gKvWFqXh/5T6euQrbhZjGpsHvGjB2ZR+UeqPtN4h7HnSHk3AZaCFTK5yl7PZtsz3Oad1JmLt0k/URPcwC9VvgviC3s88FDKRjc7/fiWDlIlQ5pUADoUbvlhGOhByOxLkjnwt0RzUKqU+CGrQaQusjPNEoroIbrCDIwscCO0iO6ycIxjGcAd82NaAYJFvhB5MoM90JZL1Rc9ZfXKYb3TYsQyeyYc/hOL/SNM/Qd2nhCA5W9oUYgkcQEALCNTaTrCYNNIcy1H1mqMUQbNFkPDwPvkL1P6IREazENEmj2PKSvjxhxhooAy6250DD5LHJ1aMhiJLVtNyaMHQ+bFLXwSjIIFESJKp54wXoZ1gbqcBF/whSMHznaq2tLoHYvJ9zOqZW/i4EU+qB5BsEiWOSBlEUzIOJdyThmyS0TYcWMKJxMslFD95yWoBzX9Z/zPKLjXUxVeB8ofBHNEwP1jMlgwIFwQ0RRMY0AQ4xlFJ0RxMD6nprlvaCp400WZGeumcX6YJnCAS5wWdc8qF8OOEKQQAI2YgAU890556lAgOI2PHidVThFlkST7LxE19ebOD/+RQQMVoE4ImRBy/UMY7FpkCEsDhHi/kCzZ59zEq1rCigzIKPKaAARQooiGksIEFbKAKiWj0Ivs0lkdFBtIJibSSRiFoGs+xjTJIIAaufOE4GYjRzNE0JFUknEVdMoYXACMQBoDGQtQxgTNgwwwiKConn4pUdYEMjGF8anCcasiLIJIbtrgCA4gwCgYW0omhkyj75okssbYEA4JIiP8M2LCQTCAgIfxAACYeYtSJ0DVndr0rU/WzVzMa0yH0cEb3XkEEBmghGYVF1T3x41WQgNWKOOWJPQIAjIRMIQoL8UQB7AEAexRgDpv1ATSUW5ROcpS2MeHgR0tJqxAqia8NyYc0tBMKymnBXFmNLVdnS8vF3rGxKIFGAKqREC74YCH16EAUPMGEAZxhswuIQH6LgA3+9te//wUwgIcRYAIX2MAHRnCC/1uNASvYwQ+GMIIbHGEKV/jBw6iGhTW84QJPeMHWuMUxHtEDCIgBGRxGcYdTvGINMwMaLIZxhD28YXPwQwCwSAgTqsAQYDDBBmwwAR82K9dD4U+p/Jquv6r/e7zrAsCgKHwEDiSghnYgCpaHFS9PEruXWlbzvCgpgR0SgoI6PGQWBOCtQzgrEc/6pp9fEu19SEufFBnRGH1wAQhcCt4oJhkrW57iV20aKGu2JA6PDUMEzAEAPNggIW/ggx1CwAVurvkhba7jkQ3n59ct2VaGTKMfSNBK2Pa5jGoB9MpqakvcCiUOOkACUQFQCftqjAc8KHNcDRstI4M2lJyunqfTV1oAIFIQGogBI0ot212H5rkasa0GW30fS8vHueR9yJC+CWz+Cft8/vGGM3DBBwnMgBOwhaji3JLqxpX3ptJ0D5HvlyhNt47bg/T2ye6DD+3eQQJEMMUz0/3t/3U/OyPR9uK06VPthmAaNG8+UpzpM+fjmHMPH2CnleN3ZSYXHNuCZjW8CyVvXtPb19smJnWHeByD6gISGK8Ff57mwLewW0VdpqfC493s1F1bntmO7lLvPcp8q9s3RoQEC4iQjH18pIE1GSmjPl7bQfda5MphOEMcrpt6T2/owix67EjDDmN8ggZEsIUitSHToNXc4MKrusmvfpys+8jnf+y6/r4ezrC7DzSP68QRiFCLl5CN45b0+M9B7mWdj5znz7O6Q7Qt3ZQreeV4wYcwTmEFtK+EPIbPq9hR/fatxH1aXyZN3VNy92kC/a4U1Krl+zMX7JwCDINPINkC7eypb/8E4d1sPNZJjsHIN2TyQq884vp+zbcM6BRkSALT5XaQkl6e94qneshhT+3h97z40H29xNVDcYFxoxNu2MIrogiW6s9+9L2HtunDGny6dx/ycpd80JGc/E5b3yjicIQ8+IJX+AnZar9IcTv4Ozj5u625Cw7VS4itqwyIIxLxcw7y24lugARCaIOAmz6HOEARSkDs8z0GlDYH1CL72yHW86W8C6S9A6jlO7Wd8IZIoAQ0eAWrYDaFCEHrSjy8czdCQ709UkEu+r6FOL79iz3l8z+XoAcbfINQ+MCL6MGOez8SjD/tszmvgEAAkMCJckFggsGQksHHQ4l8SARNWINQiAf/fhi4h6jCT/vB1lu8nEPB1CvCPmJBfgpDfxrDpirDP2wIfbgETXADS8gHfnBDjskL05qTEQTCVWO8OyRCMzRC/DM+/ds0/gu2JtwIfpCE83uEe/CHw8uIOBy2K4zEOmQs+nvAPOSlPUyqk6O8Jew/9wsJT3AEQuiDKRQJVCQ4VaTD7JvE7Vs4WNymI1SIJNxEW+xEXHSIQwgDM/CExVCFQXAEP/BFlABGfZvDFgzCyNvCrujCL1Qs1wstQZSzQOREbHiBB6iACgCAIeiFPaAEPgiHUmTEoXDEGfpGPgxH/BtHrChHWawrdPw1Tuw2T0yIIaiAFoDIFniAJcADbdDB/ywLiW40uutbRWK0Q2PcOXXUOoP8LIREOWdcSGhcCARYgYgEgBF4gTd0CY0UPY4cxhLUQjFBRn0iSTfrQzgTyfFjR1vkhwKIyBZ4yRBYxNBjCJr0u3+cRUn8yIHMpp3cKGVMCGa0N4XEN4YEAAdoSYgEAA7QgaVkSh7sR285S6nDwgXMyQqyRD3ESgDQSq/jSqLzRFAghCV4gLBcgQWYhH0kqbQkGEi8ySwsRqoUioKcy7rUu7sEO/+DBWz0g3gIAwR4gAdwAD5YShIizOQxTHCUylakRM5gTEwEv3SETL67PGjYg07Ag3BYRHDABE+AB4ykCadkPqg8SFY0L1dMwf+4jMXG1MStRMmu7I9viIROiIO0W8tP/EzEE0bR9M13A0nHC8rVI87wy07lEKl8KEQ2CAV9iDqQ0M0Z9ArFBM3R/M3S3CThTEbURMLitMvjxEuR8IdL6IQ8eIR/WESWi862482SrE4hBE48hE+e3E7VtM/IDIlV2M8+EEwmCVCoe8546kicTEydTNCrlM9lpM/HbFDW3AgN5MB2uNCWOM8OpQn1lE6PJM3rFD4W7ayezDRaRL4RjUGN8IYNbINp+E9i2730JD30MMGEc8/IOM3Tw1ElFCjZm4h5iARNQANbKMUnBVAuC02AZE/rdFGbWNL5M8laxNJbfAh8EAVN6AL/SxCM1cSLFe3OrlJAuHvLHbHKo7LRh/vJiItTvfK0f8iHWDgFM6CDfOCbFOVHLR1QnwxIJk1SvAjTBsw/7nRTMjQJg0oF9IOHNqXRSalQmihPDD1Mt9xQuIxTc1TUSWXQMn1GDDmHVMgDI7CF8ghVFf1UmlvUG+1SA33UuYjUE1TVhNRRSz0IdtgFQjCCrPBGOrvVlqhVIGpLOi1VO+3US8tTrmvSZmTVbvsHwAMCNZBDZk1Vm6ROGG1PGa2/arW2BRXWbf0ZcPCLUziCJviuVCSnIe2KL7XQRhVTdH1FdW24a53APa3APh3Yf8g828MB9fNBccXXPytSifi9jjrQ/0o8VYEFw2w1TnftoX/QBxDbAxZQNgQUUlWbTi4tUHHk0Itl15Pk2Bb8B3/Ihly4ODF4KZXM0oddjoh9jCMFvl59i19F0mB12WeFn384qaS7gm3AwJw1WXJFWXP10pWtVGttWTI1WlBdjLP5BCOYAQ9s2uCA06pFCX3VWn6VVLN1CaH9WaLFWkSVpG5dijZgADjg1LD1jbEd1hDhWYeYWK9S25ZgW4od0xx9WTBcDNS6hURogCZYtEfxStLQ28N1icCNXIn1WcL11+Bk2Q/NyhB9QbJ1EvLIrlsQBReYASlsCLyV3Ga9XJuwXJyV1qmk2r1dV8+lS9AVQ9FVy8UwJ/9n2AIIgAOJYF3QmNys3VxdTVmBrF3KZTOMPUe3NVzkBZPFeLJlAIQEaIJtmIjipdBxJdI5Lb065T6AHcmrnV64rV6EsDNQIIEYGIVr897KOF71TV49RVtgjV2RGFzSc8zQtV0LzY006gYiEF6NmN/UcF3ZFYr9FcFdVVlT5V27Q18npd4EKjbIgYPG5V4EHkrnxRXwzde+bYi//TgHBon+Fd//3d0AjhQuoYdnuAVq4IQUgN89TOA3XWCSPdmoXF5HvV/TvNMarWBtzdrg4bcZXq8D3o/Xpd8dfuAe7k2p5dUgfs/OBeJMpFQX1pYH+V16gAOC6OAmZuDWFWGIFV//IyXf9SSLIX7eIt7YUA2e65UHTiCBr3WJHPbUM97ZNMbcNRZgh/EZn1HhaM3dLQbh86ERO3MH7pUAPaCLD75gjajfLfVhKo7g5JnQ3ICJQs7Q+UTk/9EjsguGc/iHP/i3es1jSbZf84TihoXaS9ZQ2i3MMl2ejvDkUf3cUH4qPXocwOAHW8ABEnCmndDjENbZBiZhhjBhxXPR5XlKQparCfXQLE7NdoWTK+MVatCHeyiDBrBZoThmbnllK4zlKZ7lGH3meWmOVNGJ3NgYsGEa0KHnpKGaesZn5qFncNmOf6jhGZjVu/mafCZo0CNofJ6+g8Ybu1Hoek7orREdpJDo/2dxY6vFXRb2Q7KNpi/+h224AgjQA+hliHFWi0rOVfyFYOYlmHdm6b1hvlwu12suWkm5Mjo+CFSm17Eg6ZThY2X2454FZK3dOHl5JpiOWpl+2y52GDtjh4MQZmJ2i532zJ7WsmXmjcwF3M1hZ2IKHqOWZVBe1ZjiOFI25X/wZnC+WbWQ6jgp53A9ZwLFZJWmVizuVy0Oa1rlOL8A5oOwY4DGi7UWmLa21/A1ZDWe1vKla0m1a2xu533cZvL8h2loApCuDMCGKsEOxrdm1JQGYhTeCK9GZ7BmbFyCCX7mjoOgAwhYWs6w7OzB7GXVbOWN685u3uaC4/qcoMMbkAI5EP+nHuZzA43WBqDX3kjC/mRSpeW5nmDtvGjdzej1awg6JpUwDucxYuUme8STxlbO7tcvbYQsMIP2gIc5yII1SKshM18Kbm5eFkwjIgZ0IJVXiAEXICw/dWIFpmo5smqFaOZVdNFAWIAz4AETgJ4cgIE4CAILIAf0TmxgXWyUM8U6gZyX8GYIKAORZgnh3gmTlmK4TudzdVEViAO6xABDQEIBUIWVWIBKYPDljkAMf1qkXqo31Ova4esU8OsLvO6SbTftHtj8HVr13IcbS4gg4AKGOIHJwoQFOO+RqOjbtWbR7qB/oCSG2GZfOYh2EAMmFsr71uH8ht39Toj+pkP1ZIb/AMiVLEAChvCEBSAAAcCDopoAGHgBGMiCFZuxGNPzAmOwPfdzA8vz/oKGaLgFYUiH/yoEDcCBWsiwP7+wRnf0SOevQJd0Paf0SocxF8P0SL/0CDMHcggAZsixHVMIcIiAOeCHTVgAUOiiOcgBTIB1orLt9aYn8siP3SYgUimHK5CAQvAHGKcJDR9MMB+vwv7jw7aJFU8IG6i1hMiEAlAIyWpxLmbuKAdRxqIR0Xiyl8kJVCaCMR6tHXfYGB/hn/ZbrD5ho4gCHtgHTzCAYZi1yYKGAfCpWVgAn1KzJw/Y2+4ohxELdCCG98aMaSACECjmHS1j4yXumjRuXR5fZK8J/2iQAQRYgOPSmBxICDzAgAhwgDCoNH0/X1o/PYzCCTuJnJxI7Stoh/vk4XsldzQ2dqCGeJsYLiGxDJBXb2vf5bCaUGxIpaH5B2FOgfhFzpYfqIV/yg7f7B/u7tqeK2C/OY39pCzLvBmWjeVRgwa48FaNYpfvcaWX7Q+fWgmm9hfn95yhZgAAF2EQl3MReg/kelg+emJvUTFnl6Ce0QYf2gefer4wGISZB+b4h9QWA6jXaXH3+qj38Yzl7rR1+iITeZjlC4P6hWWQh5V4Ct9OAbhnwoT/3mSuanMvYXR35sefN50/ZMlfkm6IkinBLdQuMcOPasSfe9DXb9FnZtL3b//TLznUx+jGwU2TvwnN53yV8/wnpvvKtXsyp07P1gjQ9nAp11PcTCWH2BLCd8NqEfaa4PDYRmmmd3yyT+R9j3zExc0rv6ZQwIHNH/+Jo33l6P6Gj2nETG7EdnEvlH1/lN7qwE21Bwhqt57hA2DwoMF///QwEMMv4T+EEidSrGjxIkaL4PZlrKiwI8iQF/lpE2kS4ceTJrGpbBkypUlz9VzSrIkyokGWNncinPNDIkyX2h7yLGoxqMR94IxiVOi0I79suJbNO2prhotQ9g4iZepV4saWXb+GJEkWKM6zAHSqPTk2o8y2bWGylRvSJ1qbQ+3yfAtAqV2nfiWe+2Xs3dH/f3QgiNlXdzBfl2FVQo5s0KzcyjbrWr6oeWLczkbpis6I96ZeoqXdpp0IWK3g1hfZFSPWzfOrGC5e5US92uZk1r8pYp4r+yvn4VyPdwyt3CXp5xJPL08tHaTf119jg5S37Bc31RMVwoGgRvVj5tczBjf5mW9x2OqNJn/+HqHz9SKj66cO0bp+ns2nXVHcgTSPM7lc449nyeCQwigSpRegSO3tN19p8Z11X0v1KcehQflRmBF/6/kHAIgT7TUiRdktVaBgIeETzS3U5CPgHxJocc9EE7LIHkeUYSiahmSluBKFKYr4o0dpeTjciUcitCKT1VVEIE0GdpSPQNEU5Nk0/0SAwElFPlZZkYUvDdlZkdutuVmSb4I205mJHfTkb1HKeRGVZ7pok5YZ8cONL8ukQyIjGjTRjkVm1glWkMKt16ZXUoaE52pK0vnoeE5SqCeAdf4JXYwhnROMMe6QWM4VEjCCkaOcGpQmdnvaRSlTloKEaWmaytrpnZ/+5BtNfVY5KmtPhUQbMed09A8nIPSwTUaxykrrs7bKheto2qrEq2i+/kosuJ2BWpOxTCJba6ndfRceifeIAcEfuxI7LrYkeqsWtzDKVa5l4o77X28BnluseMcO+GK2yoZET4ILPmuKCzgkc+m9v+bb1L5n9dtXxyIBHJnAA5d43cFCJazuwv8Nc5iPNLdEc6O+apiHpJUDbyygfh/vpGtHI/NV8rgnS5dyS+n+uG6L7YKkzzW4OEMPdrnFYMtJ1nK6s52TlpRZyBgHSPSvRj+HtEpKs8j0TQ6DNGih8ry0mEPfZnxtpO6F7ZXPgO4ddJwqLSmr2cqhfZLaI7KNotMgnZrqSw5C6JLWj3LdZM9fG/dv4CcNzmnhUA6bs8qgt7xc4x0x6+xLejDGI+V3b533hZkH9ne1nce0qcmeGjw6wQibbtFrgYbknS/cmLSNmGTWVHmdlzdtO9icj73v54+GnifwjIfq58LGgwSxggy+xIkGRJSzE/RnSg+s17dbrx/ZhPveX/f/QCeuu2vguB0Sl2RGs5eUgV5FaV+V3peX+FWvLUKTn+d4V7T7mSh/f9vf9ZqmlH1BTWpUM4ktXDADrB1QdpajnZqotzkH8i8k2RMVBVFmwe8prG0KwRJUCLUMubnHdWXwCgKZpEATrqZvNQFa7jIYwYHl7IFyOZxJMEg/nMQmIjjEyONUdRLmjQk5RHQfCmulQvnMbz31G17B8LdA4Wmvigi5YkXYYYxgsM49llAUo7xIOo2FMVtj3BDuYNVCkLwQfMH63RpL18b//YVhF0FeNlbWEXkZ8CxB/NEQ96gcI2YpkBdx4gp3x8TggVItUBSJFKUjPoPA8SD0iAYuqKGP/5a8omIXU8slWZTJ4EmHk6Qq43XO2MZDqvGLIEnlcNx4JUdKJIBeEstiflhGJK5ml95jYCgtOcjmSLBsMTzaDNElyWDGaHEdnJpLptEDEpiCL7kckTWpyRNfisWTjdomXLppP2JWMJFJG2cytcS0qPiiGTxsiSUksKjIvJNC8bSnTegpJGCqEnv6RONahOXPtAG0V6lbVxZpcg8tuKozDQ3QQ/9oJIhSpJSAFNxFh5nGfhqzI8gEW+quSZHXyPEX4tgJDqYhmpPqJ6XYJCMLlSjKUW5vNacMyU0ByUgiKgVBvrhGRztSHjWshqjrMep1JCopXOITI4Ws4UxluFHEZf/VOFNdq0HoIYxYzpIm03gQb7paU0z2UV8qdRNF7WPRUerUpWR56jHbutK3wq+ZAhHGl2jCCB3BTq+anF09/1oplvaorBc5K8v4qda9YiSqP1tl1w5yznkwUyVlKKlyvHodsPZSc0glq1JFAtqlffNs4WTj7RjLs8voUG6t7EgyYmAx6chWOrR9jlj1FtgPDZapvTXcbxXpVuEO91TEQMdBjouRQkigDIq1S3Of89xN2valSZ0iTAnb1NIg1qbnJRV39fUOZYAnKa3tSDu0AABOcLZDpIVnXzmm2VwVGCGGBewSrStacMIVlfdNFhLl0QxfZGOn/8XIK1IwrQClVzn/6x1OdGv3XjNWl4nzFU19M2JalyFxRrEcoH9Nshg6XNidB3Zogocb1vaudLoBja+E00rhH1dkxgLK6UmciY/TdYR5JBhFg2lS4uGc+DcpTuGKyYlkF19XdBWGao8Tk9+nXSMX6FwcRdC3UBZt+TddLiKRIYxb+EaYzBP27ZkT6zcomwRuh7ksKz8skQL+IctwQnT0gpza2kJwzywec+/+jN1A27eTawYJOojx3QqLNxkzuJqjH81LPmb2qO61tJj7nGklA5rJFHEy4z7dkXe8a3rLvMhkd1SnOldT0phzdZHDXFFMT1DTZra1inqszJ0gqHzH9nBFRuqqVPOE2KW5/3OG8rxZI/9GmDB0Nvc4LeP7TtsmNraRkPtHkRACIBnc7ja0Z2tsXw+50trMrQtjem5abzrfU1IsalsiZb/+WiJ/MI880Wtw9e67sZRuIKyXLetmE/zZkEYzx3TdEX+0GZ1+bLhBRqoBAo/L26IBN5HEzWByZ6rFs84oIidukMQlXCWD+sUO2U3lEAb13kxxeWdgziaZd4vmHmW2N9Ht1Oz+s2ki7wg6anMb6Ta8EBAoQ8Qtg3TLKN0yXxajsgUL9X12PN06B0Cf2r0TXvd3osvUtiXCLva327ni6o7M2U+ecbVvPOptrydzBAqTGJeWKHK3SbUl1uqdhqJi0zA6Wf/GHpmyA57p/ko7ddeO0QejpXEGMhDj+cSPnrfk3Ti2O0XIC3bMZ57vxZ78xbNZe4ATUuCGPLxwnIaUjzwl9RYZCqEVLhCCnJYi2i6EP2i/+1XjDffQ9TzInB4um3Mc558fvsNS8hHjdwr5fJMKVT6PkORabByElZDtv+33+O8k8AwffOgLz3bvn5aKsjm9/wGAnlQRARagUyCfASZgAhbGYSigAz6g1znEBj0gBVagBVogNlygBm4gB/7DRnQgCIZgAZKECJagCWagCaZgB8qECragBqJgC66a/3HH+FmQCCKgCTJLN7hgFeHdDfkPDwahAsKgEBZhbHygESahQpD/oBIqIRE2oRACQD1AYRI+oQlChHoQXwCm3rTh2kTsl0+lGdqxn71ZkaK1HP0l3fx9XMz52/Rdmv6N3sw9hRZiofcIRoyJjxcexDxsWDaYT5FF4OqF1xn+iubxBefBB/Y1H+gdWRzKFP/1n/hFhDK5EWKh1h66Xmb4YGuIl6wcol0k4q0sot9oX2eY2+9FoiRiYaCUyin1HK7pgzXcgsnJBRkihSdyCijKhShuCykeUaqR3sw94sCpYq1dFuvBXZrBjaFExh3tSFfk4qPsYlv0YlvYn4I1Yrlxn+EZY03wAyiQAkVUAzSUIzSYQ0WkzOM1WY+FVGTYTCG8hTQOWxqS/90aUh+K/WInmWLAcOP+CeNBaIMKiAAG5AA8SMQLTIBCBsAbpGM4JaNE3JQOWsY29IAL2AKcDQw1qoU18os+/pI21pzoQSJAGoQZyAA/wIMI4IEkqcIAQINDll7yHZ9ieQd4iKFLjAIIEEE7MNY8nslGnkVHesxHWp8e8dlS+RnwmcQLxIFBjAEPVEQWRGVM2lDIYBD5SF5kuA4dJNxPVklQksVQkgU2xtsbxlpS3lxJAsAEIIJBzMELUIQ9RIAhWARcsgFeCoI97CVf9qVf/uVfVkM9AOZevpLM4ANhJqZiLiZfwsMVrJw+MOZewgM2SKZlXiZmZmY1+CUAZKZnfv/mZWIDPIAmaZbmYtbDZpqmaq7mXqYma7JmZ36mNqjDa9YmaLqmanJEBCiCQdgBClBEIExAX81BCUSBccaBOSSnci4nczZnczIDOTjnN8yiM6SDc14ndmbndYZCCuAANmgnc4LDOYIneZaneZ7ncjIDeq4ne7anOUADOLinfM7ndZKDetInfuLnfeYnf7YnNHxnfwYoeu4nfR4kCtiBQZxBDlCEDYQBQJHfrSUMMx6UZTDC17lNRqIhGwrRPepU7t3WvyGlbvkeWnmjS3ABVcIAGwAAOGzCQQyDAMDCRUBotCGEOPyCMiBGaYjBtoEf8RTiJ9bj5nWo3tFEWU7aWWr/XFp231oOQwTsgA2EwFJUwgAcBBnIAEbQaEQSBW0Ew9aJhqnNwOV1RYYaopAiIpFK30gUJeyFKBwuaTeu5VrgQSCow85hwkGoQi9kKdVxlE3CS2lwggRcweplIZW9H/8VKZoaZT664VG+6YjKV5m53YZmhDc4Q43UVWmURzz6RZkGaaXqUpouWNOF5NMRYyrKqU1oqUE40+tZRjkQAQm8gnB9qi6eaSiOKrLpmZuiZaQmmYlSKj52xDl5A07uhCnsZE9Chq1OI67yoq72G8b1qpL+qlIG69T9HUJMqDKuhh40QFdqRrPSY6giGKN6GZuOFbUSHpz+o0YxmTt2q2iI/8HKfdq4AuWzVmO0fuirrWv+tasc5ly5yhEdbWlnTEMMiKmu3StY5itH7uv1OSoQeZZF7NbaTGq27hUkrcwenoSgEqrIMSyThOVXjOVXHOm1+asjAixJvisbvhJdsWNk0EED6MFMWlyOIepMKWquniueSezRUWxFWKziYCx99elBLBxN2sU9XAEImMLVDWsj6ezOqilPmCzfpCvX4d/KWqtauiw+ntMHqZ4tusC0RK3UfuXIOqxQQix7AS19CC1FEK3uqGpNIA23dppaTJYY5BpIYhvVkqxXYC1ToCy/qew2jmQx2i1NpEy86i1ZwOPNmuUbAemtliuQ+Wy4wW0J8f8eN0mq1B3tAvXUT0XRsWJErM7q5FIuIVJtolrtThCuURguzj6qrwYc6C6luViQVWEVR32FLaQAT6Jt7bZu4LKtWLpto05rkrKr1zIp2ALAu2kqW3mFhcJBoQaSyP6I4DKF7BYF7Wpr7HhuPuUutopuq3ZJZFWdUbyWJWSv+snb8WIuSikvunIuvpGvWZFoaOmuZeCFLOLCM9DDBZ2uRGwDDrhAGUbc9tIZ8pas/f4s89puteIusDKuS/hEcXko+9rEKGjAFdzD6oIZ4Ops9xrF986T1qoY1yYuqpYoBrfEHGjCLriD4xXwTjwcHeQaSzXwiJxwUaRw/a0wCSOuSL7/cP+er2jEARbkjf4YMADwqCX4La/mrAk/8OBG8OZO8MTq72fxL2+FLowhrenSRDlY5MUQr+DJ7xXTb1FpcRtycdB6ccWC8cWK8e6Kr8y2hC2QAE9ScbKVMKIC8dXC8dLhL/vI7ZyYbwyrBKsGJE5eLw/rXuW6LiHHriGbHRGjXQsfMcsubvQ+sUqogQRM8Qj/LRsPMhZ7byZ3HiKrGqRa8LU28kk88s71WDsQQQpgjRpvbSq/3yUDRysr4is/jyJLBN0qFS2bhC3LK0gEbw+sTy/7shWrsht/1TCPYjFr2THjhx0XLR7/LxlbWEhYaN8CMtyqLfeuMgpnsy9u8/iK/6gsf63ASi3IdcRiFIId9uOPWjI7B7E7X+Mmr3Hz/uvzxmko4zBGNC0I8MYpZx/KWbM9J1BAeyQ8GxgdD+03121C09BEsJ80T7PPqrMDX7O+aW4cU3IXy3PvMXJHi9NFCOo5izQqVzMw/3Mho/Qhy3HcZvTcbrQyvzRwTQTN1gs6b18/z+9Ec6hOa/JF241PL/IFC7V2IUTTakCEHDVSR/RNm7RzVTRRPnXWdPNBJDNSLrNINHO63JWY7nPX2jRhBXNNCHFEDfT9GfGpfnKqUnUHG8Qo6AgrVnAlKzUHV1+bRixPdy5Lf+5U13NhV+9BPJweBLZgG28bLzVfNbUrJ/92/i52+TZ2MXk1JB8EKTsPTccvXI+SXNMEXX+jXWdjJ+f1QburY4vyPTRBCtzSaaP2YF/2Y5+QZhMzZydyVCMzUJ81X3PUNFRMHj00T5P0D+M0Jge3Ng83LN9uS4M2TYn2zn2wgLm1Z08tYfMstFL3O1u3MRe3N7t0bf+N60w2eIc3dFPIaksGWJPla7PuHIf3F7N3aGP2RZDyFE/yHXN1XEu3MJu3QIs1zvB3Hfv3dgO48+F2KDCIc/Ozgas2gs/1fZ9sfiMpBTvvPENve7cEwuJAO3wD/JboLx84d3NZh2ctg4sMWYfIcb8pWt/FOF+EKWjAdyNgMba4hr943yn/uEWjNzerd1nfOFrmOEhAaCGAKyTDLjX3tkT/dqQZeVgjeTzHcnbPcnJjhIBzRcdiuCB3tYRn7mG/LZdjtINrNISPVrneNgmQEOOUOckcKppjORhpOX7PuNi8+U/H+ZKF6omvDw/j+dDouYunef36uYcDur0ouY0T+jFido9fAepEhKKns+U6K5Hf3povr0rvt5czNpiX+HhJuVt3+nN/Ork6+htDuoy3OVQLulSn+n/zOQCMOXi7eqmLt2+Tt77SeuF+eMoW9Ft/OT3vul/QeV4dNbCDaGozUX23RGsbKbIfrrK7sF7DcJgDwKGjTo1CopBb+4azdowfu6QDDqUD/4BZ43iYZ7pqTNW092u1a2S627exz+62F6+pYzeqN3uEF7aF7jBll3uQ57uGyjo29zv4/rse3/qpf7auFzxS+DqBRyhGnbu+h7r8QbwKt3sS4bpxW3rB5UzTzuqmk625M7yZgvzLrbu/k7wgvXu8N/lLb8MM4ACia7XCp6LHNzyvM/Wo36+tjzXOM7mSOnlHgAq9k/u6dTzMg6rDn/TRS3CwK3bF7y/KexxEHHy9f8a9B/LQx/zVf7XID7HNf1KNwzvTs6vTm0b+CPjY70nZV7GV7zmxP+za13Xb39PSf72wMnS0n3Lej1uGo7vMq+Hfu3bgl8nb53zTuyzP+3zLi//EMKwD1SOEOjCD6/LCvreEej4+TazDMEQ+RfACFA+jSUCDxENQPQDDu/a4Fgwi0GeEDQRC5x8EHuyA61ap1I+LD8yB6buEINhA64dcWyBA7Gf9RTDBiibZJMQlIh38HR7J7ve+Qfx+8LcNExX/8bdE8i//kzX/86trSEh/7la/sCzGq/BwyGz/yyOE91Ot8Mc3p4g/9IsGQAiywQ9AQYMHESZUuPBfw4UPIS5EoC1iRYsXDTrEaJAJm40fQYbUOOlFSJMY52QAYQlaS5ctT8qwc5KmSZgL4+SouZOnwQEHb/YUenGHx5BBh57EIyPpQqRNCyLgBZXm04tByFDVivD/piAVW02WyoDAQFmzZWkSKAB2KNqFBQiwlVvwp0G3c5OqNXkXL0S4cvlSrdu3YuCLegm3NVDwb+KI99rdO9hQo2PLlzFn1ryZc2fPn0GHFj2adGnTp1GnVr2adWvXr2HHlj2bdm3bt3Hn1r2bd2/fv4EHFz6ceHHjx5EnTx4ojCCCB2fZkW4Hz3PHlP8lxP5v05g46igb3MeHzBxyp6GxIcOqckFQZthUC789O2dmgvgsZHbGTK+M2BUK77/2oKrHEzxIUUidSuwYxiQAtYNwu5B6CUQQhaCxY4w61GELQoQmLAjCahCxo56EqsGDjDMS/EzAghZscELKZgGgmtmy/whhjA64QAgTIYBEAYPLQpyMMkEWCEMGGCDkQYUzcghhndK0seCHLBAApTI+HAgDAAzAcYg+zjwpQAQEMHSAiSkWqBGAD40EcEytcsBggTMUEiEEAi4U6cUBBSwSI0EQCCEEhV7YgQsZQgAHLEFFlHA7bAgwIQBsEhKEBy6YKKBPzoo0lABDZvxnnxcE2ES2agiABQBSCqBIIRnGIDK7P9+s74U3ALAngkbCq0cAVQDYB4FMSmODyX+qQKIyEeYoSIY3+Lm1Ps/s4WcTNBMaQycAisiCIGsjtJZcXJM60QY8Ezqxg08/IhfE+ua7FqQTAzk0IUd7nSCQR3O1N9J5I//lx55qLoUoCx88M9egEzEwJFJrycjCAE9kU8QCgyKYRKFhBvDvOo3+pGwfAWr8x4cwqq0vhDoAAAUBaEpbuSFDOqjMAY8BYOLNiQnUbFuFbDAKDxSMjBBQeoNOat2H3qWp6QEDnrqifCGqxwFF5EI3RJLrQxjThbCRwQzQcI144DdVEcGci2WrA+mCTMBDoTAGsixQgSkbJgDwmPVZI08WWICA/EqDgdp/yqzMhioAAKeDF5iGU2huESrBbgAUmSBpgoGuHKynF4r6JKv3tvwirB/KAoZ9ug76a6YLElshVQoIQIYTXSRQbQH5gQFYADCOjQ8RDArhX4T4sQDxkWn/VwhTTP9hAnKHyJngDWwQWUAV60KzYY2GKnGgMlIMxcAGGwIUOLOhE3oBZgAE6eDzOENfeyvSFTL9QfeBpjSQsE4ha8CAg2IHwInlaoG2W8gwbMAE8G0mbRITEBuQUBC4xSYTCIBdPQoAioQoYgHwwAzYGFIATzTkaQ7BBOZkYJTRMCEKDbHDCQjEDHsIIQvt8wz8EMIDLwGADUz5T7kCxkAl7o9d/YMXSJqmuiR+hIAIeYMFpjIXdCUNhTZK2EMCYb/eCex318oBAcoSgALMBDb8wMBM5qAvPjTiOT6YQmbElMck/gMJGYQVMP6RiTn8YxgCQNYwFoCI0jQiAtCw/wcMzPCPdYRBPsw4USMQIDIGbvEyQATAGBzEhw5gYx0miMP9UFivKTLRIJPYXEH856cjwimPCowIAcERBkfFIQKs6EvqwGYuATmQiAkChTkAoA4f8ABtwYwYZQKhSDFtMDaMtMAEKlGQlRVEGw4Q4QlLRZ9hoCACC2BDQ9iQtzc4oAMLEJdpsOSAHJDjH9gowCz+YQcEOCAC8JqTZuBxlrwhAGP8QELhfMC7MQWTobbsCRfO0ggAkIGZAPDBWYABxVJVrV6chAgszhIEADCjAMwAQAnOQq2tLFSYDYWlWcRYgvyQwQARMIANEAiqGV3ULBlFAhda1pAJgGKCrtkHNC+K2juGXKsaUwpaNZIqGnJgI2jroJkAVQMOWV0EV1bbjVdpIzuHHoQfjpwNWC0SEAAh+QQFZAAAACwLAAsAbwK1AQAI/wABCBxIsKDBgv/+FYTGhN/Bhw8TQpxIseJDe/AsQpSosaPHjxUVghx5cN86khFFolzJ8qDKlhX5qYO5kabNlRxvEjSnk+LLniRzCgQ2oaVQoB3hzUR5FKlThE8p2iN3s2lUp1aR8tP29OfVrl5vYvs6MCzZjT+JGjV7dqBSnGzb0owbdWpVunLhnt3aNS9WvCvHngUs96hallnzvmVK2C/Ixj3t2kzsmPFerlgr96TcUjBZyJ/TFkUM2unioKU1+6wseW5q1Skv94Xt+qnnr6/BFjysl/bpkZxpd8zdsjXM4MJDEv/IN3Ny0rbbLgdqeHRv2L8fT0++naTxtc+vX/9tjrQ7beQob181P1k0eN9LUYfHyZpq+/nyyZKnjj8/UvVRsVcbQbxZBp94/Q1X310JfoQeTPtt1qCDAhYE4GzPVfcedvEBV2FlH2r0HXQTavRgSxHqFKJfJ4J0oXMZukcihwiWONGKFY1Yo40u4VhQilXxaFGLH71Y3nwazqhadhQKGdKC9zmJlmwwStljdIMhKeOOfjHpEZEJ+jiRjgZaaRCYJAE5mZmxOWUkf3dZlRBlSXKpWIfasekSlAPqWZaYA6k5l58I4fimhHGGNWdidZa5JJ5NEloWn8cBipulAAh6nKR/YvkZdWzRuaWjmnk5HKbrUbohoWiOpKlRnAr/1KpFh6oI6pS4DlSgf4/aaSWqBZHJq5+zMofZkbEWS1GtQW4WalyLzklUtNRWa+212GarrbVKbevtt+CGK+645G47VbnopqvuuuwuulW78MYr77wK0Wvvop22mRK+u3oonKkmAothXsL6y6myHb2KWKwAIAwRs2s6m+uVBFpHamUADylwlXIVnOfBmCpMX7KGSqditA1LRK2+Q1k8LMaQfrkxsn55HKmkDlskMlMM53wQxIMmqvLQ+FKsq8sG03ixnjMDYLPMJFNJM84lZxleoy87lrFyPauqJLEhHwsnyJ7ipuVuSH+sdNZsNv30qVHrJzaiZLtp8tWjsn2nr1K6/21fn6yGPXDgZaeKN9qrlhoz3Ml6zbeQPlO0c1A9V/3p4RUnDvPjkDu+tJmRTzQ5cJUXHuDZmX+9+eege65334JzDLbpg5+Xd9K9sv6r67jPLnftTFtuNuZHa6714gF37djbydf91eiPlW631TDlAzgA/aqd++t9866978/PbWvc/93NEj3NOHN99jdv3zux3rcP/njiN+s8UEBvuhI/2eAijfWVQpzqjsc5HvmNQfeLCvQcJL3yUW8k7iBGMeSBwOINsEvI05jyava3ACbwKQv8UgPxZ76P4OMZueCGxFJXwKtszSdNoxvBOmg80MVuaoSb3uU+wg1cRAMft2Kh7v/a8sIbxXB8HKxgDsMHvLYJz3AdkYcxgsGOvwiwhVEp4sRYFT+ofdApIVQQ+Uj4QIrkQxq3sIZDrChE7p1Fiyzj4vJoeMFf3XBsVKOd7B5yjl80gx7rud33VufGzs1RifNTYP0i9kWb5A9WFEHfL84RmisO8Y0Z5FrjDhmlPP5uj07UIw4L4g9r3EIa+hiMIOVHyPcxrYuM8yQTQWlDUeKxIO4IhjLmUZhVenFtrmwbLJsnS/o1sZY6HF5BTugLcbDIl7F03yBfycnrBU9qt0ykTh65MILwDxfPAGBesBbMvV3SkEns5BKNSUs72lKGUizGOzRDzmliEIsTOqA6tQn/xkUGrZE04ebIzpiLbKwRRNAkpjRZSc10WjOUnxzlNZOZKnH4ohn4OChCLVnIr8DRaDgbpgYB2pMwmmiEPREoSuShDCpyJ6EjBaY9W1dNDxZTkcd0J0Wdog9r4MIa/ogRR8spl4+e6Yj2myEiJzpLiULUgVHpxi92iToL4tM0mYThBpW6T6ay06nIhCpS0OcLSnbHYfVk6D3PaUCRavKmIPSn/kjaGTD2D5WyWgtaYfrWhf4ypDWtoU6bmk2vphQpEVRGOghCHJStL21qNWdHbaTPh4b1q4V9KhlvwkwVQsVDczrZUGda1Kwacasdo+NV+xM6iJh0SCjdpk6ykYto/4iTsdoJ7UFUoQqNugUUzHALNIYLjX2kbLSRLe1qWetWrdL1Jq990hgPS5N4zjMiFMqKOmTQgQ7YAE+ViAAMIpAFgfBhABNI7zCO20aiEtG0W4RfYOsIO2zKcJ3UZckZ0+jbz2pMtwd5gwr0UY8TxKEgIXgDAKrhAFAAgA8yKBRy/6q45eKnsjbFb1xzWt+dgsSif5QuDAH8EBmwQSBnsAFB6hGAWQjEBlx48AswwYp8WZWtLoRvHOXr0AzzUytyheRzA7MS9AUDHSc1IokhEoJAOIQPIiiIAwyRqQ784MEWgIEDZAAO9urKAWwI8xvsQeYym/nMaE6zmtfM5jaj2f8c4HCznOdM5zrb+c54zrM91oENPfv5z4AOtKDNXI9qDPrQiE60ou1haDznw5T/W3SbjduBQAgkECHQKBsiwAUZYMAHAKiHQMyhgjDwwzAI+IGqmWCOVrv61bCOtaxnTeta2zrW2KjGrXfN6177+tfADrawzaENaAz72MhOtrKX/WpyMIPZ0I72sgEgbVg/G9h9VIY3qs3rjMBAwQB4Q4QL4ok4YMIHpiYIGXLg5ZYJVrk4LhGG3+2k1j4kuquZrmw9wkxnRq9QYBoDuwFgAzIAQB2eKEgvEKAKABj3xVVoN/YgS+FWkrZ787Uw8TB73x8HNGH9sy3l/tQqaFhgBzn/wAA0AACKADiEDDz4gQPMIBAf2AAJJ+hAcNNacQLGO5/NPe2QaYLvG8VWLBp5hwTl0V8xOtYjo7jCFQSCDTzgATPkaIRAoMGHODRcIMwIRBwUIWqJs6/na52svIMeX83itJ0dFutDCJoNvS65IuWAgwskoAVbPJbemNT41di+Y7dvGO71fqJBLOoMIJLm7hThRBMYgINCNJ1UZ49mhX/eoHnTN/H2RSJckf4QlgbDHXaHTN5doIEyJCOIN1b7U4wq4U32GPAGvGPHDbvvguhDGri4xuVPVTSLmOIKDIjBH3TT3osHnvNhIjxIPV7SIHdz6CQBkFSXAcjHiwQv9/hD/wwgcAW/B4ivzpVpcjF++89DTvei1zDpBUJWSnoftxCxhRgkEAM63KOSzbd+zyd7nSd9R6Vv/cRhoDc9/HANp5RKcNEUYcEPhYAD5PcKvTRhmmdxAohOXGVZg8Vx8Ud9dZVLVBWBclIQybB/JAAH/zdO6Cd06od2NNV+gvdSoZdU8mcT6YBCnsUYkJdXAsEIOMAARMAJ9BSDbZd27lVvBlh7owdkCvh+TsENtxBOKEgX/zANagACJKAG5QAbPLeBPkeA0Zdx0Hdh8KeDJEgS8dQNw6ccQUgQnEAERoiEwjGGCrV5ZshcaNiHVSWCbMh7+oVGwqdSKcMZ29CFrbcNG/83cbjnUTo2fTz2gT5GiFKIeFRoEyCWUQAgUMX3EHWYAJXXMIEIie5HFrTHWEjFSJYYifK2hq64gyBBD8vgUgPxSE9nEOVABzGgAWLwekIoVAFIg/AGiEjyhKyIgJkIViFIEvpADfxlIcSXGMeXAD1QCGdyipm3hxxojOz3iqlIWbL4T1HYEVLVDLxkEMyyiwVRCDMAAVogjNv4iN0YU35FhsL0h02Yezk4i20IEfjgDGV1eYfijgOxiF7ogkZnjxSnj5LVj23Fj87njxGVWZcFcv5zW+yIFkF4fA2AA5YAWw4Ji1k0iQdoe+J4g3lYjnN1jhARQRNUES8SigRRgfL/SI8ihoPFCJHHKJGUpYw2RovVN4UWqRH5EA0FpREAYpMCsYga0IIvKEbEGHtAeZIsmYdCKSuteIkJqIlHWREW9QyORyuseHcg2QN4+G9V6W7jmGNZeR5bKXGYCBRFVxPMWBBShIsdIRjuSIHxOI8j05aoGJcssYpDKUc2mIaneHjOGHdzh0Zq5CLuyIUakAIMeX08aZUVKYmGqRqe95kbdZG7Z3gE0Uch9hjYsGSm0AQJcIQAKYZKWHhM2JlBSZEdSI7/aI47KEn2l12rWRCMMAMSIAbTMIKyqYHeWIZXCXS4CY6bSFilmZEAUEoPCFr/4BntQAcp8IXtgJGjyZm5/wmXjJmMz+mTCyidyOl26BAMy7CO2SUSYzENZSABM6CNYCkds0mJzGmba3eey0mdX/mY6TmQzYSdKhEKk0cEowBFm+mWokkSiMmVqNUWzIOPAUl01jeYefRNWBifKkGEDaAFx6lMD1qY5XkTE0qXDbWSKXqijgmesGNducUR29mdcFANJdSSyomh34ieTgigPmqazSijTkJ3cdgjurWCEHCfVLejttOjfcWHzVmAQjqldVmU+TkhnZikAMcRo0AECdAE5vekZWRZc2h2DxmgEemfzrmYyEiYMTqdQpIOLcUOoFE0wykBZeCI1Him93dUc6iHQ1qb4/mfcFqlrOWSQv9mJj31U/7wIPjCD3+QAiCQmR0JqFl4gEFIqFjan4f6pi4apzA6oEaaIFJ1gsgRLe0ABxrgAss3EYhIOs7yE7qVFZ6afvnIprtzpbqaoRCyoTxjJfW3jEqWEMmgBSE5kjQJpZViqyqBq9QyLfdSrSujDtaardp6L+eyrd76reryLuA6rt7yTahULq8weU3wCt+ymuSqLiQHhZBXLdT6rvHSLfaar/o6J926r/5KruL6r/YaQcYgD+XCCT0wotMQLu4qsN8Sr2dJodN3j59qqNDpgam1VERql8I6ciVyQikUX4sCj60Xhi7irHpFoaHVqfuZkjMIpBhroar1ojxKmuv/GR60JXITMyetCgIuQAde+jAoC4Qr+30IKbHiebGeSbNa6asySJQ6cZfY1SA06lxzMg1iAAEiWVeaSrRDA7FxRLG/+qO8Go4Z21Ubq6UEWhn7NZlyiKxX0ABEYAof17XhCaFM2xIram8sMpd8m4E2O4i0AWLd91/pygBkOn8mWrNJC7NLS6rc4bddaZIsIbV7Eh5GhmQB8w+tKY8lqrgOqpkIIrZPS6VuaqWJerqLupsvCRvWiVf/hbAQIAYmm1+LW41OOSyku4RtGqqoO6qKqoas26iqgQ5TBZ/KMZyt950elp/ask+7S5u9q7S3mbq+u7qBG5t50W+nUqkteGqK/3dMz1seLQuFu1qoNQi8qiu82cubeeGhHLkRrfqqscqijmQ+43spUjq2oEq9iKq+18u+6im4ZKF0ExS0/zC/M8Csidl754ctd4uikAsTezu5xCuzGiugRUqnT0F3JoK1DdADDeqym1Wr18K4eDvBeouS5quYAOy/nceooksW3JALZDkkyYq4GCi9MDGrgmotpliSb4mVeSuGkpuXanuqNyFFxIB6IZGsDZC4pevACHq0Foy0KRy8PVHBFXoWF1qxzzinN9sT0ZhGQVURyYB8TaCTvNvDNboy+UaY0cufFuu41fvCdpxPMsyhT7F9hTsRh3sFnwvGbvy2cEyVJzrHJP9sugF8YUeMfSjSsbT6FMVaEenquY1slvtSLcMqx2uKvj+5vo7stG2cnoKovUTngLALyESgsFqsyVwJxK0bpT1ZtgP4yoNHyjxsym+3tiRhgsgrij0wu7UryrRywhz8TPs7xf2bx6J6tiAImb2sxCbkgxWBsHzKvEVMcg37t0HsyZSLVRHqGKG5zREsxgSMEj30oRDBCTiQzV6JGtESnDuUyJ9MyO81zn2ry3QcxqaazB5RtRDBCL/of2grM5z8iUMbwYrcwmQLymaLwQftzxs8xh6xX8I3EQQNAnQgIGAiy39azygswbhsE1ysktAczxTNsUZJEV060DigAXAQtLb/icwP4cMMJMT6/BEnfTCPDJNRK8lsSRK2yJcHYQkzINP3ECLBAdLLstDkXL7G+rK2HLNePLMqDJp73MkgUcZuexDuLAFqcGrM98O5+9R2G9XLXMq3bMy5bL0wHCZb7bEfkY5/XBBhrQZLHUhnebR9CdXKXMsQ3daZ/NZ4XNW62b6zXBEGeg6Xl9d7rb9OnR6ADYNrvcuEHddnCNfOLNfDO8Oic1fxKxCQjSmTzbUiTcuNi9jkmdWg+dNQC11CndMWQbDXZRClHRqHTMWno9PmLKEsPNWAxdmsrcefzcdzp5R1dxCj0APZ7M2JuNu2G7qqncVubdLB3cAtmtLhjL0D/4zKAzGWZUkQzQ0BZdAOfwvSf4vTIuTbro0SPT3chz3YYTnNdLqXVWQQo/DOaoDeK5LQDg26vQ3OQzx72Y3FLszdBe7Zin3BAtG2/fUKw9zf9gbg/VzIqZ2cgo3Pmd3Zmz3fHF7f6BybqHnXAJDGDUC7rSXdbE1kaR3Yq03fj1vSkcvPi5ylQT04vmkQaZwAghw5Fi7j2VfZgBvjIT7j113jxC3kxt3g1/e6EJiQYhDFyeAzQe7hf/3ilr3h/FvHxf3MEh3NvDziQdOe71kQ2yAGRmgLaYrQZ43lTEnk+nnZF57Pv03OsA2skdwV3EsQ7aAGWsuuCHPl700Q7I3II//d0MLNyJrth0t+5LF43GnSP+wMAK0qAThgCm3+X29e6Lko56pE5zfezF/+uwq+03Pu5G5oDPLkTXSgATHACZs+Yp2+04eeZASO6klx4Pabvqd+51v+3SuBpDdJAjFgCbPukbXe3Vme4Vot6gHu5UwO5ledwSudxDDRpSJRCCTgAoyQ7JsM3QjO2y3tEcDAEwfBD8NQdgPBDMeSq8ws7ZD+v7/u6cF+yiNhevndMO7M0eBu1sCy3qBuEcNgAhaAAGdgEHjgACEQAYggENUwXgvABQ4B7y2O5IWdIXmO47INE48KVAPxCkQg1uCbGiwOgHI34DSBBEgAAKpQAC42END/UACVAACIEAH2AABcoGLMsACboKbMvsW8Lu7O/hVf3OXXnuMskarwieJi4N+lcfKpnvLlbhEGkAkvlvADoQgRMBAIoAgAUGkCgQQRZ/GYrYpDf8XIXe0TLc1kbhGVDABp3gA/3iKEfu/TrfKdEQArBwBREAUEAQoEwBXQIAAHRgAOBgBjwANm5wB28Ph8sA+SP/mUX/mWf/mYn/mav/mWrw7kwPma73CgP/qkX/qmf/qon/nwoA2pn/mi3/qwH/uyH/v2gA2pbw+iLfnlAOhEYAuz//vAH/yYXw3CX/yZzw/MEACYkQUtTxA2AANnAAML8Ab8IAC9BQBmkAPguxsI/5ADNpADPgAO4j/+5F/+5n/+6J/+6r/+5l8N0MD+6w8A8D//9F//9n//+J/+2MAM+Y/+AAEA3ECCBQ0eRJhQ4UKGDR0+ZKiNGcSC3YIZ86YNnDY4EnCEwkZR5EiSJU0WHHZSpUp1/ASwAgDgB5eYNe3hIbMJAyIADirFzIIEwL9/NWMCm2BU6VKmMYk2hRpV6lR46qZGfXpV61auXIt2BQvVHrmwULOWRZvW7Fe1Wvlp64rvmS9uNQuBiMGJHz+2bJsSBey37WClZwkrxXb4quC2MNgAqDdBkUBmShtFsMqDJj8TdoYKRtrWsGKwVdOOJp1aKmPVTccqRt1ads3Yrf/fbuWXLVe0fDEZuUjBCHDhtYFnE65NOPFxo6zTIkLARQaMmGxUxGRShQmCQPwAeEKQJYcIeJ+VhlabnLlS02jVr4cNn+nrw+/li3au+vZVecaKvYuJExxAoMM44goLLL/7tLIvreXgU7AsUMyoY52YWGkkpk3OYAOmmlQ5I45vnAItqfQilK+9shpc0L0WbSKrPhRfDIvFwfaLKh9pbrHGu1eIkEANvmrrK0EaT5sRrQfXS/K+0dBD8kj2rHKvSSkZPJI+5Ky8cjUu08KxKXF8aYYeAJJpooEy7hnOrM/a7LLGL7lakrk5j3vSxCjjVFHOOJHLMsYt/8RPvjCVoqf/mWDQAaAcMRq4Yho4/zKSUD9lqxNPS0k8T88q/+wTLBs3ZerOwbQcbFRSmzMVrENj8seaW6TRhx84IOjBlklLNXDVrlTtKtPZWm0tz0L5pHJFYi1dFi1Uj/XVq2bdgkspdH5RZh4A7sprVwQrjVba2YSVbVrYSoT2ylB/NbfLdrt69sRwxYUvTLl8EQcAS2IgoRBvWYXzXSmBpdNJUo2VF9k958WSxngXZthLQ6vNDRfebAFSyH/f3FXgIwneitxiD0Y34S7XpTfiqTzW6uFPVZa4XrjeIaYYeaa5ogExytkY3ANhxorlmERWTeiXa4LyaHWTvRRoNx0WNFWjnZza/yh+oJEml2zaUQMCIpL5N0EFq5YPZK2ITo1sdjtNV0qUtzL74EDPdXot+bjJ5Rl86NAAh1fC7hWruu3G1OBNEYbYbaZFVZvJuWUcnNf15FEmmG4KISEGS7wVG+7IJS+8bJLZNnnpxD8f6vFBUadtan2o4TGUGEAoZMilfG6YdU5Dh3B0o5JWFtTF19bdKdWlLt482br5ZRlkiIBADXtQw93z5OOeCm3SGoe75NNpfJtB7jWFmu7isYcqUV+6KQPSabAxrHMXrzda+/gO915pxb+PfHyjXA7e+QTGj2vM6g8e+VuYqlej5CmvNfarj++Q5qkAnmx4KdOd/2Bkvgy+y/8dwVDGI2IQHMDsZ4EraiD6ogJBQOGPdPxrUfgWo8GRlQ9yAjyMXHIhix5IAA5tessJ54dD3jnOhb+jYNNMpz/U0RAAAFRiE6fFDVwIQwwMEEM7zvKPILLMiWmrn+GYlb8KLrGMrHMiFBlHv7b0pxh98BrYisTFatWQiA8UI6EQx0TwXbB7KTxe2z4XNx3d4hJ44cRwxPaq+91RNSxMlQSPksQ1KoyP/Qtk6aSIljGJYkB/UGSvGBlBNhbRTpIEAPCieCQZruyL28skDJ1GMHosIxhf8KE/noK7UbbQkamBpGhQqcpKWlCWs4zlJeumKn3Iyg4S0MI2+MIx1vQykqX/xKPojjhBQfbxmEBLY9S6ObhRXQsVLMCBKXYZOKVYU5jYfGQe/7THM7LSj+IDpA1X10Gt4MMZuvACCBgxJCHGxJ3pSWEYtSkjzrGTVS9U5oJaGbPzJbOe/UNRxe4AgTXJbyoHPU1CxyVP/ATOo7cj4yq9GVFk6hN5v1yKO4jxCRY0QVIFXQpIh8hPU5LvhvH7Ss9SWkwzqhSTLh3nMvOTj2ic4gk4GMVJqTUskfaUqgzdHYmKIlTjEJN4xmQpOC1qVKWKSRd7AAEocQoVnaIQnsAkKZKC6hexOaeuSKlrXvW6V7721a9iq8pfBTtYwhbWsIdFbILGkljGNtaxhe2P/yNoUAYtPraLj8VsZjW7WeNw1rN6zapWhQrRi75ookHL54vU+FU0+kVHp5DC16TqqjoWrarZ7J35iLRb0pJVovecYWpbtFoMbjIm50iFG2oQVYeWpa0MfCtpgonQc3V2rs114CST+ttvwiycHGwtAGopWX9tDC3PFdVt47lQhs5VtKhF4nbvc1rCVRSpmhxkrE5hhTLow7xgqm3a1AtX9s5zqKzdX1i9O1aios4dn8hDEpKhS6Gh91cDlm5cr0RP384XuK4U7oKI+0fW4UMUnQCCcFQYFQt7BcOKmW5It6ld/K60tORkMIKdlptTkKEMBGVOi60HUxhreGAHLq49u/+rsu/ecHDygPAXdLXijwYYltEtcoHjxOEGK1nBTM5xkiOWj1ho4gl6+S9phJw7nuLWiGPsbZdt3GGx3nfJlhKHJtwAB/9utXFrXsyLDxPjnRo4zjqes5yPqlpx1jha6YAEIb6wDX88VGZXJfKgjfwxJJMYrDdeZpg9PS99XAIAXVCnYKhcZUy3eb25hXN8HR3DD1M0g6LG57z+sYpU5AEQ7aDUxFodXquWa5iURLRpaw1f+zIavAejRyQo0YZt3KO+QbZyI11NYFjrsdO5/jSdF2znL7eIi5LQBBtqYW1mY3vYxnXzKWecSmSLWdl3ZliT98ksUjgiD4+4h3dsfRz/QK9M0MrZNI24nGxa41vXuA4us8YhbT+EY9QEzzYpM41wLbvr2xEPt6JxTG5QL4gok6DEGpDBbnC729hY1nTHN/xxEFuy5HV2tpOPzItBEALgIk9NwVdz8MEQ2q2x5uasuVvufEO85pxWxx1OUYd2CJzhQc+4L7edYZkf+dD2bjjTH05ycReLKJXoxB5c4XAAv3uQCu22oWXN9tbQN9jXc/rAIUQUZuzBEYuw+s0PI3TBwZzjcd8yzfWe6Ku3NOf73rs9EqGJOyz76S6349azjHiPf/3iXhY8mMkO9BP9Axtx4MIbqkEUT/jbFYGnu3Ozfs2NFz3hL1o42Jce+ojp//ul2wPMZChQAQo4AA9+9wPwhf3y2rfF6NBFOo1jnxq7g67Zw2309IP2D3IsYAQtAP8IDLCG7CtdzbN/Z/PV8vz0zturuvew9g+X93YzVBAUAAD4wU8BPMD+95i3LcOzva7jNM9ruQTjvaYbvcaTE6KIA/zTvxaogDUomj9DP+pSPwe5PXNTvPoDvbLrPfq7NoaahAfIP/17AJ4Ao+XLPGKLN5/yNgMEuaIiPZzDvmcrqX/gOQeoAP3jAAywhwpkwQDMQCXZQJPrwBH8wBoct8f7v9L7B3jAA03AA16QAQeggAWAAVgwuyEUMAF0viOkGhm8PAQEQQV0QvkSH8DYBP9KyANk2Ie9gAVFgAX/0zmMczuMGikCVLgkvLuQY8AmvME7ZBzAgIY9oIRFYLm988IrK8KyYL8Lc796+zzGgz8bFLHyS8BvIQp+uATKmwY7xBMLzENygrs3i8G5EzvZqD6UCrH7GLEDXMOn6Lc8KIVFHMNLYz7NiznOmzkyXLx7W8Vo8T01RKmnmDhK8IPyKEBdbEF4ezVUlLukk7/DaEUEeUX5iMUZdKXhQLc1qAZc5MBG1DYXjEZ5iz56M8b1uEZLu75MxEFKCQxgiLRHEEUkJEeN48XDk8bEA0YPtMRKXLRBhDxeCYx8iIROkANgm0YA/MJHDItIdLFJXEfmaMf/1slG+NjGMuTEp0A7N1i3dJwNwnsaiAQLiRyyVKTGYVSNiwwtNBLBP+zIopiHSOuDgJu3kbxAGTPJYBHDsvFD66PBQBS9NMSvuiq1U0CDZNiHVVk1qSDJvyC6MORD3AtKV7S5Mxw7o0ycSumGTvA1ALjHX3RGItzHAfRFr1PFTWRFy5NJmFzAlKkUfJC2L2BGYiTFXTRHbuvHzltLrWzLatSjmJSczuEHWegEMAiFptS1vHzGt9vDtGzGlWTLunNLoYRLrqySSnmwPEiDIOw9xzTLveQ6yezDf1TCgJTFQSLMmYyJfDgxL6iFSmMy0XzIs6RK07RK1HzLJSTKEIzL/26EE3HoBDOAg94AJ9t0RNxcv59kxL9kQna8TKx8R1jUxFlsE3lABUmbsLIqy9skzc3rS7KkTMC0TMGcJ8LMKx3RhC4oBHGszXzUuvDsxfFUy/KMToucTmzEu+A8xjb5B+Rygy0oB2jEQ7000NK0TzC5R8YcwfcTyLCrTF8pxhmqFHpABUcwAk5gzsHbyUJLUPFER8LghygwAAOYAthThRdwAAcQgnoAgEOYgBmdgGHILnU0vxTZT3e8Nf+kpqzQLytQA9qkT/340KPrUCOsyq7Agw7AhmrAAD5QihxgAgDQhg6IAwDgAxiAhi5tytyL0N0zTwoVtbqqiQ+KMFMYy//klE/aS1JIdM6pyAEyiIkx2AGlgIE3iAkbOAMtlYFqsDowXU1hnFCnzDEzjQm56AQagIN9eMrJdMjlLFK0XNCwEAEpBQA8OAGlmIQT4AIm+FMtHQAHKAAm+FLQiABMUNVNKFTScMkbbaKxyiuDyoZbOIMksIWXhMzvlNQQrc8RHYwOEISYCIQOUIpACIGgQIFZAABoAAcA6IUOoMAnMQAYeAEYsAFs0NZt5dZu9dZvBddwFddx9VZoYAZyRdd0Vdd1ZVdyBYB2hVdwrYZzbVd02AVIYAE4qIZ45dd+9dd/HddhANhwfdeBNdiDRViBRVhxLViANQcAeIE5iIk4oA7/o3CAQ8AOoTCKMLhTQeVG37xEQYTH/4QTHUmFJyCCaSBSWNVDXi1HX+VHYG2LLPiBmPiBLIgJZkSASQCKHwi8H2AC24HO3zyOV33UjwmkWY0JcciFPfiAgaLOST2/UvTOF6TasJiF6MiCBeAFAACFAPCOKegANgiDApiMLKiCM/ABAyCFG4XQQZXQMTVUfQItAKCcT6CBJihQzNzV9YjKUpnK5lzSrmCFMOACLmzWxyBWLggDUIgJTzCDLDiDynBbSoRbMc1PNLxBpQWAZroFNCABSwBIU2zT9HvTiIxTO7lK/gTEkC1K7ONc5kGFD9AChkxN0nVZfZRawdXNcRxa/9fVT/TcMoVbLHBJFF0wAhcIhY9tWb89Uug73ZNM3VHkTb5VTeYdSKAs3rOomERggP66XDbN3fmEWUqVWb/ET6KdDaN9pZcdRaLY3ppgh2BAhRqYgdkMU/F13qvV33OEwYbEUeFVC/bNyAoMjMWqCR1qAwmggzWN2vKdWgTtW//lX99NX+At2h3FyP58X8USlE5yARxIBrnNSdn429sJXA0c3Fy84PzVUQHesA72M/glCzdKAwjQg5XF4HA52qY44Z+JXp9cYaCs3gcGWRcW2epap+GwB2zYkVggARGGYUg90MdsXgqWYPKUPpakPg3WVdYEIwAdTl84hjRoAD1w4P9grNpIdd8r5svz1eIA5mJX9WKWzV6pyYpeSZRgyIQoHuGKbMzSxcAgLpghfs4WDt8XnuP5ay/R+opYmRUxwGFCdGOdrOD4LDYrvs8tblXFIGAOhsLrYotrUYZSAAEcqIV4xN39zeJVxmJNpuK3xd64zdyt3Mw8hpN7EYec+YONVONZUs42duU3/l9//F0kls4pRtpbnqOvqJhosIS+mYYnus5axktB5klCDpnpHZbV5VEztGZiVJaAgROZwogrgIA/2CBKHmZLbuU1JuZLZmFOJmFPruMexr1CBKqsyIdnyAVuiOYemOZ1LshKNuHnbb+eLOTeZRfqwa7KBeT1vef/9tVd4QQYthgTZ9CGK5CAQlgKXx5deHZnWAbmyKxUfR6N2XpJWebI623pOy4OujIQygkGdmAEDSCCbZiPalbfax5fN91dFWboP5oU63pLlv7l+FtkZrHQff6K1+GRcOBoj3YNnt5hn2Zlku7feH7nQnzJJT5qy51lzO1pcd4+p64J5lkGeohmHBjoqlZlkT5oec63U4Tjcf7qh76rCfisvtarwPLrwBbswVKfc2gHSVarvdrewWbsxnasy3LsyJZswvrRyU6s0OLci34KpA5psr5qs+7ImSSgWdEHTojit44KkL5dubYNhJZEhd5mQ645oNrgB+bs1fZsZAbO1mmo/6z4oGy5B0nWg61Q7d406Nam60DO5NHMwXV6k+fGzNs2bkKtZ6bmbcYwEB2qi1HwY64obutt57nu6q1W0LseZ1x2ZLvy5toeys8mU44xyK+gomfIB34oAwZQA7D4biM+biNNbh6262IGaiCObrF+aVouawqtHiNxIwB5BReYgVzVb6vWbeXOauaeYK7Wagum53BuyYmmH8B5CqbChWzwjq7J77LYb9YNahJ17YmE7bPh5nJZ7y+mbg8fIxH/iozGBwB48AhPixX/5gwfaQzvb/MV8DiWbvBGcPc2sOYyjlqq6ZhA8bYQcvaGYMX44eZIYSUdaiI+5kSGj09Wqs7ixP/OlRVqqLRXiAEXMIVToXAxD01sBlEiF9Ek32Q57mRrBPEyp6uLBgBS1hYAgIMGUAP4DIsrt3HW9u/x9q4A/2/VLWIWB+cE5zTqeS9/8oVz8I5kwIEUGIU0bpk4H2vyFu8Nx2SrdXR8DPNSV+Q9B8qw4ZiKkQbkPKArsF3CUHQ7ZnSsi3QFN2nzPs1WP/DclvOZMy/jkCmbiYlpIAIQ4NDU2HV8ZvULB88sF+qT5nA9r27CIHMF/6+DjAatqYmbJoK9lXZSL3ZTR+5Vn/Plvnb07XBLp2Nl3pJwD4wx0ZuYKAeOhtrWmHaKfsIij/cjz01tr/Z5d/J6X2qze4/A6I//KQ8QEhDo4wh4MNTyF0/JFodT2ZZ0Yk/qV+924Ht4wNARXLiGmgjuSWaOi49x2vp1p4R0dwdzRHb1Me/zJy95ouBxgXtwKV4Pl9fmsNhy2ujyjv/yQ1b4Cpdoe0dpLyGKRPmFc0gdAKhybVT3kE91Nq5oO/9VPI9lA9f6ZG74UO5GSKYVp0iGGYgBCcf6uC7pnzZdjkddj6dekO9spYZ1ZhbO3x70mKADCBADUVcMoad7tSh648H4bBf23cR73Nb7kX/6btRlo5iGTx8FH133R6dzJD18IU56JqlxXjf2mw/jGSmhbMAF+jaKQpAAXG9NSvf6Rkf1d1f12q/5pT/2/wx2enlMEsBocKPodwlghKrXzL2X9yo28vD++pin8UkfcpceezxuEJPPGhM3ilEAgR7QaeMnyIE3+Ix3/hJufppXem7HcYZH/rP+fZ4nE3wIvDJgecU/fsmPY4Lv1dlH8vEXwscHCAACBwr8948gwoQKFzJsCACeOocODUqs2NAgRosADMpTFowdQlsuZthCeFAjyor2yKWceLIlzJgaKcpMyU9bTYkvc/LsSbOnQmxALe4cmvOnQGATeCI12hJizaZGMUplaFCfNVzW/CGkA6GMwqJOHa6cKnYs2rBn095MW9At3IVVhwqNS3Ct3YtFlTLFm5cgVJlzj1JFafDcr/9m9BBO60FilNy/CssOHSwZqGWnbd36veyzc866f0F73rh3KeHSCwPHzAyz8Mx/9Jr9OpeQkYYr7ayqFkgZM+neKV0P3Zw2uHDDyGGKzrt89Om+yQFHFPzcJfG7/LLhkqYPYTstEhhNFP778/TK12UaR7s+vdz3Fpvblc85emr4rF/bv5tRuTvEKJNOQqOAgMM0OpnHknrwofdXe2P152BBEzZEX1wWOtUUX/mlt19L2en130z4PJNLNgqp0QAdRC1oFoUe2hXhhjHKOBaGcGkII0EdRrVjTiAOZx9V1/0jji/P5JNQMjPEUJKLvZ0nnY2tAekQjVNVaR1cOXJWJYf/qP3ooJDKWUniTPIYU8w7CunRgBj8GPZig1uGeGVDWVZmJ39dXoYnlQP5yKV+1Vl5J5oa5SPNLdbIyZhjozw6E53A8TmkZHpidimmbnl5HJj4jVnoqLGJeBeSiilkiQZNlPNapQ9yWtGpOWnq06yx+SkZoDcOeiiphNKaqEazBYOOQvyUAcEfgsUaaK56ZYrTl9Hq1KtAn7oXKkK/9hkssNjVKpA/1tzinUKvuIBDMlE9e6O1CY3LHrWgxiutp39y26OYwk5Xpq4jzgsAOr8oM89Cb5YxaWvvlnqvvNgmdCtTEFslsbYS7itov+H+a+i3ahFrkYm+iLPQNkSAwAmu/1IyaKnFEU+rY8xq7Traxkl1HPLHDw9UZEzb4RLNkqu2ulvLqk0Jb8wDw0TxUTXLnC+vOQPg7Z1kgpy1fyNbFKAx8qy1bLN7usyj1BXOXG3aalONs41hQpscwETtBBtMi6K4lkgztKvl2XW27XRLULvbtttpZVxj3KL6S/fWneINE5LO4LOWwtsGDvPgEiNkuLOIm3azc1Zj3emHkSsHdEwdfbRWOU2sbK/SL8uaNuE21as54rk7tDjgMcrNtGp1D+t1RfpQ06g/a7FKxKu0l7a0zzX7rhHoDYt+PUPAmy284x5Dfibh3fyyDD1zkZ2hw49b7zlB2cO6Pcb6Nt7tzv9cp44o8hLN5ss55pKMGDypPu0T3/vWJj3c1a9q9+PX3IRjvIhlZF78uMa59DEXPXwFbrVDGwMVyLvOkc6AD+RYBHszwa69ZFzuCMbBRhY7DbDMg9OzXQotxr2KyC9Eohvd20p3Qp3lsDQrNE1TTmWiXHDDa5xoVfRs6Bnqua9p8BtID4fzwx0GxX7gw18RPTPByf0MOdzAhZK8pgYJlM2LNwSh1LiIpd1pjH4lZN8Qr5Y/1PUsPl4jjprYREaBNKmApbniQqiIQCuKsI69a6AU4TO86qlQdayLj0PyEQ1cZIMfXmMEBLTAMDdOEYfEg5gc80RHxj3yjjTL4+nMBK7/CvUvcQpJFT48WZV7iIc8wkHkZExJSR0CM4tzsmMQTfhFCJ5SjIa6pEsWQo9lfASJCxEJu6YDzIQokmdxLOYqg0fCZOJxmShs5mUCM0h8EQQrt6BG87yWufRsEyHd1F8IIRTO741TcaSUZPi8KUF1rDOaBDHfwZB4lhnWkJ4HFCgj9fnKVpJzohSaZBWLR9B5/QQfzgCgLudioB5E0aGbu903G8nKfqLFe5syCjh8sAAMBCIhlVBBBDowB4HUAwkOmMBOgXjOYf7pHxCRD0WEJo0lFRQOLIrbQ/GZUomyjaU4+mdOsmADcjTCANBASATYAIBMDGAWADiDCrDhCQOo/0KoRERnhjByxIuBrU3WVEhjUmAKMEWVj1PNizEphUx/OnAoDmiEQGQg1oHsQwBt5ccCMgEAEeBBID7gglv1GEYdFWauC1liihS6EEtIoAlIg+pJN2utVDIksFGiKGEjWRNzBIAZAmFCFRAyBhm84Qc8kFMBQCGQMPggs8CIgCqSC4tFxhU2nk0Ikp6BD1qODQKF4FM9CXJPvyaQqgv8a0uxKhNmBAAcAskCEhCyBhMwAQZI2Ac/AsAKgZzBBsY1gAjy+wJs8Le//v0vgAMs4AETuMAAhgYzDBxgbxQjGN2ohoJD4aRQfEPBFr7whQGA4Q0buBoJ5jCIQyziEZN4xP/DKDGBNYziFbO4xf49sYtTzGJz2EMAZgUAErgwqVkIwLb76EBlIysQLgjBuHuUpXNG9lwALOoW11DoXP7wFa5cKrsD2S6S8wnYfb4UtuEtLFBMYAeBhIAPBAEFAeohEBiI1QZnWLNYMcrcDRX0uYhRFZQVwksJWCJXVvaNMDOKSnBaVMtflm1N7BCBOTChA/YAwBtkAIB6hIAHgeBCAXgBAEQgIA5ZcABO5AxRs/RvrrOpTdcW8ooUIChafwYAlgNmVbi4llZbhKQQjRKIHA9DIJsIajXIkOO2CgQRSMjCco2sWuvU8iGqQ0i5MljGkc3T1X3NMnjjUmsFDfbQub7/aEClCpxmC2SC7BAQwqYtlXY0gYb3enWs7dZtbXM5abM2ikvt7SBRi7svp6pbyU7GwqasugfbQOW1ZW1oehc6207Jd8VgeWSF+/s6ZRKaklhoSw7SYZTWTi1cXU3oqi784eJNDr+5S771CCmQdlV3GQUyw1Hco2kJl7eXGU5yh+P75L8Mt8qHRO7VgExvnTQJmigy0oNb7+bDmjet6x3xnJsczPsGOraFnjvW4JJhgzSIU1uEO6dfC+pu2XZ5zE4Xn/cm5Vm329AdAhXXgQTpVdmGY/Y6OLIb9N5skXrU1A4UiAfenG8l6sXiLhF4jCMrW6HgWUZxtO3xnZ0lPzvg/w9H9Z5b/TNnKdJa3E7xa8lRHIlZDAUX4lQ9eLzpIEd8vFi7ELRf5Nau3DlhyAh6TDIT9ho3yv/E4fF1TgMHLniF7FUD70DP+d0j/253Y/ttz6/zj1gfvci4qFRyRG6dT7xCzZN/yMpfTPBjoT1vzM8TwmseZnMZjOhxnng5BmhAzra76pkVc8q/XtDOV6k4Xd7adV5+VAXygN4/KAUCLiADNqADPuADgpZcbdQD4h0JIB8EZqAGbiAHdqAHZiAAkMMHjiAJlqAJnmAD3gQKriALtqALgh42vKAM7h5NGCD8gZ4CzqAORlc+UMVRPaDkmZYODiERluBKFCESJuELqv+gEjahE5JgDD7hCQKRDeJF/D3dwI2FyyUEVHjN6g3fqznSGwmOAKIF+kWG+oUG23ENRfxE/1xh2cGcUzSZoyjEDy4UEZCA3k1N75Af7/mdGWZe6GzeACIas1VQC3UGHBpUQeVEqqCeHW6UQhgIEZwWH+5d/zVf7D3fCJVhT7DfIKpHDSZi3x3eISqeRRgLsljFUSlE2JXi2GXiqA0aAPKTJ67fGh7S9cFdKkUbulyMkMSOHsZhH8pivxFTLXYZIH5iLnrGIvoRaxUM+ggMiIxUSZUf/30QGfLc3zVc9HmbMoEbGPFHI8pEwInL/Q0EHTyV/GGiNnLOLRrFGYaF7VX/FO5p0y5m30FoH3cQDXYMRGDM3NvZnDEGHS16Vydy4+A1Y1GNo6mQIlC8AzEUgzy0XqoBZESYAgmQlEEiXEEO5EFumTdGlPSF49U5JOkVxfUYXUoiBERw3Cx65Dui1DfqHPSRJDiWkzj2nsCIhe9UznShI0KYg8pARky+mx/aDCEG4kgioz3eJMrtIjT5R03Q3fGky0ZeI0jK5BjCo0KenyBqz1IyIwHiI0ranWUMjPIwT0smRCF8hTn43mol5SV+pWaE5fyMJS6WZVSeJXWBxryYzzS2JUKIAZ/Vw7O1YyzO5LL5GSeKYTzWBCiK5U4O1V8qpv80AwDBnUIM0Awk/8iSId0W0aVo6qU84qUPpaFkMiSvOE451mWeYNAvkt6qSIAWYKRcfhxjhpxjJqO+2aUa8uXPjeNrKqVEvFBCHc9ZqIF1UUdu+hlpUqVpFgdqapFqysRk5mVl6sxUYmZCLBE3PGRCbIPxQQlu+h9SfiT2OaVI3iN75mRTDie/oKJ0zl4/Fo1yFsjkuWRiYmEx7uZzcor4fU51HtN0YidrQkcZKeApbaF4IsTqEV2ACmh07t8y3mV8/t9TJiQ97SNGxBL2Ndk1WGT2JYQwvgJDhGZ9umNX0iRO2iSHvmjVGeLPFQaIYmbXPShBFJxWOid6xl6F2hJwUmeGbuLtQWWN0v/EjfonAExTNa2OQvxBA8DB4vUnMbJoKcFRTUZdka4WrpkkQO3EksahO8GT0CUEL7lblU5olQVpZm0p5nWpyB1pjPYlTx4KQqWbjg4ENjHdmv7oXKqnd74njELmkCKocLZdPuqER4GU1t0GBIgBSqiohWLpZcQbkx5qT8yjvNRjSerkSd5p4XDHUiGKijTnpFopLMaRmw5oa/rm1F1ocNKoovolStQVOY5neT6FqlreYrZoY86Kq2JRgQrWgcZEdqam4WlW3kTD3jzHIFGiJWoEpQopqwpqpsJpN7pnSMInt9aqqFZEdE3XqQySlIkdTFTrm7oegAJqbyKkoWrrQib/qvItKkGoyZOuqx+h6RWoaUyo67DSKlxg6pVqKk9wqkl4qrciKbhaZiYxSh1W6ogwSQy0Wk0AbBiuVJZuo7yCpZy+66d+7KuGq0LcGSRKbIkiBGnFCU9g7Ghia8F2LIZ+q5FuaLyaJckSxKnZhnFCY0Iw53X1hMtmI7DyprA+psYabEskq3UuK4j64nf8ociYaA+4gHm2bK9i438WLZvaScACAMLehcJeFb3qoq0KxLklp9amHkKYAghUolEM7dZurFfK7GmKrIB+KagClK2eY8yKFkJIKZU6hdxaqmQQ7KrKaJzSrJfS6c3aqcNiHH6uKhmlaUPFbdZK7a/SrYsS/yqXMu6c2mzSdihKSiSbYBuxeGaCoEXhbu6lMt9RNi68jq7iFuL08S0zaRIucAOJSi2xcIJtwkXrXmu7aqLstifD1mzIgm5DQhA3JElQog6xAG1cDC+7cq27Hi2sFp6srmbZOuNehIAx5GvWkFE7EEEKXG1aWC9BFm/shi7y1mm3kq3Amm2PLMGT/QgZ2UIKcKRdsK8OtWrGcuziJu/xLqz8NqxADAMGUNkhQqqk/gUAcyXnBmvebm/7RSai1i/4ni2UIgRzMkLvEm7m9qzrHi7sHuP8fq4Bwy8CP658OqypEsT5pq8zda3XCvB1bmqxvlb3bvDtkm7OpuRO8C/c3v9w9gqrDh/rwfawre0wcyRoknmwuPwEKKlBJeHwliyfltYuUzKv9opuAOKuDF8lEK1Rn2VxElMozCau5xZwAivvC9Mu5JoiZ1ZIOVSt+iKx8Qaq+6qwhsYvDAfyHI+xEJfxxP6MEU+rEZUwbF5vBRvtBc+uIXsxWXJw8yJyyg4EKA1uH/WxbmIvKIexINPxCnMeJo+sJqMlCPOZ1mgxX7Wxr9otkYLxJC9vC2eyHe8rQeSxCyTDAJebI5fm3L5uF7/xtuYyKReyLZLxLrMtn5JAJX6t3A3zip7wXyDuLFvyzCrzLTOzMobqKt8VJ0PA4FJzQ0xwev5xR8pxoZoyIdP/bxDjrCaTERqjrBqPshLL8toqbU2E7c+M7YzOcx0zK+QRBHn+8jUPFCyjFjtvpTuzcBwfsDyD6SE/8+S47RHjs0Y1tPAs8Q/zMN5Wmd6OdH0EVKIUAgN48kLn8/uGciR7tDjPiBNzGxMvrRSf9FkmimGm8SN39Bq3KT9rLi0DBUC/BRTj9PfqskFPTkL/jQkzdFBjF0hrsEjbMkk7LjzXK3GiiUYzcksDtT6z8UOv5ykn80S7cEXv7UVr1uRIGUsTtUsD8jqL8ksv8ztXMjLPayor6HyiiWFebj9L9VgLdVkP6ll/sTdntRg3c1vn4Es8NWLz8V3v82FnKzffLVZj/1dJb3Zc3QWDDoTbgp9ZUzZdA+lQR3Vm1/Jic7ZW67VMsEEIhMBiEcQPwABuw8BOTUJuw4BtuWEC9su5trNkYMNXTXUO9wY52BYB7/VYrEOv3UdSo0Q9RPcUG4UiOAAoeMJhIQQoYAImNIIAVAIA8MEJgDcmwAMVGoSPaIG/lvZlkEFuIfcWC0dvHfOgIRIf5MAIy7QzO0UjwIAqA8VlDVlxKcQchICc8IGkUWVG8AV5fmZlw4V8+7ck2fcP4Lfz6Td/S/dNo0SAD3hPmEBlAQAeoMBCwIAZCAQfFIADiMBisY5SPA9zn/ZfVDh9x7Jq3Hdz5/df7Hd/5/iyGkWI+//1UGCAIAiEIHSAQrBCjwnELIACNmT3mCEgMGxAAijAAoTALHS5l385mIe5mI85mZe5mYc5evHCma85m7e5m785nMe5nJMBD8i5nd85nue5npd5pKn5nv85oAe6oHv5ifv5oB96l9uWCowZANTBCyhEFuyAQnABD0CZQVSDAlzAANSEARiAgxQAAXC6p4vOqKsGqBtFp5N6qdvFqbvFqqdNqqdFq+dFrA8FEkyBQEQBEyTEPkRAkidEFAQBOQvEPYD1Dx07sie7si87s8cFKBhAHcyBAQgXKYiJIDjAo9FXHAhCGBDAJDQ7uIe7uI87uZc7QigCD/gAYgHAMETBQPD/QYkr+Q/kABNsgrnfO77nu77vO7/3u7//O8AHvMAPPMEXvMEfPMInvMIvPMM3vMLr3u7FRSOEQR1kO0HUgx2EgSIIDOQFM2H+nlOAQyPYQTUoxDrEwRhI1h0f9NFWoYbUwybgAbENZSXYQY3PMFp6vFBmIUzsAyjgASh4nCqwgRlsPPUl3iMlSsTDRRxEQBi8gA8wzD68gA2QQQe8mT52vM5DYxJ1J084gAkIgCcohAzIQBgsQE19vLqhMzl+Hn0uhA10AAK8gcd1gAgQgNHzjz5ufdbn/HoggQVEgI4hBC84QI5NAK5XnM8ykIeGnufwgwUkuTYUgCpMiiFEwKN5/8ICqLfMJF0bsr3kqORJvH1DqFkBjD1CVIIDREQdnEB+Tg3pV40B6mtLqJliKYSaTUDea9HsD/sWl+O8qFkQDD5grIOvDYCaRc37xX5lVmFcAEMAJD9vEUQcPDoAgEMAoP5Fer7v70vXWytQnH5CkIGkAwB5Za0Ngv/vD3ZN3P5C6D7iMX9rWqHTDD+JNgICBDlme30gOz9AABA4kGBBgwcNZkIwkEcYgp4KzAJQJ4Agg/8wCsxIEGNHhB9BhhQJ0uPGgRk9jlRJsIAng1OY8APALwAslSY5/tOoc2VPnyNxFgz6s6AMNh8nKCJ6kifClEuh/nyac6rPIFxkHjRXwv9MVKFNmeoc6pUsUZNDq5Y1CIrAwBxdCY5ZYEKGg0kXN+LsuBesWr8A8vbl29dvS4NcggiEF4DZTcKAeY7961cy5L9GkSqFWjntZMqcO4+8mrXgOhs/9n3Gu9Nz65Al0Vb2W42xQBF4DMKDBk1A49WWmQZ3rTbwwb2uDRd8A0MgqQGpgT4uPvwv6MtHESaNKjk09c2yZYccbbBeDh72qhMe/Nj7ZNg7I7P3bIMLgEoGwAFQxEfgMADwfEjMOLFQItDA9rwaDDICF0TOJYHqqAQAZgoABQAmfiDtNbAKPA5Bss7iUD6iMBOoEf4G0u67sBr8sKynCmyRqPEA2CQOAOyY4SGHetxrKqjwXEzQw8BG/IuUECZYAEUufhBIBQcK2AEbp4bssLsgN4TNwCuXksGALw1IzQY2ZIpjgQhM6CU6vixTEEup2CSSy5GyANMACcPwQSAe7PStJwW19PDNldhss1Cf2CgATD74iYM5UOw0ABgQ49QLyEHhLNJFaHg8CBtzRLIOUzipHBWeatIbVdVVWQ1yzlY/DAgAIfkEBWQAAAAsCwALAG8CtQEACP8AAQgcSLCgwYMF/yksyASaQXMII0qcSHHiwooYM2rcWLBePY4gQyb8J7KkSYwQT6rEeHGlS5MpX8pESHKmzZs0axK0wMsgNpwvWwIdGhGeOqIrhSJdCuAn05NKn+J0KhVq1asbo/L0iTWrzq44jYLVGHWsS6pmK5ZNWxItW7Vv32rtWdBtXIJr74YUq/dg3r4a7QIW+HfwRMGGCSe+Opfr4sKLJfJ9/DVyRsSAIVsmiNlw5c1AG9eNrBk0wcmJS5tuClr15s6DP6+eKZozadmzD6L2jDu37c2uLcPO7Btn7YHD0wYHvTt27+KsgT/PnVzvdOghjwusPnb55uaZr1P/by3eNPe45bGT/bx1NGX1CMH39R75vFn6hu2zTQ9fLXu6v6XGn2nyWTfga+T1h9xsByqYk0HtBcibg6cddRuF2yXooH7KYWiSdtEJ6CEABd6FX2IcdnUiYCl2N6JIILb41IqGlYheg/VpqKCMKr6Y3X+OieihjXLhuBiPUtGoF5KM+QhSjBcOaeF7GDLJlJJ3WZmkkxxBSSWGRO5nJIo69qfljLPNw41xQLonJJhTvrlhmfCdeeVq2dySDZsQArjglxSGqdyY+dGpnp1LESoVPcoQI09obUro3IiC3qcoi4ZihyhSly7FDS7U+DOUl3I6WGl3nS6ZKXSbEpUqUPMo/1PMo65G+mepCp4KFpaqSkdhq6Mulqc1/CRqa4aAmhrnhFWuWhywoRnGqKMzHhsis3BaxmuWzvoGrXGDfRpqktZ+m9Srd+mqIrpvmRsUu2a5S1tf09JKbp9BYhvospM26+ucq8FrkriiMlZulNkiDLC2AoMlr0wNg1Tvrgcnmyu/4Xn4sErbsrWxSxFvJG6xFOPrpr7KMqxxt+MFnNbElposacZSquyvzTsyaNbIZPFZUIS3onwxzgsrXKfOXcHMUsd4VYxrf+pixbTHLM/2cVJd8by0Qg2TKjTUGM8XcldXlzT1WGVbVZXS/p09kNf9Jmzx0f/m7LJUWrddky1/+P+8k5/IPg1f1AavXLeZSC/FNkVckzTNFQ3A4fdAQAf+9eBhG2g40XSbNvZAeTPONQDtqAEBEclAKnPQce/LOeKHd94aUotL1DhJdGiAwyu1rm556ykbDfvrmiaOU+gWjc5ICi5YYqzv1wI/tPCyz82q8TPVHlHjALzSgwZ0VAt92l5RmrmJn19F/pPpP7U+jMeDSrLoXD/egBjt3Psz4NHTLLfgxYvdobC3knkYY1YZaVzpTpe6qsDNf66j3gAFGEDPyYQfw0rg6P4gARyMYl3jk6B6COfA9knlfeUrGnBeoj2/NM4SLkgBI1AVQuuN8Hw32pwIvUVAkSBve1yzBRH/JCC5mO0vX9IDG/EmuMRn9ZAjLTRI47ahhfuVQ0w1BCB2SKi/X1VtNSh80kl+mJN/LBB1OTziyZKIuSZej4JvtKBIojgShRQCBDMwhebUODOxmc+NToRjIOUIEjJKkWuciOEM/chH1kEweDaM4w5bRkiN0LFp/xAiEYVWuf4x8n+Xq+AkrfZEiRiyjlRMgBbyx0n+hVF0fxwlJWVpnlIe5JKEUQg/4MBA4XXylcmLZSQHSUvQALNnGDklJu8Yg1HMD1e/LOZquCi+m0lTOLYcCC4BwLVRzIAEjHimxaI5zOJQ80omdN8Xa3k3U8rPP7bogQTocI9ayWZ0ueSeC7MY/8ot4rBIOiwnGG25TYVQ0Yr2xOfb9Hk73DzwkxEUKCkFycN2FoQf18CFNdpGBwn0oIEJVcpCRgeZh+4RlGwUpUSN2cOC/gOGMeBENYWCz5EqdJ+N/N0jp7dSdgJyopUUiDJzGU/wlZAkIkWqTfV5yIbypKHYaKhUp0rVqlr1qljNqla3ytWGGqWrYA2rWMdKVrFGtaxoTata14rVs7L1rVatV1YPij+4WpWbNblpQ7mZz3tK9am3c6tdB0vYwkr1q4ZNrGLtKtjFOvaxYW0sZMs6MqzukoGTVWoum6pXhzqtn9A55/Os2VNsugyXdiRBTGmI19Zikq+bxenfkLhTJf9eE0EUnSVwhqoQU+BAA32zFPdIytDbPSinnjxpRLVIzNLmyDSotV8Z6rmuvNbUui0x7nFnu8batvG2pv3pQEHD23uYrgnTqO5NIfbZlPpTvEAFL5ksI1eLMNOZUtPu5ARCTuaa859YJK1/x7sYnvXmH6OIAQkKkV+mjqq93r0hfAks30IlRmn3nEYTIAAH6pJrvb1D7jGBKMwBx9e5RzIMRnFxjYIp5qK8RO9RQcwpCEMUkiamMIrn2xcDIjAheLnjDHj3YRqPlrt9VC6OQVvRCeNWL2TMq281UAhxPtjB+eWne0ML4EEFNMdPfksUSXLQ6VYLyyAUcYUXI9oaf5n/yTre2Tu3x0s0otPIaUayI2/MUzCzNLcnBsslLZGC1RoLzV5W844j0+YQexHQccaKMotKh/Tp9ySgIEMc+MWPQJBBEARRhRneQBVosOEMsIitnnXKZ9sumsdrxtRY6FgOMUCgriF9FSIQwAUbwECcPlABGUyQhWJtwgBZ4EEI4NGUCfigCgtQhWtXnVz0lRjOgcb2n7M2Z4R09KNIhRSeSwIDNgCgHhNQBEFIQYCfMIMADuEBFwDADxPYAQBskIFAosCEaVPOlbGuUZeN+Ggnh3dtjbLXQWDoApm+zSZ7tQk/BMAKgfxg3gNRRAQGgoBAAMABlRBIFpAAgB2QQSCC/wiBv/kL8FcnptEPfvOWs00UjN6CGlYeSDLkqYfKoOvSNmFGALQhcpIPZBgE2AQAJhEAcwtAFcUyQw748YI4CAQTC1j5x2WQgxzYQBXaAMcwwEH2spv97GhPu9rXzva2u/3tcI+73NVeDWjM/e54z7ve9853s4+974APvOAH//a/E77t3ijGLryBDbVjQwwMuEI1Dk/5uKsDHAFwCACiEIWCvGEBKEBBB+5NAFAIJAw8AIAMzA2ARmw8KhOYgyBmj4197KMats+97nfP+977/vfAD77wh0/84hv/9+ogx/GXz/zmO//50Nc97qNP/epb//rDnz72f2+Pa9xcH/bwvf+QXxH+7Zu/+MVCQCYEYoMzGKQew1gHAjwBgA54HABIKLYQig2AObyAHyZlbTUTcLJmcM+FFD42DxHhW+D0HGMCdEuBBEUAAKpQAD2xCXMgEMxAb1mgbwDQa/wwDAugdIcwAdVgDzDgfgGYRigVYZLkcn3hLjbXYghRZjkHZOwzbkMxDCgwAQjAem8AAwKRAw5gADAADNshAxHAawMRBQvgADlwFCsIUAMIgwVIgNwyFD6mcAXRUXbGOF6BaFXBDGFTDdVgENBADgUBDpqndf2lbdM0cKxVcFgYF9BicziHEJxQaJyQHn4ohvAxhQHWgq3GRHXYLjixhQiRDBsGBwP/ch0R9yKCmGiEqGTDc4hUMxN4mHPtUAYNcAXbcCAOCIEeMokEt1xwuG2YmBaIoogHMX444lk6SCGmOIeoOHO65WcHqBKbeBCm8E2M8ICclU6D+G+0VYgStoqZaIApVkAHxIUCsQ1XAAFqcA+EohORyCUjoWUuyGXMuItWmIUm0YsGwUHgxjF4NYtOUoslU4W6eHDhaIcn4YoF4U0pYAmpQorauI2K9o6WAXOqI2CpCI4bQY4E8TjUCIDnoo77yI55dovd2GTKiDYiQY8EEWPTcCnZuI/B1I8D+R1y2I50GI+IyBEzaGV7GFM0ol3EOE4t54+MFpIPaTffaGEaYZEC/6FhHMaQQKRXHLk03IiM7zWR8bJOqkgRBikQl9UEoQgVaNaS0PSSH/mPMpllAomLPkUROAkAjIBHr3AiG/lwPwlLHomVswGQ+3WJJLmMEpGUAFBUfwCWgPhiY2k7NmaJrgaTsLaWrKiVz1gQndgAq/SAcymWdUliZRmRvoGWN+E2FAlpWWkQblkIGtADtoAf+ugXh2mXQYmX38WXbAmaj3kQOPkKOAAC4WQkmUkTm4mY1DZi2wWRQtlcenmFA5GUgTmYMFKYrNmasmWM3TWb/1WTe1mbMWgQOEmZM3CZDRKWCeSbv8lyx+iZySiaRQmZqpiURVUI3rGaLAGdh9SZAv9YieOpQsbZKwBgkbnZDsvhnc8Jnk0jniwom9T5gufJLUmpnMzZJbyJTPCpasCZZOW5ZGYZmfdphwlIENvZnv2ZFf+5UPJJheQ5nzRJlFjBD9ZwC9cwP+sZHO6ZHQ8KoNIZnPXpjRbqMEYZGQa0Cwqnnx7aoCAaog5plRMqoeY5lXqBh1SxoI8Io/Ajo3c5oH2Go8VJpG+hiD/RoX/Ik2rzoDNaOO5opIMBm5xZHOSIDZQJbqXxoUERom4olQVKIFUJpSN5oFdBjzyqQUwKMl76pCV0bWEaZtY5FOR4D554BfGwpT4KMW0apBQ6pHEKj2bKFPTYlcsZVVvjmHABpBH/WowEqpg0B6lvQY4aBlwKkRxcGiyMmpjCuZhj+qZXKalpQY9w8InlsBCwkamu0qeNSon0KaTVM6g4QY4pOQotgRmqyimsyqklOpwnSjYp2hWuqJOOKBSCkauJsquv+atdwZgQB5VEQaWuCRjkSAcQwJRR4RbIeiebuqxzOhbOShvQOhTSGpt6YUDGoIAAkGANtxZUsa1o0q0Bume96qnEaZPMChInCQDlUEVqgGc/Aa9bIq8jKqB/mpdSapuyqhKuaI7JUBiCtUIECwBvGKhUea9TGqw15315+Iv3OG6XOq4x6qR+aqOP2qmRirJLoYjtoAUJiWeNU662o6zzymr1/5ob4cpeMieqQNGLjKA7DzuKoyOz20OzBUuvsPqZC9uX2Akrf7lzVAazTEW0vUmyrXqKJ3uzkWaxLtGLvDSYB4ZlVKuZE1uxPMscn9pFN8q1DPuXezhkRkaKYytFRkuxYHq2IImxCpuw+sqx/CCNO7kWmTm3OGi1vJq01bm016m3HKGIHIRegluYhIsXdWu2KhuHjIue+XqbfvsK34SPsqgZk2uY/+mmapu1iGuffIsR6VAM6dqJEFAGAOhXPDm6dFm6JeuogIq3BLm6E3GSP4sDQetCa2q7sGW43qq4ZpGz77Kzl1uRzzgNRBC1UbGtxiuyfGa5Wou5m3tCGluQHP+7D2pgRdW7pwYbG5V7t8+Ltpkrjha6hYkUA6ZQvms6namRvvZ7sErru/LYtBqxYtbQr4Hrc+Z7ENeLvySauib6rcDqvxiBrvOQpcOLSbxywGWrvtsrpu3bvxu8YtcQT6iZVPV7GNlUl6ZbTTWqu7HquxD8tez5FQIrERaMvDVbbfqbuPxbkozrwbVKUwWMETOMu1dri6h7w6rLtgMxD66bDi5brDA8woFRwmN5wugEp7zbjPfqwQ6rUD+8EUGcG5FLwFWavDn8FswLMth7E5N7IkpsDMDQAyH8xIp6vsThGxDIkvpIxUf2qkZMm1PpwS6cXV0MEl/sOUolx+nYWnH/m7uuWsQmq5bvKA+uWwopALfYOMghUcga4sMQehGRO8QiycePvMJhOoNNLMhQfBKazDDW9VrEJaIL5VS8AFWZVcu2TFWIdcu6fMuStctq1caAIAGQ68sxS8zGnFgiWlWJPIxcA1jFfMzQbFi5HM3U/Fa9XM1btWKyAMfBSM3XjM3grFXJbFye7FmgPJO7u76CCmfALJjsGcupfBZSPCmv7MqHbK52m7+jvL9InLG5pc0u4ALOhF1pjLSeURx43FdLtcjnTKOirMKGGEltnAYJ8K8ErbkSa8fX1VcUPK1Ha7MK7KsMjBVrjBsrdgkgILwLXdAJfNAXrM8QjcP9vLdJ/9TGWyAB3HnIcywSq9yaeuxmUTrTxxk7J32tTRnDSNHTm/nTjvbQjQzJ+tLGSQACnDC7mLwUSn2YTB1zQX3FRdovK8YHDCAG1piOLE0RWW3CjIy16ZzBcioikowKNbCcOm2gpIHAdPzUCCvUGJ0aNicHEtBzzvnV9/vSLb3PMu3V+LoYSowKLEAEoYjUDRxU4LnVAZnCeh3Rg2FzaEC9V42iFgWflp2We63Y/pwYjf0BV/DC8Rya2oLXBo3YQznS6pMYnE0CA93aOjw7hp3XbF3a6ty7R1oMkPABZC3Zu53RQny4fSzSygvaOep9VLA7yM3BvE3DH23Dsr3Azz3Zcf+hxMVdadVt3cot2mtNxG0d0rk4FjY33RlZzxVa3pV93qHsyDF9xFgB3iyQ04Jcpnfd27F939xdxkxrFhh1CkqArUnl348B2yDd3PbavepkFvKwC4NQA6Bbvgxe2L5RCT8QBJjwfm/gA0jQCAKBB1WQ4lXwBvxADipeBfQ32o151jJxxufivEhx4EdQBqyNzyr12r4BCgbwBm9gAKRQEFwgAoIQBwYQcorABlDeAVkAANAgAFDOBlAn489qxcGNxVeRDhZeA/P7FzvtvaFtGkzQbwDABJ1HEC8wB8VicgRRDQRw5NAwAPxIxnytFzbOMTQuz1dx4Epg0ZD456o8z4D/8QJ1IBD+VxBhYAOzkAkTsH4DsQYvIBBVvgM+gAewrL3qjbNpi8JrOxTpgAqDkASvMLv+seEunRsYAGoAIAgdUBDMYAIOUABM8EwiYHUAoA2fFgcOYAZatwARUOwR4AnYgA3DkOzM3uzO/uzQHu3SPu3UXu3Wfu3Ynu3SDg3MoO3e/u3gHu7iPu7Nvuzkru2xgOBlUA3n3u7u/u3m/u7yHu1NMe/2/uwQYQKcDgB2oAIF8QJhAADmAAPCLhCeUADgYBB8gAFaNwGToAoQz2zaLeDOTeCLixSlngdJkAz+0KOsjr6+wQNhUCxc4AMFQQD0BwCoNxBMMIEGgQivR9/o/wzcbr3OEncNCM7f/PnxdZwbhhABEO8A6sYLqQcAL8AE5MALIlDw8hdyApEJk1ANoIAC/abl4srlNS/cMpEOn6DxL/wjPD8f0GEGFmAB7gcArODvAAALORABUv4RAFAJRS8Qh2DrIZAFRGf1OtvVXU7YK8EPsnAKVAC6ZmPoJpHWUyzzDm3fmf3jL8H1ebAF7/whht8WiN6Qik+mTv3bmv33sqAJVNCHNz7qrb7cem7ag9HnTonjIQH5aVDWox/fQI7d+XzYFB/htF0VdsIPsaAJXjC/ey/7/037ng7hoL7ByW0S6QAJeQAIV3T1pA/yxI/Bn36WoV7FoVqQn48Gpv/gYcEP1Q0O4A++3RW/5+T9JMufB2kQMWU+4ZQNnXrfvHyf9V6ur9t/vDMe9tbh4BPf+AABQOBAggUNHkSYUOFCg/DUMYQYUSLCf/8mXsSYUSE2jQD+pYOUBxC/jhMrlkSZMiFHlS0lWnQZU+bFkwUt8DLIcqbMmjt9TnT4kydMoUVXZvzH75ImNMmMDuz5VKpOqS2JVsWqMarAmzmzar36tWhQsRi3lt1JNeLHkICynkXrUm1ck3TtHjzbteDcuwbh9k1JFnDCv4M18iWsVJOcaV8LG56IGHLBsJMdV9ZLUPLgx5YlCvYssHPohZspMwv56J7Y0aQPmrZc2bXRvDj/985uPfsg6NC5dWuGmHQpmnJoff+GPVn2b5+1vbo+zhwAb8/RkS+sOCw1Xeuuk0NeLn2owcwDv3MPL77hQ9zp1ZtXKHxxO7vdSZ/n/F6o89vQ3eufjj3/ANyIon96GSSPUlar7z/x8APMQQLNwsw24AacECHqYpPwQb+SSkQTPsoh6S77QoOwrw4zXKvC50g70bUNlVuRObUq4mWQQRbkrMbZUjSRxZj4uxBGH3+bEbwjf4QKRBHbKTHCJVFUb0ohPyTPQvgwvHKgJA2LkUqPDtSxFA67BADI+tBMicgtjWRToC97jBObivgJEQ8GaURTTe7iLMlNgfzEKszQ5pSS/81/7ERwEFf25LNLQo0DtCNB02wPUERVtJJDbPKEEk5Jq6wUrCxf7K3TQwUUlcWK/nFFx0czHVU8VdG8dFKpDPVsUxNv/TUpPjSx454oWxVSV9ZKRcrF/pDt0tcGbX0VFDO4MOQkWZWtilfDuH2LWQpPfTZVTVk1l7lX/4kCgAoqeOCFOkQkCdynvB3M3m7FpcnZItNlU1r01H0VjwdaQLiFCkywUN+i8AXM4Xv5NcnfNwFGU+C4IKbtVRs4ACDhFg74l8tkSaW4RXJLPjNOjY0Ddtc7URgh5IQNWOdik1mUmLaUVbZJy0FpDRjd6mK+904fQE54BQei7Lk5pPuKev+/n4OzeOidr3y5LI73e9UVLBAYAeEVHiADaoL7RPnq+LLGdGshu2Ztaqnv5OOURPiI4IEHHOBiH5Yx5rlttw1cWeeWi9bt65nWjXXHe/6BB5RKqjk27sbttqvqnzgnMFeiMzZ6cRrx1kTPzF9bu1bpQAdQdLlZpPsyDl9tdFaIPB/PdXUPfztxraHluvRIwXwVzycja/1K3neCXT/ZiZ/beCWRx71MjJ53yXG6uOcJeMLgBj/Q6Omq/a3z+70zT0glKr/N9b+K3yrxEQ8a1aPP3ZzT7B19H/yadzJb3Q8v5BtdtKyHvAYlL0QjKkn9zMe2AhqQMgicXYbSV6j5ve3/VTkCIEokaKkOYmWEKCnhb6ZHuOL1z2vrAsADSZSSEzaLgq+z4AWFpzkWVs+F6lsXMBLEI5XUcFy+U2EOCbLC/blsgXQCIkyWIocARnCAhaugEse0QyMCjXEJfBgMtaOgKlpRhSmciuFyyETTKfCHYVyXYuQwQ5l0sWI3/J0Wt5i/cjXxixkcEgxRk4dH/MSOL0HjUw5JEz3ukSDlGV4PaffERIEtjlNsjCGvmKFF1kWPbDwe6d74OBiChJCrq+MmJ9TJlzQSlNdz4ii7ty75MEaRqiQQK9fiSgxSb5KylF8p21IVXQYPiY3jJRfB6MNlmu9VAOCHLJjiFGLiEkDF/2RIIklJmCM+Umg89KMom9msZwLAlIAooybPiMckSo+WF4QhLZfzSgbGcpwVg2E0mWILsWATf85T4+veuUSLPFOe/xwIJMHZxhbes0UwlEdb0lkUfx6Qnch0p2jCUpOTwHB8ygQkgTbYLSsdVJ9oeAUq07jOY+ImdjDZykEPekCPKrSiTeKfQ8fnUXl8Ig9p6Jw19XNTirxUo/A0aEH/Is+K3ESmTIVqVKU6VapW1apXxWpWp+oQrXbVq1+lqj6p8AqwljWeZkVrWtW61qqOia1vhWtc28rRjcK0rgjlyjeJ6khxhvRD7+zpTyMm1Pfs1S9G3aNdNXqVcuqQj4OD5f8ffWmgfMriFFTgRL4Iqx7DUualZ1UqaBvrWG/qj6HM9OtiyxnYLUy0n5v1UBbf49Ge0NajFn2s4kLpRp3OlB/XuKwlJtNZvmIRh5/spSQ1SElOYWeg8kBFHpJAn+HCVjrEhUoycxvJcPI2g74FrhIYISZg5jKgBqQnFCXbQ6YKRB67kC41PUNc78UFu6LRbmn7eNpfgrG9vz2FEvTAJJYCVLZrTG53G2qy9gLgvfHVDX21SdHz3i+9lVyvH8EbYDXYyLoePrAFL9xcew6owQ+ugXwj/OHrhBi9Cebvcje3YSV0OLblvWaFxTfiX+XUP88txiBqMIrCsnjFLrYwjHf/u+CjDRTASrgCdTlrZAIfV4s8nlaJT3dbeRQDEiwYb44LTEArKxHLA9My9jz621tQgQjbWCWVvaNj4J15YxN+ykihB1UHe/kDfzCuTm+cxysrObJ99d87k3KNNvdAxeYdc6DbWWiQTnZCep7lQPsMiQ/AoaUKjjOSd2zoemb4heC9BRpiwM9Px3iodD6cnWHm4ztrusuo+IAYXFvkSHMS1m6T9QtpXTdUowEEnMAzhXsd6jIjuNLKvTRz72ZrYuC6Ce2o7/fkfJ9fXy3YdRs2SX1rDVzIQQJhzrZ9t01eQpuZ1OpFtMxm2mdUsIAIxTkqmyR8UZci99mg7q+8wYsL/z5AANAErdO659vtn33bdmm2pK2LgYoazIDVCNe3wi1zX4/kN6F6FfR7MO3cjpbzt7hIBATKgFcyhxzEzZZOPbLQARlUwiCg2EEIZGAIgbABBj/3wUD4gAIRrCHf+oVsqeO953l32RhJSAGRWS5pS/Ma5swZwwtYMQcEgKMgHeACNPhAgGEAYAo+MIQhGiEQUCBAEaCYQCD44XD1hTuQJa/JyS8hgSgbk98AF/PVfzMBngPgBXEgyD4EwAqBOMDmU+BCQaqABIGcwQZzfzeGl24V0A7E6VvQQJj93uolM7vd0llHAJghECZkoSBR0IEgsvCCepjdARaQwdoBYAM2kP9EERgoLgAsAApoFN8e3HW1SKXNvtDC5OSyAEEPMunchC8b0oKfzTACQA6BZKEIBcFDCGzQgTAILhOkGAYbCAALwyMeAJlAQPAXUAD6F2AT2MDGMPC/f/733///B8AAFMABJMACNMADRMAAhAZmSMAG9D9v2AVjSAMGoINqcMALxMAMNED908AO9MAPBMEG5MAQJEEATJMSREH/M4d6CABgEAgkcL2BAIYBYD91sABBOBYYYAMAyIEzEAhEAD664yC7A4ukyjtGk4UZWDWVwq2/S74pE7XZ6AC5AwATsAOC8IQCKBEZ2EGCQAHE44KgA4Ax0IHgsymXEw9Ms62a6DL/ZQCEBli5fqk+HHu1KHQNNggBRRiDCMgZO+ABAFCHCZiCTWCDAiAFAOACPlAEJliA1YMFA5gDQVgARTBDkEstkVu+ncK7k/AHa7iFbCACDcisbiK9Q6O6fnsPfmADGQgCVSCJRjADgYAFJrCBH7A5ADCDHJCBKpiFgZgEHrABPji6jzOt0gs4cqIrNiwGZXgECWiCXZs6X7O+wDs9Ecs8EjO1h7KrI8SFa7iCc5ugjJvGOsS+OrvGHoM4rKmt1VpGVwABHJg+UxFHOrS6anyxf3tC/XiZNTyJk7sGOGgAOmBCRJpDNDyycoy1c8yybKQsouCpZRyGGXABU+C8grzE/5ezxyTDR2OUMXV0SJO7hm4sBAjQgoGUw3k0yCrLyFHbSFNkMk3cRphow3RoAlHsHSfkSHJcSXNsSaXzroYkKOcLyWvgBBLoATi7yVL0SWlESGBTSDRjSE17SGWYhzIwOKmxyKqDwqb0tqe8MyK8LbeyCH9MBiW8uG1CyYtsMa5sOK+cNYgLS54iBmWgBz2AADEwSRRKNqHYN6VUDo/Lq2J0SdQSS8Z6pk78xG0ggmPrmKyEtnqcNHfrSXjzrrhkR6q0BA24tqRxTMDTych0tu1aqJxUvsIMyrEcynvQAgkoBJLqzHy8LoZLGSF0TSfSNLdyr2Wch1dwARx4tDB6Tf/S3MqdTMjJ1DwFiqln8kcA0AM4fDicHMzrI06nNE5svJKK4Cp4yk2qVEwNGIW8REvoXErTA01rrE50vE7sZJV8GkoAyMzNBLe01MpBK897FM29SrcoCpAmOYk2nAd+sMqDEzb5fMzhrE+NvM+UvJ2aCAr27EYA4E3fhEq/PE6mnM6uPM+F/Kyr4Kpn8k/mdE4NNTB6NFCMorQEVcstW49+bM/uHEURbbkUVckDZUkUnc8mqwzKUQeL+ND3lDIYPUXPhEwTlUwbLdAFPSCH8AdqwIVsEIgAHU8LldE5s0PqNFIh3bIcfRV4cIe5pAcAsAUXmIHfBFIpvVGMpFGevFL/2JQSuIgjRrsGgWjOOIxO8jzTtbzQtszQCTUSN/XQYjCGdACAcmgCxmTT2BzHIUXFIkU63arTHhkt/ly0bnQIHz3SEr1UNCXS0GxU5BPOBpqnV/nQdRADCGjNO9XUTMXTNC3ONf1UmFmq5GlPACjLVVPQGUVVXF1UTiXG/XrVy4hUSf1QABhJLTAHElVULKXPTTVPV31UWA0PWX1QAGiHKxDFkQNOAlVWRGXL2XTLAW3TaBVV3RSIUTBKOMNWsAnOZ8VU6ABM4bPEXI24nVpOgQBIOvCSTPxKbT3Ug8xTb93TfQVVTRzWaegBEngFgkjXz9lLdUJWbv1XiqFNmQFV/3Gd1DgVCEtwxh/dz1ul0odNVXf1N2eNUnFLj3GlSoG4BzEAx/UAWYidUm6rUgwlWcrkoMeQVowF094k046N2Y/12IWbWT2t2QpNmmANSnnw0oGYU4VYWKzk119dVWa1z04dTXYlJZytiCXFBW4oEUI11IR42j1b15L9TKpFUKvFzxQKS5oaVgAYhejDN6fV17cUT5u1011tVrUNWuZzrnoVCDW4SogY2/CkUOsMUhh51zP8WXxyD5Sdh4Go1bNciMJNyhHt2+oaWoAtWsTN2s64k1kViJHES6CoW3A9XPSMUbStUb5t3GxqW3h623v4RuG6CMsNpLLFW+lkVSt1Xf959UjYBVwI7c14/IzTjc+7NVreZV01/V1V1cb/qIg2/FKmDdGMwN1Z0t3lpcbepdnn3VZEGo2tJTedHdSafFHsRd7nTN0y7d7mbVXw7VcPGt9/eIelHYi47YG51YjsrcioxVqYhd8+PYuBuk3S6tWk213mc5B/WNJbMF8AuFeV8F+VyM+y6EvM9V4cTU7cnDcEDkxfDWD6xZr7pUuCMNgUSFgKXt+6A2CzTVbFzSi+MsLECtWAtVtkXAuu9VqC0Fj4ZOGXDdn5BdoNPp1hLEylQmJJrQnGBV6xXAsTrt6BsMpTdYkKlp/t9dy8FdkqoSukMuBQlSenUjS5MmO5euD/a6ClZIiBGEiGt+rQM5bjOTbjJKbjO8bjr7LjPObjPm6wxqrhL27CBHZUGP6rDrFf/B2IQpCA0p0JLNZLLVbdxO0NxIqKciq5JV5iJ1bV2KWINC6IdtACm/QJSA5H5d1i5tVbgjkrO05iluPkbfVkipDigniFFDhKoTBlEpJk9z3bVWZlxTJN1fqozp1kmvINUC4IOghI8CyJXZZHVD5mLpbhE5Vf0pxlvKhlgtiGg5W6sWjhIXzhBf7lLmZUQvbUOs3mg1BmghgFDdhfqYBmGxpn7i3naj7nEFZgoz3ghdhmgpjgqphnUtTgJxbabpXYb01eSZ2INA4Lg0XYrBjo/5OUZl+O4Uoe2Ws+tH5WiH8eCEuV6HCuzYrmUzMF5tZF56stNY5GCB4+CCgVi4m+o3pO5fc9aedN6bWF4onwaIGYXLSQaYKk6WlWZXPmVX0uZEhF2oRwaINgBJJ05lIWaYodaotuV3w+angVTDW7CCkOC35gWdED6qnmzKouaWrGaGvOaQJekaYuCDaWULsIai9q37Muaqze27Ve0OPwaoMg3agGZyGeWujVVaPOa6RO54HNDbcmCNWUANvti7nGml62a5s27KrVa6XGKYboa1vGZeOVa7JuTLMW2NW96fjN7ETDOKamBgg+C7v0NMiQbNil7NKm5OpY3HgdWBAuCP+vDotqJWXZFu1sJWnbNunLTtvUhlbZ+AuHroxbzmXLmG3qI+0cvu3YyO2tNtli7m0vfe0GsLFeGW51rW7Uve6/zGjl1s/RAwAePot2qMnvJI3pZm/TDt/BRm6URmyVhqP6JYj7XQZ6OAtTONfZoG/urmvjRmvcTu/9lqB1xjjnrgy7xFfdOPBoXHBDXtbT9l317u878m2DANv0lZHxZtjatu7jxmvMdnCTrREHbm01Pgv95V8DN3GoLe4Uz3D0VusWP1rr8O3KgAMIqHDpuPBBLmjCLuIB/l4PH4rukHARbwISoEj1OHIsKe+FPm/wyG4RzrQjCXKDiFsi4FjmuHL/3t7ymr7ntM5nrfbyYDoSCRfygCzNzN04jdNchOaXiUVmMPfuSO1mFY42wfbX183ziN1zhSa5OI/xYRYITtBMaCxxQi9sIpZZPRcXPmfojAhyJbbXBtADwF4VSl9yJb90RM90RT9kRu9a3DyqciACKn/J+x5iqS10JicNfmBCwSkIXkdwH9dL+3iHYAhwV9eoMS/zOjf0O0/Uq2Zz8eCHKDAAA5iCzFGFF3AAByiC2nsDDCgABIjBapiAcY+7StTu4M0IUG7bfwjonzT1Uqf1Sn926eADDMCGasAAYSSIHGACANCGDkA8PAAFAJiFCMADAIAGASg+aMgZumNpzl7a/7aFdVlnSA139hHG7xX/jRwgA4EYgx0oCBh4A4GwAR/c9zBA+AEAh9rbZN12XI1Q5mBFdiLE+FqveXln8PcQAX3HgxMoiEk4gTBgAhmoBoLgBQTIBIQPgAgoABsou7OIO0yQ+mPl74ZcklqO3SGngwsuizNf7QTX8bued+boAEEQiEDogIIIhBDIAiQ4gV4clBOIvOkoO3DQgT88CwdAARh4ARgABfwbwRT0v2qQhluwhgGEhsUMBQsU/MY/wAV0/MiX/MgP/Mm3/MvXwMrHfAw8wc3vwGN9gTkQiDiAgYJwgENgPcoDAHB4gShYnUZwAHN/c6TqCCkmCZmPvnC4/f+G/Qmv3+wcN28VH/vfyIIfEIgfiEF4EAgEmITu+wF+UAcZEAJUioMQkP19Vq0liflgZeYiDz7CfPdTX/Zvkc3JmAUE4AIuQIBeAABQCACSmIIOYIMwKABKLAIDqIL8F8Y5QIIzAIgqBvDw+/cPAMKEFnglTIitIcSEBg1GrGjxHTFl9BBOPNiwXBMSpiBStGjyJMqUKlcmhKeOJcyYK0vKrGnz5sObOm3S3Onzp0OgQmcOLbqSVRgusBBCY5MwEJcwoBAKYmOVTSMAwNhkITMVQE+ECyPmtNjRo0x/1G5dkxgWwCgQPcpFfGv07kqXeI3a3euX7N+ifQP7LUtYKNr/w4p1vh0L0XDDszcxanSbGCGdBnTMXl4cWK9nnp1D44VMGubg0z9Nq0bd+rXJxgwfV+x4Uy1by5fbNdEwih9n2HtBC0eZurhN1sjrjl5eU7nzutFhywYceaJOyhs5Ym/4KkWPbSePT2dJvLxb9DuhTyev3iJ79M3f/61OW/dtari4Xe/8p0EZKblHn0XnqTcggfcliGCCAMRX3nwN8tWZYw3lZNtkwSyzHVjdJXSPFhpwohKDEgJgoHwRmqgggSUS+GB7KwZmn4WS2YRbW/g1ZIsLOEwzk4oytvRSi0EKCSNyLtKHpHNGCnkTjQ55WJN2/V1WCARioOakjChCyKWJ/0wKp+R7Yib55F5RGoQNmBHhSNJbYkhgSUxkEuhle202aOZrdqLH55ho4mVfR4AmVKWOCSUzwww/1qlng3hG56d6hp5G6XSWtgapoAJSyJBthr5pJUSMQKCFaJ2aJGmTnC4pIabRaaqaq6oGB9FCHgKKqGURlQFBIVDW+h6ry8WaKazDIruisrZeR1KuiZk5aqIITYODC8kw1mx5xSbJrazJOhsUs+MC1ZNBFZK7Eq/chcWJBFfcs9Ox03lbXL3OzepZvsXtSxq4zpbUkboIIUltrxCpASxw9Abs3L1jPozcv4r1K1zF/Jr7E0W2FeygSu26C9E2OKTwyrkTIxcxdf8qYyyuuRkv5jKaNgLwcXwIu3tZXE20gxjNsLHcZ9CtyTxj0aQdfVjSzIaF80kidxgWHA38IVjTpw29adahLe3XxbB9PePGPk1582wWWqTzyA3xJtKEZSO0Na1dezb2oHYfhjfYcm8bEdQRST11Yt/NdVfYxdF9qd6K8R13zDI2Tl+UBkPENuEQ/QEBHA1DXvbiAE9O2ONYb1w64n4LC3ja6wIwOIYIgSgi2KMTFnpoiYsN87io86V6qri2bjkAmE8NUTIxNFqf7Z8RSXnzhfHurO+CAV9T5SBjtCFnl1kiAapI+407v9HvVT3H5huFPmLXy5R9NfrxV9vZCwdLmO7/Qj//Xv5GT28r+87lvkexDiLv0BCH4NSTkrnAFhZTH17INzMIru9/qgogxwa4pQIiRC24yMa86MczEPhsguPb34EoWBQMri5y5dIgkDi4PXoo52yZ0YPnmKbCokjwgXJjYfB6JzkYxlB4HpyfaWIHgLeNQnQnXNAOhQJE7EXRJ1OEEhFJRKFVINB1mUuILcBDFyfKrYc6/KEFO3VFnmTRUxFZwinm57qzFUICAarbE4uExgb1T2lDbON4dtOBVVjneA2RE524lkfo7RGKp/sjIG/VEAwAAzBKXNTyiLZI/lVxNWkU1BqxF8nYfIosSoSXFnKIxzKiMEWN1KMLTdRJ/0UKz5SXsd+3NpnCVzIylrAapSQVMrx/sKkh2+gBCVCWS1Y6smyhZEkfvQZJYD6rlu4yTFyI8DNjzdInZsRfN3XyzCIK8YXUrGZDHFOohOgBApuZVDh18k3xOfOTaBqna86JTmEa8iH3uIJvVImveN5knszjJScfaU59doh1ugpjeFzJTFgqtJnllCVD0yO8nvzBjgmdaC8rSlHqTVOfYflHrg45p5BuzKC1Q+gufclHQSlxZx2zGZyuU6FpzCAGpdhmTJ0JDYuOyx6V5CNBYzK8kTqtpmc5yFMlORHHwEteMlAES80VhiwQ1VmZQEGyksqSArTyo0IaWHOwQ56bUv9kLLgEgAwQkdVxbbWrtvpqWM1FVlmKlX7cuRVOFdgdC7iCCMlMSFzn6qy6MvWuYEVqX1WyV3vC0i7diaoIz3IBDfSAGdD4LDRgwAfQkra0pj0talOr2tWytrWufS1sVcuFKMS2tra9LW5zq1vQIsIEu/0tcIMrXNcWABjDPS5yk9tabTS0uTllTvfOcgMFEAAiBCgAfQxgAL9dN0HaLVsBqtug75prACsi71k98paz/TWYANCHPjIq3/nSt740lQxNPIRZ+/K3v/79L2TVixZ01RTABj4wghOs4AUzuMEOfjCEIyzhCVO4wha+MIYPTAoysIE1giADHvaBEFDgwQ7/dsAD49bLXvXYg8RfiQgiwkCQhICCw+Com2VXXB5gBEIQAgUAL8xwBmYwxcQmJjKOR1Pg6UDDEDOOCI99nJB6zGEMlZAYe5c8ykwYgAs8CAE8IBIFEZgBBj5AyBg6UAQhMEF0StwvfXyAAQeEoSJnsMAYTIAEhFSiyzsQQT1SrGQtO0cRCAgBBioyCwRUgQkOGOomCiCESauCa1nW8XJmQQATBKAiiDh0ohOiAxiEwQF2aNnZ4EzNHdSZHyY4dUKgEYBhAIAcBlAFP8bQ5hR/8XiYdk6geVBniNTDAZMAADMIUMkckAEA/BABigHm3F5F0yeBNkSoIVIFISCE2QDY/0S2+zRtiYx7OvvYByw6HZFACyLboDAAOQAQiA5Qx5Aa/TURHYAJhGSBCZ4DhXkR0oE3AGAML8jCG25cPgFXE9/OEXZEQFGAhp0AxQjYBEKmEIUfn/G5Di8OtiuCghmzwQbfdgAXzLCUlpEyQek2SbsTUnLgqCMASCbafD7uPn4IoNIAMEMOGmIOAxgCAKAQQBj4YYcsmEEGGKhG+TQ6Mp0XB+IQUcQEEmKDM9QjALNAMw9o2R97vyfkEZmAXAGABxMAgBRIMAMSCnBlTboXPS+3SMwRwoUzI6QAnsBy9zJKAFIgJAx8TwgfHGCCEpjgDA3hhwrWEPX26qbaQLF6Q/8q4YCEwOANPPc5F34g9ntbfidmh0gHBIGQObwAIlwwOd39SqC7VyTvACBD2BHSc8Azp/T0ST1ChMBViOyDGfywQNEb8oPhP5Dhgu0r5hMCDAHkBPk3S74PuDD69lL9NadviAycAoAw5D4hdXjs9vepHtpHxPZ1OAFCpq9wnPc+soF5PT+AsYC/k0L8tL59BwSaJwAHKCCA6kWdWqkX2REI5mHCHCDEC5gBABiCA4RZFuQAP/QCArwYGTVUAibV6a1DGEDdG5jAOmBDBwRC28UbNLxAFbAc5ZVbebDfGawcANgeNhTAvnFB0PEeW9lfYGCDDDgAAgxbIIgAQjDBAhiRwAv4XA4UQAQgQLPlDoagVfe9hhlol3ap4MwBgCdggAU4wAFWgwxE4RS62QcSDkENgxYaQNjl4NfVgw8swAL4289BYQH4wBRFlRVeYWg4gBaqgFhgFTC0Id/ZwQJMQAiwAqrxoapREzaUFQCAA2SoA9SlXwzaCjTYw2NI4kCNizbMHwDYAzSIWIalBDwMlYUFBAAh+QQFZAAAACwLAAsAcAK1AQAI/wABCBxIsKDBgwZ5GPz3b2G9eggjSpxIsaLFixgRmsuIkCHHjyBDUtwosuTBhiZTUly3T6VLlyhfFiQps6bNmxQHLIw58N+6dTiDCrWILaXHoUgxFk1qkSfTj+TsPX3qtObSqVizItRZ8ChBn0C1ilV5VaTXsVPLohVYdS3CqG5ttn2pNq7dmly/VgV7t+/Euh/P+r0JeOxcv3AHmzxMVrHjknl77v35+HHhjIIrN77L+G5izRw7m7wMujTByGwnhzVtl/TFzKxBusYqOu7n2BRri5yN+zFqAJn59l7LO7fu4QiLMz2O9jbyk0KVP7f7Ozjl6VqlR4SN/W9f5mOdd/9nG3186eqqzac1Cr679qHttYofH1+p+pL63AlFv/D6faTvnVTfcwEGNeBU83V3YEUF/vdOMNLst1N/q/1X3mIL9tbgTRkmlSB2HUq04Xj8WHNLNkPx15V/FgY1YmotivhdaR9OF2JEL043TzHG0GMQORUWRE5LBK0Dz0QqfsVijDblyB2TOb50o1A1PjflQVH2xs81uGTDT0Hw8LAAAlkYtEkHGCzAhFTMyDDBAll8udWEKwbJpExOXglalirpeVOVyPlJEJ+s0WNMMfMcxIYK8EATQSMFvRAGANhEEAg/QfjADzYdGBJRkj0teSdd7I1qEKGL0SiVhYIOhCpo3OD/Qo2cBr0Qh0BVIFGQCHwIhIIdAHQgCK4+fEqnknaamlKeyg46I2iADteqQK8+hk8zwegX0aMCsSFDQYiowAUPPACFwhsC+fDCp4a0a4hU1iXb7G6lzkvps5pF29u09zZ7Ti7R6COgQAhgIlAc6xL0hghhFIHCMADY4QAbXDhgQkQC2JCDDTZAg83HIIMMADQeh2zyySinrPLKLLfs8sswYzNMzDTXbPPNOJ88c848o0xpz0C3zEw1QRdd9M9Gm7xz0kwDDY0zv3RDNMqh0OEPACEI8iUZOxBUzwCepFsmAJNw8cYYOhjbVXr2blbSkzFWGxK/LumLG79y38WOL87k/0NrT/eUwQAdAiExhUAysAGAPUXBIwAoAv1QBUH2lADsnGtT2LbbZtHtV96Bqdoi3nfmE40v53TESQoz2DIQKAhwgcQE1QCgiAGFh3DGFAWAwo8gTIShggx/FwQqW6Ju/hGz9oIemuisXtjig83kc9I2V0Cgh0GqjHEGMwINg8eX/AQSxhm8CMQMxXxcrbZemitfEvPzOo8Z9P+RbqE/1OQiTkd6gEATysGU4wEnefKzD4baZj+MeC4ldouN/u4jj2AsAx/FA44tcEACTkzFgMJJoGzqVT98VSaCrJkgibh0DYTwQw0MEMM9sAJCBIqQQSRsVgNfg7/7qBA7O+rRSf9WN4NXaKWG8rohReinQxM+BoWm+eFzYmWNDAKgHFqAABzGgkQlLi+HytphU3qoHin2hh7LIIY8TvIHCfRgGmjpohczwsQwOtExUCyNGWPDjVxQw30FSUYPQMAIt8hxjhepo6nEWJEHmiSPoNljafABNW11BQ4Q0EI74nJIROJwgc27o2IgqRlJauYcvgjYQUwRAxeM4i6d9KR3QFlCzpDRPKZ0TD6e4Qt0HASLDVCDX2IpSxyBcZGiHAwpK5PLwbDjF307CCM00INkDIaYxcTSMUfFyNzckj7SA40+pIG6g0yDCBIohGOwmc2CKJKbyUTMqvIXzso86IIHoQMErkD/QJDALSXsbKertnmnbk7EkSVZ5mOa6Rb+4eJ/BnlFK0dhxUYyhF8BFWi/3obQuBhUIh0NiUIdw1C0yKMYy/BRQdpRBgiUYW4XpVtGBfrOgsazLyNVTEm1siVctNAgjABB62D6T5fMtJ01hdJNPTNPH9bzLvMwhjESVZBpNCGd9YnpU46azaTGbal2yelgdjqVWM3KIG1swjb8eVEaHitUSdSoV1v00e18U0FPXQsa1WiQDZLAEvGJaUgBwNViztVCde3IXUGU17GIIxfSKF7gGlCGwGr1iG9FXlxpSlCl2hJaTS1jY7NCyV9YciCcIAEOXBeYtnIxswfcLFI7+9XP/+YrtLgc7VRQqUqCYFGLli0qUgory8P+J7HQAe3odJsU0/USqNScRnsua0jYhlCjp6ItXcFqG9yC00VoeWY0CXJOCTCioiClbnUzVyfsJke7iOWuW8TqF7LKZJwPNYg++QkewQ7TujaUK3yPK9+10LfAI8zKPfFREIm6Er3bce1/2Yss92qTlk207Qm9i1fwTqVEuOBGQQLnUn70V8LXBLBsuzrg+yB3IYu1EXNrIg9ipLQgqW3diYUbRxVbOLsYtqOGn8hhxnoYKT1FkW+zpwcepwbF6/Txj50VZGQOGY9FlvGRhUIPZUy1IG0kwjaY49/zSHnK1Gqxel7clRhbaf/GKTHr3wQJAsCSWb2aIa4njetiBIcny2/esk32Ko+/YVKTd4YyaPSMSD6v2c9iOfCV8RQUcaRSYANhpSudXGbcMHqOjjYPm7/i5kDB+SOlPW07xNAAOJj4NXiOzae9GOrxjLonpZbWqTOCDl88A9MCsQQIemCLRA9WJrNWYq3dA2n5ANrUgjaJc7uxZO29uimKHk6yb7hs7NyaPMqNXrRFIl7rDSTMY9ZNrJ+zbRF2ezrfBk6u97XrifDPf+QlQp153OnxtDuB7yZQs7MiabvYFyELJsh+2yHcfpvn3/ILOHLifWyOFDwuBy9IiU5EEAdTVN3Z9veZ0Szx4VB83nf/q7dBgqhSAJC4DCauzbrVA3HllVxDA8fKxd2ScQAkmVZELLbMZ07zkU/55rg5ebjpOW6KdJmvAvktHEKuF6rfp+abQ3pslH7b5TZdInIeyDTf2HCi/wfrDFSzqHOOoGfr+usHuVa2BlLeQlj9yRUXC9pDWWV4TlqZbqc33Avyr94CQJ9aKEdRHa6svdeSo2n/uzy9jhPe7PIX7BiILWaQAk7cnfHNcnyGIc93g6NcgjPu9XgBoIYGiCHmxnFyi0QvZNI/HuOnT6Fup51pFxTx82ZvvNF/rHXWcH3DlCdMRKhn7nuwWg3Av3voh2/h4pvm+EROfpMOcm+IAiC1OEhG//TzPmH4tRfNae67TSWP08CnPNoVvOCXysHk8cuS9la2/ehxv3SnbrmnPwUA00QE0gU3wRdJiqZe1EVd+Od3+ld7/Nd14lZ5A8FyAnFOGmB3BniApeRwgtUQHwgcZdaA6/eA+ReByDeBygcALHRuEnAFDLeBsrdQIHgWbXWDNViDMEJY1Ode1lca2Idl2mcVhCYQfuV5Mkh+UnIUNhgTHsGEOiiCAkGCnmWCDoiC2aeCNpENsuI+rSdDVMeBKQSFVSeCOWiGZMiDa/OBYMGGbviGcBiHcjiHdFiHdvgP2HCHeriHfNiHfpiHfhiIgjiIfRgVhHiIiJiIMZVqDGEKLv/gAqagiJI4iWcohVWHgzCShujBhj9BiZ74iXQIiKA4iqQYU6JYiqjoiYaYiqyYiIVXDf+wag1AB61Yi2+4g8GhVVCoix5BhbVlhSWIhUKohSmxS8+FDTkmdIchhoFyhkxohnj3hCPYg9j1g3vCdk+xc2txJeV2RU2gPTMHej50WbtYjk54Wb64XepXhcI4Su6HeioxTuUkgNQkfk/CjAoyF7lIYWpofhWGfhvVOZFnev0nWimRcFaVTqLYEdLnXukYX+v4i+0IeEPIEQ4lDl+SVmOWhww5gwL1kAQWkepIkBLIdCFxUvJHZ3aGhwLSkD8Gkn0mkhBJkilokhwBgAL/sXBtVRf4aGEw+WgyGZI0mYU2iRFR9WUblAKjcFll0ZMvSY0CFpQxOZTDWJQWwYXX8CUwBIZnsRTiCJAG8ZNrJ5VASZXuWJESUYSO+HtQVhROiWZiaWtqJ5fsx1RoiRCPFVnOF0zrtpBgiRFxyWxkOZZmSZHEiBCpBgDCVk2xxhCA+JcZEZjeNpeCWZiTd5gGwVv6QH8Q8AfrdlHxZhfdswbQcBCGMAZsAD4C4QlmYAaQ8z6SET/oZ42aEYRniZkD4Vy+NHYFmIseEZpuAQoGwAU/YAHaUBBZ0AFnwAQLADGCgABhEAYGgAiwuYP/OJuUOZl1GVbvqHsRUW70h1UJ/yhhwLkWPxAnAAADikMQGDAsAIACcwAA5ykQU6ArmOOPcAWWtFkZtmmYVikQ+PU/hQABYoZn/VaeaIEBhyAQYVAsBJEDXMAPwOAAYXMGMgAP5qCe1WmJ10ly2Qlv2MgU2ogWbbFgGLiSe6FoCDoWBRA2APAG30IQw4ABBiAAZiAQ+2ADBDAAOYBeGbMxHaMyI1MyTVOkRlozS3OkSrqkKpOkTNoySPOkODM0UlqlRXEy1HAi2EAHEtAERIqkVhqmNEMSC1AJAnEGNhApWWAOoBABnpIFMgANwyADXMAu7gIvbIOdg0mXltl+yZcOxaAM83CEMecUX7miYmEClxMFTP9AENUQAKrJBFGANb0SMRdzn7F5fnoKjOzIc7kXRQKBk4gXg4IRfIiqFWYAA/bADNzCCjcKAAtQBwCgDSGgODmgK/zwAwqBqdaZnwC5n5YRoh7SnaDaZVPlcQZ6gKeaFeaQAw4QO1+iCB0gEIZgAWniAxChCioQARHwArOwofGinx8qcNvZXaMjZ1+IhpeoG8uqFeCgDhGhDUQyEOqwWZIJontamRN5mfdBaGv5Cskqe+0qcvyoWeKar9rZp3Z5H5YWDfDwfP3mlAPbHfdKrpwqkZ5akNORauBnj03oku4EmRxRsRM3riVbrvNFrI7Ra8+wDlnUZAoIskAmshdBsib/Z7I3i7IGprJ+MW2WoAE40Js88ZZURrMWYbM4h7D4qrDciR3iBQ71t4BEW7RGmxNQyVlKa7H76qfI0X0DKLSXyDlVKxFIm3Q4m7RMa67DcU/YcAXiKWFTe2FjiyRXO1tZe7Jpm7JaYiLZUAgSUKA3iIZXMrHYUbZbd7Zmq7PNwbNjEUTQgE6MUGZfSS9zS7cFG1sHe7EjubULaxpY2Ub8BbcyS0eVa7n4abC/iriHq7h/ZhpPNwz6hqIoMbmjUbpkW7csdrc5m7c7WxpmpQcNAINSO7ocQbjTYbjGp7rJy7qRxrhMIXe8wEFI6FrkqHIii7zXp7zZy7zOVhl5Cbxc/+kVC2i9kIm9QKi958u9BOe8QpEPUKMKnLeU5AhlPQeW5nuNuou2nNu0g6F6MKQG9zC/sFG/AHm/tYm++Mu7i9uzp7MKLhADACvA+ki+f2nA/InAB6y+Ose+NUE9agABwiTBIEXB9ou7hoXBF6zBbWcX9+bAOsaLi0fCBWzCxYXCwarC2cjBKhF/aSABIUy9DUnA6GfBN6y5M7m/ajsWPSULqqWM0shpMjzENLxnNuwY/cmvWhFEaZAAP/zEsifEcDnFjVbFinHFXKsVsRILquWxXvw8g1e5RGzFZDwYZty5U7FXadAAelCoOZhVURzGl3tdRzfHnyOsSDGiEdFHsf8AAuHnX1MLxlMWx2VMyH1Rx/z7vFCTBgywPTC8IJD8lIEcYFhrxEKJxHrLFJYmCkArtLQ7Rm88t5JMx5R8F5acxO0LNV+QgXysrkbxx5EsxqA2y61hyEOByALxTKIAAgR4izDhy6B8upibuvmbuArcukEhj22QTnzcyvfzymMby4U8zatbzc0bFA+CCsqcbmwoF87sk8BMa8LsUcRMJTrMfdSAC9msTm6IE5/szqG8YicszstLzt1LY8WACiQAuOvMz+3skO+sbPHsFrV8yi7RU3Kgy/ssFP3s0P+cuQJZeqbcuy4RVajAAgrNzb3szVULzpUc0cQxz0FBX7FyBxn/KLjEOzcNjV0sTcsujRYTLdIlgUafYNLqjNLNrNJGu9PDLNDbS9DraxKPdQfmZdMemSpITbNKLc9Mnb5OvcEiQUmpYNKKt9C0kdMaldUS3dNj8dMLDBL/QtPnRdZZsdE6/dDcptZiwdbWnBG7lAo00AThQNUkatYfadfuhtfZAdM4AUXsoAt7UGfbXNU1QddnbdgAh9hZodflfBHjdAo0gGhybRiE3U5o/dJbncAh3dYV8SCDkAKWENlKSGpXfb2WHXGYjRWaXdATUSKnAASha9TLMdrZVNo+fdvrgcMi+myAOggswAmwPVbCXUzEvdbG/RS5/dQRsSWnQAW/Hbca/x3d91fbNlfdTHHdXo0QXQYJzf3cJAXenjTdeU3eSWHeK3wQ3HAKVgDaIbhQ7o1I8J3Yp53BXV3fBIFGjlADlmDTetTfc/TfmS3fAKLYfxJa4pAKYCC8+x1JDO5FDo7bED4U9J3DA0FJnYDgkQ2PFGi7vMqhvrqpH317GSuB6JAKZnAF6xDaGj7b5SveWffhG+7Kt5UPwtAJNPDaOJ7jKa7iYcnjA/ni+xfjJzTjZqAFN57h75fkSn4aTA7SMNXkUO4Y/hALmgAEcQ3c0K3jFbzlMM5WXr6NmvEOn5AHSTDW3p3aX5TlxqPmT87mXO7mjsHbQFAI9mDlgYbleN6Pmf/aoYMc4Ck84EIxD3GeBPGwipKtUz9+Qx1+3KQ8lXbOFNlwCkpQSP9A6f5n6Hie6dbt42jO0H1BD6hACElAqqv4n6Ry6AOB6uWt6qY+2Hdx31ZQ5gxhzFpB2YXd0dK86WXZ6TiBD6JACV/AyqNez/V16SKE6/Ot6yvo6C4hDp0ABoWwD2Um7HNN7Qlk7RHO6EWs7DKRD6LgCM8uuAMh7mW96jNs7C7e5X3O62LRDd1eCP7gcPI+FcRO2noOgfi+5vqOFezu7BvBeAFPFeQuP+YO4ti+fcgNEu7gCG6gB/8+cw8f3PTOJI0pwcw88b5c6Xir7iIR5pTgBXD0lR+fFAP/byMe6Mg2718mH/K9avB+zhTe8AluAAjwLhExjxQzbyWza6jSWIaZkfO7Hhqxrekqf5OyoAlo4MS1UfTwEfHM9AYYEAFcMK/U8gMTYAFRcCRsIAIloPYK0WlOn+1OzvMJHxTpAAl5AAik2pBa/906byGegACgMAwdIKsEUQUyQA7QIAJrMDKq0PgyUCb7vYkf2Im2WPmCSNWWn/l1iPma74lVjwa1EMB7SOmdX/pyyPmmj4oCwajdEqMDwQNjABxMcDhsAQ0DQArsJflMKMp2i+yEOfUVQQ+QMAhpcOIXsfcGwvWPYQNr0BCNMAEFUQm1egYhgPtscQYwUDxHMQCN/9/4XxKuxx73J/jlMiELnQD6Zi4QyM/qTy9CtsIQm4A7R+EJKMADOQAD3yqCInArCg4AACEgwsAIzLAdRIgQADRoCR0+hBhR4kSKFS1exHhwWEaOHT1+BPlwY0iSEAGUjAgtEqE231Ayq4ZS5syDJ2mKvJlT50RzAADkAPDv3yEMQf/5VMFGKAAePv9lKgDOqVCqPgf4xGo0q9N167Z+BRtW7FiyZc2eRfsVW1qxS9m+hRsX7Vq5ddvaDbtK05paVN3GJWcP72DCW48WBksX8WLGZ8Ps8BmmaVYMgXyykYEVCRKyVw0fzvqva2PSpeMqrvvX9Oq3qFmbBU04XyRKbf+MVq0b+PVuw6td8wYOdxgCMnEQVFqIYBiALCHw1InAxqc5A5g6f1XN1Wtw7qR/w83enft38U4Rr+rk5tXt2HJ1lzfdnjF5+PWxgmISRJFPclzW7pujCCQC4ccnWM4oy7PQ5BNtO/seZIu+tMKD0DQJu5MvLn0iccSPdvzKEK73KiwsxMIuJDHFuBTEKrwGVYQxscEojPHEFE1Mi5X0QgERsRFrjAvHwVAEskiwWJwKu9GMhJHIsmhkUi4neROyLH8u0aQNf3r0UbAo2arSrim/BBJJrQxbkkwIx2wrTDXLYpM1N8MaxhE3XLmHyy7fPGvO0/gEFAAzXUwzUPHiBAv/SkPJQjS+uPjBUo48cWvsx0V7s/BSMgdlsFBNeWv0s0/TCpU0P7FiBpI8SplUUbwsHfXU1kZlklMlHaT1tVJblPXSXRmTVRJN4iiHH79Wg/XTXknNtUhb0cS12UzxclXaXxfzsxpICHlET9OS1XTZuaSt8dnQPCW3sWurbfZaxOasRJM7pjl2N3AvFfcsd9NdzNwW0eUXsXXzBXRfwsLUZg9HHtGnXnu9bJZgOANO0V/tKPZuRonfNHjGsjLR5A1t9nH4YXI3HqtjjOuyOCiAVxZTY5ixUtkuHNXhcJF7jGWXsXsXRVmsmmdmq+UXiSZsYKSHlstEUCjZoy9KufvZ/9Cgw2IaabOMflnrCGUmOmvwwILnDk388JZqiHO9Wkavu+M62rdnpbZtJsV+Sz5VHMlD6qm7qzpQu7fCe+6v4jZcSrBnLnzCrPYxG+20AV+b1sGzajxxq7DrVG7NzVI67BuxcmWQQYbhuWfWAgf0cpo/3w1x2Jmte+nR+UlEEz7Yc/0t1vns/aTZWZN9eNAXhznzPpEZJA9XUlfx9zeDVz7x4o1nFPmVqy/rEE3sIPlvCKVXk3rsS7v+fKy1x5h7saAZhBBWSyaRfDLNV5+x9PMnnH2K3QfLIihxB3ikLni5qVysfMO/fnHuVgxcX+1EVx9sKMwTk0uR/b6EPwgOZv9/HQwd4+qzCajRi34w0mCUONhBu3wQgiFMHm/IMQcuvIEZ/zDHHSiBiVYdkDApZNIKWSgXFzIQhtvbDSYcQIEKUGABZ4AaNloVlCgB0UhCHCJcisi/I7bvNfWIwAhaMMYVEIAPO/uLDwdjxSJhMYtFcyC03uiTLv7vNZN4wBjHCAAKvGFqanxVApW1wDmuKI7n8twL/RewwgWCAnpsAQAqEAaHAdIubASSGwu5tUP+K5FGXCS/CgcKB0CSj3ioZBUFGS5CbhKOopLjHOvIyNcIwgMV0CMHImCOVDIJkzXSpCvHssX8zVKUq0FGHgixCRss0QEoIEXJLInAk7VSmGf/Iab6jJmurHnvDkLhBy8UoQoDYspIv4xRMK95pE5eTJah5CZpqqEwUJSTLNN0zyrxZc11XgeWiCzkNslVMwHeIR/Qg40qqzmtfiaonS77JBfhOdDFkENhm9gHzxznS30CjZ8NDUs2zydQayGGhHuYhkbBpFBpqROkm/unJwM60ZIORh1mW8QJE8rRhZYGgF4TKfZI2q7BkMIRe6gGQvPG0oh99KVbCarxhpqrXdkDD5pABD9UGiSmss2pT8VKVIc3VVqVihV8q8WWVBcWfAKmo1b7KlgF9dCjvZGso2oU7jQhCDSula1dtVxcwSrW2d31U4jqRfOeJz6u8rSlgn0q/2FhZ1hNxcl7eMioX+8JWAUyVK5QpWvXJCpBEaIFfvLrK7Y4O0jPfjasoY1oMWlK1LMIaw7hUGqJVsvK1rp2rjF1p11nS1WzJGxh/jBWadoqorcKDrIvleznKOursmAiZOTYamOW67vmtu65II2u5qa7KAnhjBJ9QK5mV+rYpvbWteFN3HgNRZ9VHLUW2VXubvfp3s/C13DyDdR39JE7D6m3sefsLvC+21D/zg3ABfsKLPgWCrUCZ7tsQSeMXApd2M6UtDHEij9ypyX8ykm/HuWvXBv8tgfzSTF1ykMoGtbWC6clwyraMHg7/M4PIxFSmpBDO0q8mxqj5cajS/Fgd/8s3B63Lx3NK8WHDOwx9no1yZFdchZb/CZhzStP9inyWY5MohwzOMtD3PKX6LEtQAyZO2E2y5grVOZ+rthraWaSXt5QC5LN+cRwvTKHgQtRD9sMzqvBB23S4A9sHJqKVQ5soHU86LpqebiGQs8ZXrGln/4V0p31qW9BS2nRyrbJtMqHKGqj0U4n6s/OlbSZSR1bbV76Td2wkyVw02rsvNq7sa7zmVmIZxLpA0tt2IZqeG3OIskZQnRep521RmwIeSM9j5hU/8jsawUDO9rCBqGtjcSPWFDCC+Wg0LJDw+3pLTjYsy50ahxtl3SoChCMFd62P83aUIv6tfDmsaE1xQ//WWiiC8lQlLpbxO7yufvbAGeywBdFD1TkIQ34fp2+EdzTjPkbpgt6YMDlvShZdKILo0juxDTe7AS329vXlLbtTh0lihMiDVLWF5I3/tiXCzPmE5T4m8TRCTDoWnUKN8++edtv40EDD4IQJCjwgIi1ZcIOioCHWH5e2qB/CR+q/kKyT4X0R++8vUyfHSkW4AMVyKBAWRlDBKIAgxKQAwD88IEFfCADQWgd3IqceY260Qkz/MHNKdM5yzmuLuz9IAv8gEcH+p4VByACAPuYQN/xIAK7+xPksYz4yKOUD2FQ4gu2SDnd/Kz0/aIddg5ADgCiEIWtmKAOAKjGAuopAzYc/6IOwBjm30EZ+BSxIxVuUMOMFbdyIDn7QdBe2T4EoAqfkAEyWSGFBUzgAOkAAAMh8AESFpAJsQhg9rPHhjnUv372V6Ma7Id//OU/f/rX3/73x3/+9a9+Zuzf//8HQP8bh1jQBCAIBXIIQPbrvwRkwAZ0wAeEBm14wAmkwAqsvwW0wAzUwPnLOn6YvsjwgbfzCRlAAlCwAwcgBQDoAB/wCS4AirAQADaQQTZQB3uwwRvEQXIIDBzkwR70wR8EwiAUwiEkwiI0QnuohiNUwiVkwiN0h0/IAy2AhybswSSkwivEwizUQmxYBy30wi8EwyC0wjAkwzL8wX3wiQnYDwBAgv8qyIpZCAB18AkeCAN+gAEy8Ak7MAG/gzhLI7764IdrOAUgsIQKwwuymzexcD4wcziYiYIgOIkIQI5ZsAxyEADyswcReAMAwAyfQAIh4MPPAyiRa5pE9Il5gMIkyLakSbzma7mG6zmMAYYOUIEJ4AwAwIOi4EQEsAELkAE5NAcZEAEVCAHgCynhG62uq5BsOAUl+AOMgwtEZLj7aUSYsQdPgAWsgAfFqIZNmIWtIAVS6KitAzFltI+aSwITmo9WrJFFrA/o8zlkNDVzhA9uOAUpgIPDqwtpZD0Ucz3fIkck+kPewIdlcIQasIUpU7nVMzsr+8f3ksdaG8jXOIdTAIP/MtDHQ2THGHFH+IBHVwpIL5pI08iHZ+gEGuAEhcw55mvHV6TGWKSYyeESPXGYkLSjkSQNdDg+IkA3H+LHhoy0h0wcEGkPouSdqegRm6QlnFyMfIiGk3xGlRwXluRIl9ygasQYt1AN3FiK7NDKpVDKY2LKwniHX3CDHkiG1NOVjUQhq1QhrAwYrizKHpnJrjwMTqmXSkMzccMLf6CGU6ABODBEUGHL6HHLIIJLfpHLzzgWo/zKj+MVomwQyaTMyrTMy8TMzNTMzeTMo+jMzwTNzJSHYNiDGhiF0ETNxkzN1WTN1kTNwHDN2JTN2XRM2rTN25TM20iSyGQPkHvM3xJF/5kixSBRo0BsxiuQsrb6ScXjOaE0nKrYyt/klTNxi7CMp7GMi3koBkhgAUaQyn0szAw6zCtKTMWkS8+sTdWsF+ukKOx8i2y4hS7oAXWEj+V0xcVbR+OpTaT8SkqpyYgcKb5MC3pYhk9gAT2Axt2wz5bEz8Ugu4Bhz5qiR8LghvicgYSEswWtygYVGI+DzN0cxdArxcbAh2YwUDXISODQ0LbkUBvx0AilrQmti3PIBTZIAR75ThdlSOY8u47zOBglLvcUi5JMBRpoglUkkRU1zBZlxRcFUKES0LBgB1+IBAnwzhx10PCsn/Fso/J8IyAtqyjdCn2IBl2ggfnE0vzc0f/7bE4f9TcwxSsxxYp3CAZUkAA6SNH61NIK6cjy+MhNgtPDklO/xAUvIIFXUMsYUVLxZNIh8VDgnE7Q80MZLQt5KAZU0AAtQFIgWdQtbVSNdNI+3Ev3DERckAMNMDpTdJs1ZdA2ZbxQDc7gmlTRS4t5MAZUYIEe6Eky6VQ+5dJM8tIsCtTKEjduwIVEgICoVFWh2dPx+VVgCtYhGlbqwkkC3YUauNA0JUyqZFFXVdM3fVKpGi5uyIVEkIAySNAa6VVn/dSYgdVIDdFZHVGywAdn+IUvAAGUW9bj4dYl9dYsfVcQFU4RJU6yOAdfEAUNaIKe3NeFfLZpvEqYLKRpJa//UHJKX2gDCSiEPC2SdX2QPhWPP53YcB2rRWKHXxCFGcABhGvYqWTVDf3XDg3YM4nXUTVHfZAGAFCEBkBXbdXTfmXUmNVRcBXVYdMeOl2GJCABU/BZCmrWj33WdIpWFqLY+UIeawAASdCAK9hUQQVaTxXaJv1Rki0ssEkHn9gCDbjSlo3Gp7UPkMWQqe2gqg0wjQlEAJAFEsAB+pTQh+1HQHNOJSvacKMWelAGYwCEBMDTpq0Qj33bqNUwuYUguoUwmzHWWMCBpWXcJHXb+oDbN5NcBqJcF7MLAiWGR4AATd1czv1aX21X8JxZQqG1AA0ScQAAabgCK01Xr33Zbu3R/1cd28EFvLco0WAgBRDoAbHjOr8FSlBzU1EbXY7JG3TwhWcoAwhA0NVtks6Fj88NDpGdo+hVEza52FXAgRjAULYF2N7119/9Vugl28lCi384WWcoBAnQAnta3ueD2LeU2PCNX+k6C7/MBXG4Ag3gBI3S3u1tXXYNW0eN3c6JN7ZC2lBAXuUVWIFk36B13/Ul2lgltOHcCn6whlvIBjVoADr4I/WV2Q0G2w5u4Q+G14GV19DQTmNAhhlA379ZYBVxXM+FXBwLXf4RX15lq2zAhWu43/ylkB5OkR/u3iDmXtWT4QwOYYL9B8MlBm844JSkESdmXRd23QcG1eAFYb002v/QMFZqGIULVhQwbtwpFg/vtbAhzp8i/hLF+IfilQcUVmFXgeM1kWPKed3lM+MZltVR/Qdx8IVoSAYdTt8nYeGh5d+/hbXAxTLhNaJ/yAd7dYcl5thAfhAoLg86phI7Vh88jhJs6IZGVocu3l0rvkkxdmAYpmSADODnLEldQIc2Tt5YluWlpGWoLeQ/ieCQY6F/QNp88GNgRmSRHObHLea2PWZJ5Z9/IGBxqAVIduZnnmXm5VGHfF5c1uTh+YfRXAZ8KITU7doJmWSxBWc2tWV4ht9y/px/KNVsaIfcLYTBBI93huBoBuJppuJ6PuNSG8obpodRIIHkbTSCEWX7IOX/OZbiBobdQ7ZiNIYdoYBPa+AHOEhhoaC2uxlktSHor6nmmt1o05WHaeiBFGBakZbTyhXoKD5p2sFomqXhe/6HNfaH+0XOqhhpI5loQiZjd81p2TXnev0Fd7gHLdDdvxjqji3p4DBlIkPl81HlkP0HGo0Gf3iFFNjbqZlqTq1q4Ljq1wDfL83liClJX0AHAKADCOjZ7ChrdT1rGaromi5ocj7o2Y3Lf6DeZ9AHl9bcdJtp0rVoaT7qi65inU5kpPkHMvWFcwAAS1DY5IwgSm1Pvqbom3bZx1ZqrxEKpMWHexCDZNUpOkps6V3sgW5sQxZtCdYabKaGAuaHV3CBlU3Q/7tW1Lw2mdg2ZuNBBBl4gU3cinoYAxiAgTMoEFjIgejOgRQ8RntuqXMuhnTmh7lWg9XGnNYeX+B+jbQ2sf9Nl1kwAD6ohAggkKwggxLwhEzAgE3chAgQBEMwhGoIxWfWaMUs1WsAgHJoAhDwYjfxbQb2bKOe54CeHS4IQU7MjKygQ5+YAiYAgE3QRc/jb4SOlX+wVWOYBwDgBBDgSWA+cB8W79XZ63h+38/RATPwiUmIgK0QhBAwhEDogEm4cAIwARjovur+65URijUukOtVVj858SdOcWRZ8UrGZK+BgTcoEFAggK0ABhgwARGwAWgAgGEIBFUIhOgovwAg8wAYCf+TYIidUHOQ8AZlCAZ0wIZQmAEXCAWX8IgzX/M813OJwPM9rwib8POQgIlAJ/S1yPM+L/REr4ieAIAdwEMAmIQJ2Aob4AKfYIIf2Io3eIH9zmgOt5pFzgVpuId7WGJ/8OfQntf9ZURL/rUn1xoHL5BOzIoOaO842PSsYAMY4HTIvuKI2WNPBoB2gOpUXZYkD2MWh9kFL+PZQW88WG8CAYcTAD4meAFPwAQRqHQ+wANPOEHkZifrbp2ubmR9AACG/mWFNPY4fm2bFm5qNh5FsIEXiIPp8AFmAID+kAEZIIOsM4QcQIEcwIPgA/fy4eRogGufAOk/Vsl0F+R1L+UmX3X/V5e1ILccwa7efAAAw34F70b1ggU6ZPddcQbe2UZmfJlsg7fsy5aAJtBsBk/1cgT59hX5FofIgb8iZQ6GdAYA1I5qLGX4UV7yb4H4d8xq7NlqdyZhXBCHAtFt3tbWn5fooC8N8l6NtRbWtn62f0iHYlAGevAJ7ub44aZVmHfycA7Kca55igcefAaAJPYJASfwsBf7l9fgmOfgmfdgg95wwMaxLD7cEC935GVYib8ngF72spdnvI9hve90vp+zpcAFang7OFBtuZdtuodmu39hxb/ltN97gmdpn3jkHe7mvv5nmUtwk2530+8vrP9eoWDkaCB3AADl0nf3sa97xG9V/2VHapK3ZmD69ab2iafueTWCeqd1+M9efZROatrOJKEY7NlvehMC4+P/2dS36qH3yKI3nqOfiovtBqzQA57VTRUFbyNOfgXnfHr2689PJ6E42WfA+AAf8ALHJ+svj6JWfd53bMYHCAD//gEoaPDfunUGFzJs6PAhxIgLB+qTluscv4KcQDRpN3CgxJAiHWIbKRGkyZQqV7JcWLIlzJMxZ6okZ48mzpwMCeok2fMn0KArBzREOTGh0Jgf3wVbhs9gGQh/BBpNCvHlzKpWt6rEynUkz68zbYote3CrV7Nq10IkujPsQaRsIQ7kZ+1WNoPJZsSw9RHu2rQttc5VK7hwQf/AiCWSXSxU8c/DjicLdTtRMUKFlD/OK2aMnsFCErTw+7tYskrClK2ingt59cLGsHO+1tl6Nm6Wlg9iluv4YzZc1wy2uyLBElXVhnEqz217du3Zsp3DjI7zNvXsEXcn7q0Z8Ud6yojJM/gqRY9ppidjF9lce8z2Zq2vng4/Jf2Z8u8DbY45JHdUFeWbawNxgws1GRWkRwNlJJefWPtF9B5/K0n4FYSO2VdhSBm2dCGHSv0F2IgEjdjdegESlllhH+HTTDDlFVROEyBwktxsIDpEYYgj6WiVh4ht2ONDQVpIJFclvoUjSB89iJKK3hX4zzm5RKOPQaNw5JGTOTJnJJL/LkHH35BhToSWmUmhpNWaJJrYZUFRDvhdWR/l48wv7CwEh1QP5vbjkmkeCRuYbJUpaKEmASrohDyxeSJvb6ZYFKQszjcQOr44k49efNmCo3OL8sZoV2Ped2iaifpI6k9tvrUeqE1CSWmJmdV6K6653ppPNL6g8xcjEJCmK7HFGkvsk8cquyyzzTpba7LPSjsttc7aVC222Wqba7Tbevvteq5e1h2KmPEk504EAjkQO79satA9WkjAyJPZiUouq6sSSuZN+Z7Fmr+0OeoovtBSGufB6dL52ED+UJOLOAvZ4gIOyYCq3b0CBixRxj2pahaqZn4cUsf+KunkySdCi+5R/wv3N5A8TeGjIAAMOggrxl9uzLGp8IUc5sg875xVl0aFqzKsLMflsk4f8XMNLnkZRKMGN+KcM008BlwybfzuHHREXA+tk9KJqdv0QJ19tpCWPZRzMX8Za+2v2Fl7vTHYV42NW9kCnZ01cLdYQzMAakCgR70cyp13hXVndXfAjPu0N2x9W4o2QcuQt9A0OLjwip89Lk55Q44rBbnJaJJOmeV/D/YRN7lQ489ClmjQEdwhjr66QaZXh3q+kpfO+2StM82Si3i+w1BUU12tu87EA+D7YMCzKjxD1EsvkvGAD1RlNLR32lfoYe5OvPYrYR/Uz0iu3/v2iHUv4j/5POMrQ/+FCJsRnGaez3v6UmM9Ur2vIAGM30Pmh7yPsENTnIKXvOiVO/NFD3090077iFTA6SFwLgpMDUWk4YtzMOQVFJtG+QT1v9Ud0CQb1EkGe7TBFnZwIR8Ey1JkxhCboYhuFQTgBbMTwxDNsIZruWGHGkYNXERsIVS70QRV+EMWBpE6Q+RQEY1oFiTS5SPyIIZTCNe2tz2PUSskHQ3dM0BGZVGLYuHijuoCteEwBA4NoEO5dnZGyqWxQ2tElOrc+BU4XiZtxjDGPBgyjR6QAHQp9CHgttdHmZyqX5ELpCC3QshyHShBDLGEBHBXxnztcW+TnNAfU4XJTCaFkH+hh+ZktJD/qBSCf/3T4xTRWEXnXLFCbWRlKxPWsiJ9RBy5kAbhALAX8kVxY6Uc2ynpkkqRrRKYQEHiX/CBJ3c0RH9iyCMVIym9aBZpmkCrpjV7csO/nMMXV2JIvOYFznASTZK7zE0v+fPLdP5EgX/hFf4YMjEcoPCRfMzlQfdVya+hk584md9f2vUuhvyhQfO0oDgxupoX5iSf99mnQ3NivL9UhIkNaYcWqnZRjYrIngr1mSVTB7CQ9lOYS8uhUxpynh5so4cdfObQyLkjc7qvoTSFyUjtggtuOISHPv0pQk15T9x4FD4gPWpMyjaQhKSjGMpIJEOoNgpbzg2aUTXrSzEY0+AZ/xWrQxHmX9RhDeE4pG09NWj8gIrLtApxrddrq1tTgq5XKqMYYK3jHSPF0fvo1ZlTlY5fCQjYwI4kSqbppDkasg1GmmKlRmzs1h4Lm6pq56qUVYllBwJL8qhjYZy4XTs8+9mzBlW09YksGyd7WolYFgDicKc+Lle4Pj2VlaCF5EaJqkHd7rYtr9LmL7jpN810zgWfKq5xabvX5C4Ub8xtrkNUlKlnYMlsCgHlFe4hW0Eel5S2pQxps2Na8AKIIQDthsJoqdiQtpdVQi2KcmX4XfracCHtasYDJ1ILT0XqqP0l1X93EmAiDpjACANARS5Smm4Kax/rBeaDzfjeycSXOv/ztXACAcCUZjylKvFkRGbxms4QS5GvVsQtIGeK4sreRWoaA8BAp9HaZvKTxmmK8Jm6e0kd7zgkS1jbqABQ0ZslZJRF1q5jbcxLHKuSyU2GiD8OkMyBFEel0y1rJo3svxFriMvU9PKXU7yTnd51uuBVMwW1jE83nxPOcW5IgBikhsu4zsFYDq2eqcrnovrZjevIxCweAo1MAKMh1dDGdpzYBA2MtVzVqAZ4e+HhehKvHsMg1GJpwgxyMNQqvEinKibwgghkIZlsWIANJhAEmnlCADzIdEHsOrCPcIEL4J1ALzK6ukbAANX8+UEcvlbAALnRB7VmBgJIsZB9EKASAKj/BgFAkRF7qIAHv3YuAOyoh9JIKjHFPnaySc07Zjv7PtCWtlWorcVwF2QHZFgIPxaACQBowwCqyEgYuBCGcydwsyTobOIE8u7mIlvZpKP3RlM9k3vjbdrWrEYAQA2AKTCBIYgIAQ9CwIaCkCIE5lh4RCKQABqEgQy9GAbOc67zKGRB5z7/OdCDLvShE73oRj860nEeAU8kvelOfzrUo67zQKhA6la/OtalzoMzZL3rXv/6zwcA9rGTHeighkYAXlIFJhAuBzzgAxc6UI19vKARAID5Q9qhgAsIoO8FMEABAi94wRNg8IY/POITr/jFM77xjn885AU/gMgvHgCUvzzm/zMf+cJrvvOJt7znQw95zou+9KY/veInj/rVs17xBhGAtgHggzAsxBMEqEdBYMAGTBggBzkIQQSikEzitEO9Uf4z8pOvfFLZ4N/gcEAmAMAMUACAFQJgBgD2EYI6VMMQ3v8BDCax/PGTv/wBy8QCflCCc7NBBQX5AQaY8AITsNogeDc//vOvf/4MAw+KMMj0GYQn4EEjjJpBAIMq7J8CLiADNqADPiAERqAETiAFVmBP1MMAxt5OCEIY8IEtgcIYvEH9dZA5VIIdnFpD2EMdhIH4sRwbmMH/7VccGc0tcYgq4IG3OcQmhGCMGYQiGALDqMbRWFlQrAMm2EG8MQQ/4P9BGCDCQkDDG4yBB8pXDZYPET6hIOCBpK2BGcBC77wBGQjC8H3ZP+wABizAGZAIGWDAGIhAFQiEIiAAF+QACowh74RACBCAEzaED7xAGERAtMGCAzABF0TAG8pWpUBKiIxBBEyADwyfISBAGNjACyxEJhSACQRFGaXMFf4ECoRAAfCBQyDBCYyBBZxBQbCCA/jAHB7PZGxi/ygiRKhCAYgAtWGDrmUBAhxcNTjiGYhAySnfP+BeDpgBXPyDOSCAJ/wDMBAANPyDDKxB9mGAINQQ7lnAHi6EKhAAphmCBfADPMBDQWDCAPTLm9AKk7Rbj+BeGDxiQ8DAG/BDPUxADNb/AwpwQSb2x4+Ni4Cg2QUCgAiIIkM0I/ZVggPg3g4Ym1VV4Tn2EI/sAz+QggA4xBrIQEFEQckJAgYURCUgwPKBRDEaDfolRgiIIQGQwkAgQRZoETY2RBw0GwDUgwC8mg8uQKAUErm0icb1RDsm0z4IgBcCwA8opMLZQT5qImSgjKT4Y04EZEPgQT7yQwGAgj0IQCWYwRlgX2nVBg2qo0RIpEP4W0EIQghUHwJ44RnAJPI5SQ4IxEQEQgdQBTxiQwA84z9wwQ+wZDYaxBjsgEEInEGYgwigIj8WZiKGSU8yBDMEQP2tHT+AggnAA1SqSVIOTLKshVMyBBtYZEFMACLA/0IAmMAZRIEDaKWJcWVY0KBIgGVDvEC0AcBIAsAbDMACRAArfCRBhKTJWcBHqEAdwEMAAMNArJ1eNsQZ5ICCTGVBrIMMCMGYmYtiWSaSJOZCaEPIFQQTVIE9oIC3TSbDxBG+HF9ZZOZCvIFaOkAlAEMACKQMmMFWdtHxaQ1rMoQMxCMANEIEAAAmoCc2tKEB/hlIGiNDqMIAqMM/8IMDiF+CCsQOjEFxMgQfiEBBMIMAgBo82IAP/Gd4Nlg/SieRUOdCGMAmFIQNnAHaGYABEIAA2CRSguc+vuhXkKdBGMIEFIQ2CICoDcDAAUAQrCQVwmfByOdENkQR+OgcVCLe2f9DAATlWg6Ebv7DJOBBaZgAG/wDH0zAPvxDFPjAP8CCATBpB7VkQfDB/5GDAYjfJPKDPexADuAeXFmmUnYidVAnLBAmEggBAJBCASQhAHhnizok0hBZUpCnIFQjPDhANZ5BJQrlG0JDBATCe8ZnasrKSMwnAJhBpCHCBEBDPbyANNaBBYCaIBCAOghjGPwdiirCP5iBX/5DJliABUSAE/4DNMwaAhBmB/EAiqIo9gWBQvLBAkxACGgbKOwqr5aLrCzlsDElZQSCsbqnImwkAAyDCkQAAqzcQvCBWnoM0iirt5pFERhrAkbBFIylA1gABnhCQQCDCliAA/joaRbNt0ZMXEQswK4uagTY3cghgAPoQKnuAxIsQAdEgBaOn3IcKDQY4EBUQ6nSVD1AwxQxa4VUgyuyRVVU4WLYA8QyRDUsmgzlDTmkhT2IXPwEBAAh+QQFZAAAACwLAAsAcAK1AQAI/wABCBxIsKDBgwb/GWQC7eA/hQgjShRobqLFixgzaiQIcaPHjxRBitzYcaTJkxfX8UPJsSTLlzBf7lsn0GXMmzhzRrRpgZdDmyex6RzqEShRjUKPijSqtKlAcPtiPnRKVac9cjWrat0qkafPhExFJuXqNCzZgWPP7lTbFKpUs2zjEryaVa7doV5/3kx7V2rfgnz7wv370S3MqYQT0wUwOPHNfOiI5gW71/Fhy4HvNrZ80fBLxJztLt4c2qS7X9QkG+ypN2bm0iQxWyYN+6BnlqBrnx2tOyc/a7jEHZ1cMDfL170t0q6KPO7y5ABuozQOnSrv6izpKTNGTynxlpWxZ/987rQ5W/K9pZ+kLv7o9fYiueGyJlCPFtUFWVN2Df8ieqXmqfVfbeqZxF5/Ob2HYEb5OPPLOwBM0wMInOBHkH7FDXhQgP1pOBSHZHkYWoEjHbhgTAqeOBE7vziTDwCMaNBEOyKuld9XGYanYkKyOVajZSQu9eOOBqVIZEG/BQdAOVdIwIiJn62GI3j8HclRj4kNmViQIEFppUhGfjlPMdwBYIkGRGxDnYnGefndQF5+BGJ7WsI051Z1/sXlR3F+6VGYR8pnDT/taAFBIfy0+dCai07VKJtStmann3UldqdWefa1Z1GZEgmoivg4COEoJOCQDHuPuuRoqovaeOGUcGr/eWl1nQY1G6XRRXVYrSp+uiA7vjzjDwBqQKBGq60pWpNCiEF6o6QvzQodryNJWxaum5JE7YK+wucPNbkI94oLM9gCZbOqPsoYRGyqiyGVk1K6LUjWNjXvWdlq1CeuGHUr3jzELIMPP3RAIEaiZqGbYaPrMowQqz2xKvHEFKtb8cUYZ6zxxhx37PHEDX8s8sgkl2xyqlCdrPLKLLe86FUux/yxoP8kg0MKo2Tc8Lotseszsg5Ful+8ft7rUb3eYavrZ0a3529y+DQTjDwA/CGBFvcA3bPPy/KMbG5aFyd0jlUWjSVhTW+V73hpY/e0bsA+o880RFDYLmj/8AOKJ6vW/wSOJ6AsCg40hEMjFD+FQ0MTY2PDGy2ubWOE9HBKv8UvS2+X5o80vnQDowQzHujwQOa8IEIIMKizcyMOyGCCCdUAkMUEtBOgAwDQBED7BIFUOtC7ser4ZeQXTW7hl2tjtO/lE2XOWToB48Okk3fbxMYL+9jzwhsFyTCGQC+ssZJA/FjAB+4D8Pjs0I/Le/ZfxDuVvH/x9+a8Y9ngcg0AnIDQg5qoCttAZMAGgazBBuMTSA7MAL44EEQRC4AH7gQwiUyozncCAV7Xyja89wmmcrti3knu95eoEUMe9xADBP4gutEZJAS9AwAfRFAQWKggBypAggQH4oMqCAQaEbCBCP8mAIqV2CQCYWDDGtgwDHg48YlQjKIUpzhFaFDxiljMoha3yMUuevGLYISHFcNIxjKa8Yxo5GI11JHGNrrxjXCkojmqEcc6alEcuZCGOkJRqlrUo4z2EEgHYhiIEBSEDCbAAxsmgImBVIMAoCDf+LIAAwwCIAJImIImZ6GOTnryk6AMpShFyYxRmvKUqEylKlfJyla68pXqKCUsZ0nLWtrylquEBjlwycte+vKXoxwcMId5ynFIIxfdMIcaGKAGc9iyHgKBAfcAwAYZFMQAjQQAF3wwEDao4CCVWIAlNcgzDlqpfhIxHl5AyDQRmoSEcZFHMJqRj1fEIAbmMooLJUL/BhsIxAYMJEclBBKBQKykCEwYnwmmCYAdAiAMLxgnrDZItA46Rp06QedR5qccjcIGnmrJHzcIZjCE/WR5AKhGB2xggw7EDhQBWMkbFoCEHDiAFALxBAHGwgYT+AAGDhgo49ZHtoqe04OaYSduPFoakHJFO8aYh81w1kKUUiQQgagIAMyRiYHMgg+GGAszIjkQeFQCD4cYy5so2j6zXfRWlOLoRKzqzoI4VSt4TI3VrkAjoOxTQI0LnjmPxFSDYDQnhc2JXLuSWMvc1Sn5eMYv3EE3EFgigHSlHFEddxzIIdUujb3JYncSWsVgZaiWecc88xEjIpQDs41dazmNStjP/8qltDAZ7cNw+5fHEiVJ4ihHE6jnVwGCNrBs7az73uojpU6Ht33xrU6gOg9OoGkasIXrq6ClXLdaSrvIW9pS6wqm04ZWHLegRgohoAfY8la2mZ3IYcH7l/neBLoo0S13yZsR6cYkspMd1wxOZRN1hQa+svIsc7Pk3PXgVzTm7QuLXFSsYxU3vnhC7mzbatHvNjeu4n0ufz/iX5QANxkzcIEpGIVhriBYeEddMNoabKAHy6XEJhkTdwp2MBbbGE4abjFC7MtgGcOPxiX6MVtwHJ/5SMhuBTZuaV48WCLhlsiXAbHlRrwRJnukQQ8qhAT4qighA3azgqWtlW3rHCQLif/LXY4wWVj0DHM06UlRNvOZt8u+7naYMFiOkpZDCOf+ylkr3wrOmdJUZiU/LMgJXq6Hi+wn/fa50BLx8kXGtAx6qJCFeXb0o9GcXJQE+oNGRnWlQ+xgTPfr0E7J3zVIhQPspkvK04I0jGub6qQOup2utoimIxK1qamhAXDwsYqorOYdXZm+R7J0UYMdkWEfJG61cAE+Gy3queq6ys5m83nc3KVuc8XaBNlc5+DQAAsvTM8zJvWG/RzjScd71VumdrVhnRPoLQMZN8vZrc3tn283e9ni3nN4863vg6AbALIuBATI/G5J/26i8za1gu195F+Pt+EIsXaxvdEEDXCi0Rv/v/h+bWXx+kKbSNLmLMgHMuxz5CIai35tSf5ab5VfmuXeBfTLdxTzNM98LvxGiT441w0tSKAQWuN5zzOIcXgT5NS+5riqF07ooyPdkiaR5zL4WGuw4XrqACCn1dGScpd/GN9d97pAcJy/bJSBAXQw+9nRrvZIB93tlOY6sOU+96SDhB7LIIYryJXPrfGX2RxG+12wvh5y84ngWpGuzaVRsDLofe3JgTy9ea31rAv+44QHwGPzEQ1frOJmK9753tue9qr7/c+Av/fpRZx61Rs+I+9okZgpHivQY0f0Gm95XyhvIMtzqve+B/tFksSNkp9c9sY/vsEjT3qhv333re49/zyh+ogxv3ZrmPeLvI3P/Db3+rge533qSXgOXEijSVDviIGDjXyg4375Q6ciRVdqR+c8AAYKpWJrxZd+iLV9o7dm73dbzqct0Jc5wdcMZdAAead/szdi/WcS7TduEeh+cDd4hPc0SbIKjIc3HeiBDph8fweA32clA5hxBfh7BEEPxmAMgNAAnseB2Rdu63d7kmcXIdglE6gvDOgU3SIfsRBwLBiEQshn0/aAU+h9gUeDrFZjFYiDAPYIE0cjCwh9H1gttDd5AXgiNSiFnvJ7E1ZylwWES3gtQ7hrEFh68FeCqHeCSZcklYAmajKG0CdRKweCZ2iEabggaziHSpEiOv+2BRIAasvChjHYd3Z4hbnXcXoof3zoO9xwC7KQgHI4iGJTh+CGcCMogpsYfvMnZ8WWBhCwgczSgkdXhmJxiHJxhHyShGzThVkBLMIQA+UyiqSoPlQoc/5XhLmYiAiyiL4IAN/iC37QbokyiYyocFRXiGaofGg4g9G2hUnmiwCzDElAAjkjiMUYNKZ4cCfybN4Ic+D4ZuJHDtlwC5cQhj9zjSToc1UIg//XjVn4jQwnd/iwDMGwBRqAZ9aYjt62jtx3h1ioe1o4kEd3Dr4gChrwP/nIkMrxgsnYfZm4dRMZdzOnbm2wQqtCiXBmi/SCi3Ghi0XBi8qjjzchdi6AA7X/MCz7x5ENeYxGZ4WoiIcSGH+sCHL1GAjtpg9YIXU8qY4+SYCGyI2I+I5EF4/lNnP4oAy7kAQpsGIwQ4tNyZJy4pJsAZOxQZRc2HA2dwkMoAX7oBBfSZPeaImn2I4JdxZyWRDOSG36EA268AUm5w/MAjOo1ZQF55BAaZepiI0CSZJwplqi8AGh0ygPR21ieTRkqRZmqS8yST+uliRtQD3WWJn855FRGYMAKZGNaYIjph2owAJEgF0hU3jSZ5hOyY/IeJr/OJUBCY8UWVf1lwh4V42jQ5qudpkbsZmYcpch0pkdxWUAZgQuYAtGFDbGiWnIiRSZeRbKqTzOOVd5eRGq/xUJEiAGb8lz11lo2ZkR3UkV7tibVfmbuPIbt+AFOFOdB5KeK2ma24iavKmavumYuOKaH3AFWcOU0VeY/Gdco4Og6yk520kW7dmRaBmOIoRed5CQs+gl+gkbqfJuKYlrD1o8EcoVE9qT4JeW/AJgNaCRCEpzOOiBfUMlM+osT2mDuqmML8mM/bGXuMIikcBe+NkYHTplXIN+Gyqi/HmLUrmMVCmAVnl58wkcQDBgOylsMfp4R7qA+Vg9ixIxMmMxYTqmZFqmSWqmaJqmJJMyatqmLjMmn/ABcIAwHxOXbnqnOlNO3HamTPGhdMmOC/KeABqfAnoi8oEGVHWlF1GkB/+2pTvjNY5aijfKfiW6FSfaFd/JWFYSNZ/AAlgzmySWpeT1oQrTKinpKrj5k/6oo2XJo/DhoyrCDrqwB1AGlkUiqnVFql3Kp6iajT+XoyApg/AJpfIpHolGA1egc7Z6q7W5kn2aLijqq/34kRAZkqY3kqwJH2MCCRpQCOepZ4w6qkvakk26o0+qhlH6fAvyiV1Qa3T6EuGaq+M6luXaqueqiOlKgf3BqSxgYcuaabhqmwpae9rIpP7ppMOKrsUKN7qwBi7wCu+KIgFrmyNqEZeqWUK5j9i6h9CxOalAA1rgDxErsc0qsECGmKsarKmpiSlqodWRDrswCCnACSN7E/H/6k4VK1+VqhUXO2qrqKLJUY9PQATh8KLvNLGGmbPptLPM4artAau1UZCdmn8qebMipLQR0bPrtJh4mamklRw2BwZl969/grRhOa+YWa+a6bTiAbWcsXQfm2xkG6ola7ID+6cPiYnCOqjEWqiEIU95MAO1YLQwYbXMg7VDxrRUobXG2LLyGBrZcApA0GMqyawDa7eSmqpQ2Z+7ibB8q7B+axfaAQkscH2Va7nhybLSmpucy6pre6/NmK9KyBnicApdcAXxMLcjZLY8ibgborjlwbbY4bZ2wXqdQAOMULOZx7sc6buGBbxNwbiZu7GceBeq5QY9MA0mpRaGeznOCxjQ/wsgwlsdxKsW/HANkgsHgnm6zcO8DPm9Vxe+RyG9V1KhjysXOjgINZBPHtW9/AK/bKe23Dm+0FG+XCEOqWAFYqC7OOG/tIe3iRmozOliXrtbcpEPwnC8pttb7puOACwQ9HtfE5xh9nuVbOEOqeAGk2ladWuyHwwAIax+GauKjmvCZPEbkku1juHARrgDCDAB51MQjWACDoABDgQAMrA7E1BAAHAGDrAAUaArLxzDWTbDjBmg2UoV87ALeZAEBMYZPCwXVZAD61AJBcAMBREB3OMJAgALADALqqAKmyAAkYQJCwAL1SACRzzF8ksUVJy6TyG7vcgV3HAKUCC30BXGcf/hAI0gEAREEPsgAKrAD/ywAF3VTSggEEwQBQLxBpV0t7Z3iUEZkaq7mhzbFPlwDI5gBK/AwE2hyGphDgGAxgDABD5EEGYgA2vgAz6QQABwAkz8yADQCBNgSbyDCcisVTjauir7n6WMxad8FOyQCmBgnuy7uy0sQswQAOAgEFmABAVhBigQBTIQBNAkEKBAALEDAC9wxJmAAJbkAB0gAvTsCdqADficz/q8z/zcz/w8DP4c0AI90ARd0AbtzzB80Aq90AEN0Az90BCd0BA90RQN0cNQDRWd0RldDRkMBJag0SCt0NXADPgs0SF90ihd0BURybMgEEXABQkECwLQEPz/EAJ4MBBRwE0KxEAAYAgdQIi/yszVurfPTKhZnBPzgAp50AT1cM2Y08FfYgJ1IEgxhM4DcM7bIxDrsACNLBBkkAMCsU1APa3AOtQrK5KmXL06UchSUAjrCx2wrBZ44ABxUAQhEBVsUEn1IAI6gAdZYADAIBB4gAEJxAwLwAVsYACbMNasa7Cda64Ji68LyxL4cAyEkATaa25xrRaCwARhQMueMNUwfAae3dICIQiCUBCqkAVRcMmgXLDkerCQ/bmSHboowQ6dQAZz6tQke7mYe7KTSoTN7LlF3bdHjRL+EAuaYASDm36brbcQnLJm7cxoDc1qzRLz8AlukAbV2B/P/z3Kq6uq1Kq3Z32taV2UL5ENmkAF18ctUE2KfCzAEkrAyWHAIIEPqEAIWwBAJ/LdihncovzfpFzdRh3NInEOnQAGhZA1O+LfEoyy4w3eRE3gxm3gHqEPyv0FyeDL/f3egxjfsm2vkR27k40R8pDCacDbTOjhZIi2ydnHH0Lf6SHIM3kSstAJT9DKfuLgCALijy3itE3itj0R9JDfW6DiVcHjHeLi2infJirjumHfEyEOndAFNAvICYrleai5y+zYrjvAsNujNO6ZHyEKlPAFuavlSg4fPv7l8x3mrzrmz7kR3qDdfyCYzLPmdMLk7AnjOvHHFQzbSCILmuAFyTAsIv+k5+LR5sM928UNusdtEEWeBymu5XbF4r3H6NNN3BQO6RbOEeJACWCwYvyl6NoH4WVN3tRt3taN3geRD6JACF+A5DeG6amn6arO6axe4NcNJ97QCW5gCYg+YqZOK3wOoU5uqVBOIHIOnhHBD7FACV4QiHBW7LmG6kKd647e6bV91PQACXkACJaOpdlMscdOosnOs8sOG277D6ugCWhQC9Rm7aF37hbr5zkB6CUspQSRD5FACWnwIvNu64SH6xJe3ltOvaz4D56zBqGgDw1H771h8AJurQl/3mnpD5egCW0QDs2OKxKvGxT/4FbctfvOKf+QDuD+CPdwg+WetPaus+n/3rRw/rQfT1qEjgbJIHchXxsj3+MjvJwnry/4EAmE0AfjXl4vf7bY7uWNDuSP3u3r0Q2OsAa10PKd6Nu//dpB7fSbvu27XuFLofEcHw8c7vJa/9s/v+Rc25xDP1fp4AhuUAoMPo9L37sxv7Qzv7jrXhr69Q+SoAlyEA7b24p337x5n7X4jhP6/rPKU/SEAAiITutwTfByt/ZsHvRVgeWL5e7ATp3iPXM976GJn7h7H7w137Zy/g/+kAgc3/X6NvpTVvq/e/rR2/cjMub/MAyOkAehIOjBJvsHRvvPa/vim/rDK8gPsfFy8LV2n/aYi/l73vYU/Pb/8A2QMAiPoKnP/5/0jBndEV7xEx72nr5bq0AJb7Dh3G/40G+30r/omu+elvcP+pAIlOAHZw/7A3/470v8AAFAoEBsAw0eRJhQ4UKGDR0y/PdQ4kSKBSlexIgwYkaOHT0KBLdv4b9/vBzlqSWR5EeWLV1KtEdO4MaXNW3e5EhzoAVeCVfatIhTaM6hLYMW7agT6VKGIRX+43dJk5x7E38yxYo05sysXb1SVAqAp8+wLI9+zVoWrcKza0e67er0IMlqkPKUqmpVLVy+DbcC2NuXL0nCcwmvPHxVI8KxGgNzbCv45mO+kSUDvlxU7kCSmDTF2ZZXb2bSCf9SLs00sdLVgBO/PdjY8E3Lqf89onZbWzBu2xM3A84XyVGfnLx7dz193O1Pxa5n6iz8lHFPx7SVuzT+VXff7NcVyiXJypGbavyKe7+cHH1X5qxfc25usLVs+d0dbl/f0H5W/HD35wfAKagS0cSPeG77D0CX1FMQqfYMO6w+iFbjaTULL8QQQ9cy5LBDDz8EMUQRRwxxQxJPRDFFFVfsMKR/oBlkELxYpLFGG28kKSYcd+SRw+d+lNC55wILiz74rGvwogSX6m+5JC8KSRJN5iinpfieZIpBLCfbqLkHgYTNICPBfKnJLZdEyky00FSOJGiEe0Qf7NjckiIt63yptcJa21C/6chCEk+IjlPzKzptIwn/FEf2qMWfPA8V1C+ZMIu0pgj75PPKuf6sDqhKw0ytUK8gzYwkfgj0Ax6bNP30pTtbJa1I6mbzFNZNexOVPVhJGmaQPFzBRiRLSbV1oFeL3Y1TWmvKVUFimT3uWf9IkrKOffj5zUppbT0W2cGUDRJabym1rVmstl3zH3L2cOQRRwMU9tFxa+p23jXBPbJWb9E1K1pBCQOFkj2mMQ+keOe0d8FJ+U1YJXzJdMnc/Bj2SOKlKL74n33w0GQR0QxeFWNB6204rYeFFHdfQv098x9gCMkDmYINypYlVkuGaWGc/Tv5ZsjmFTkjix3EkjBENMFj5oNq/sjnnRki+emiZAVU/19kg8Zo6KmfJEkbdvFaiGkEpc4oarJxorrTlK9euTes5/zHE4GnaUjspN7G0uyzV+2ZWK2vw3uiv4UKHEF77uj4IbvP23sivRvHru9A2caVZfQIc4WQPZCRaPGMnIZcoMdDb1pyq4st/KHB0V6PMEE04UPpug/WlnSHRredqNhmDbdMoNtG1DvCqtmDkFKgpN3m1BXEPXeM0l7W93GXb2j1yQAnqRJK7pAT+ZCdX6h58K0yfW3UgU+N+gn/WecOSjaRvfPkm1Z/PfHHfwj63iP+vXK33SZJKU5SHo547nn1Q8/98Oen3VXNfLZCYEKsx7fSHOZ1gogfRQyopAheR/+BCwSVQMaEMumpzH/BixVJkDEIQrgigxqc39hAaCydzTByDVRbCSlXLstJhjCL2N61WLJBsHRQOR+04a3ExLt8PXBX6KvgZQhjDvfBzyVEHE0SkZhEzpRPh+c7Yfp8GMBFEfCKMbwbF7fIRXLthIkQM0r/ePi/wZiKD5rAoE2wqBIjHmeNXNRfE78IQSjGqo7/WOEgevFCj+wxf33szR+TGEg49mt6hSxVWgBmBi4YgiRAvAMjP+JIh4AudJK0ISVJyL9LhjGKqiFMFBxQgQo8AAXs8oRm0Mg4G6Jyhqo0ZUXkGKoeDuUweHhAC5TZggqcQBtIIaV+IGkbX4IQmH7/G2ZpJmgpWJLEBhwAwDJbUIClRHNCaqwhG5PiRVaacI4odJCpUDCCcC7TAOuA5i4/N83UVHOB15wcGN8pxoxBxQfgXOYKHCDKlphzJPwsjT/xB9DTEdKVhiwoObhwgBEocwUPIANDh6jPA6KTK+pcJw6j184dEpOOxiSM3PbABwc8gAII4IJIR/o9LaYTpc9jZxxbOdBXEo4kG+sYP/5RD09MYpsJcehTIEoaiY6Pok5sFT+fekMukcQVvuLcWqLqwBlWFXxXHeQTL5rJrv7jEEjT6VDGmkMQmtV5aGWpQF0KT7j9AxvFK0Vc5UpSDpq0jT8FS1At6c69ErSvmxBY/zsEM9eVLtCuucOrUBmrzWLejST1QBwmPgYXyu7Psj5FLPlUatrFtpSzL0WQVzUns8uUVpB1RW1q86fYj2yVrUTFqGdJ8jo+vKu2hC2iYadaqcy2Vq+v5esBSfJX4wk2K7at5Pgua7vm9jabpPGtzYpDGExsr3ulwe4qT3tS3ap2iWTNq0WB+9sikgQeiMuEdb2S3mA2bruk627FvpuZ8JaOg+FxxCBSchz+Ljc9uW3voFZ7W826FrydXd9wNREI/aKlwcoFnyBykANDIEQdZLDBDvgwkGpwQQY8qESAhDBjIWCCvW6Eb4Wfe2HYStNNLHTFaG3z4Z7eOHSeMAAf8P+AAFAobQovwEQgEKAIAOzDBD84BB6oDA0B2MHLszCyWN6oXufKt7FFzfA/ynuHdnRYrMjNYi8hvDckVEEgUWCC0l4wB/PsgAwAqMMJDgKNASxmwtkV8FDPHNyH2he/QmYwnPkI4typwA4CqcMLDrIGGHjCEBMABQCKEAUkyIAL5gAAl2FgAzYULMAdKXCy1irFNKuijJCONE/lHObGYUAQAhEEBg4yCxOEwAFIqAcAZFAAPHgCBkgAADneEGULcCHMCyhAtguwCWx029vfBne4xS3uYYzb3OdGd7rVve5xA4Dd74b3ucsdb3rX293qJhAfqlFvfvf728PYt78FPnD/ghcc3tVgRrfvbXCGNxzdqAbACfAgEDu8QGkmWAMA7CGDMADABzkQyCQWcJBAWCDMI+yvRGLNHUzS2jGEgdGvcH0dIu/6sJDzQU4BkIUfHIQAuQRAGHgQ9B0IpBIjN8inTz7mlKtuwJdZue7qQxjt3cEebpZMzcs657M1wgGYwMQCYgwLGQhEBj7ohSc6wAYA9AIBiOiFDYoAgEYEQhWNCIGdb45ybCoauo410T/UwS4rYknruOV14+JgghPMQSCzsIFAhoEEEbyADMI6BAxEkIVnNkIGIbC8qvbO9L5vlsfwTIyt91CNma/n8Ou9eYRDKOYcl1mt86X1a1Al6VwP/4vSsi8lbxNtegJHKzHMiJELI/V6/P03dK/+md9Pn77DSMkOba4U87XLdeDLR/iwfrpkon5gkqgjEoTYROuTpH3wOR9y0Bda+AUzfr0QhhQCW3Cr2O889zcO/lmTv76gv1I6jFMpEPUzPN4jwN/rvtnju4Ays7/jjsNgBrsIMmTZv9zpv735v4sYwHuZNUOJkGrBPgxUQGliwAb0ie+LPuKDut0oDHM4P3eZlwy0nQ08mw4UJukrvmkpjPvbg2TwBwcbpRM8pyKLPRXsokMjM+/iwRdcjsPwB1SRrCT8FBskHRwkGx0UnACsjLV4DWY4iVAwLiJsJCN8qBRUQu9jwv+muw8vhIsPvBRJOAU5KMHEu0I0lCo1XMOlqz0ndEHxG5UIGYcZXMFxwcJT4j4V5EKVg8PcYI/oWAVKWINnOkRvSUTI0UKpaUSne0JBPBcpRJWZM8OOyET/WsQG7MQ3/MT566aVAIYxZCBM1MM/bL9U7L5VrJ5HXIsCu5RL6ASqCD5ErEW6gr1S1KQruZQh2QtdZIgPNJSW66uIKERCeASHIUZd2zo83ML3gI8I4ZNLxDFjtL2skkblKYxV0IQ1IBhspEVtRDwr7MZl/JE9cRpnXAhoHJVzFC6S0AcCaQMlqcFirKzmw0Wc+ZIj0RAHJD0IvL1FQ8eV8IYxNC53NEH/eDxGzEpIZuwSb3y5w6iQHkkMbBDJkjTJk5yukpQSOSgHlHRJG3GRl5TJmSQRHaHJmyyR6IgOjoyIjYQQkOQFlyRJnCTKohzJHqEHSCAEQDDKpsyQmHTKqHzJaJNKqQQSxdDJS2EVfGQLXkQLrWkNdXyDZMA6eVw+gmSt7eNGTuxJxGjLtxSSrWTB+GtFAYzIiAgOSghIA3tH30NCZHzFjuTJwGvGuQTAuvzC2NqIiXSDULCSgcRIg1xLtqSJq9jJIZGw9yJHQLSwHtynwoiKU2gDfHrMbPRLmwNMrjFMD/RK7RivlUjKPACEiuTLizzNbTRLVVzNHQxEVyS/jVBH/zQgS24yTXn5yz4URxFqyIp6SAkkwMLIyzTwB9oUr+JEmONETiUax4LkzB3zzOf8CcYkQ9axztrBzuxcQs3kzuHrTCgEz4iICk1og3Ykz740TtRETzZUz7RkT+90z0b7iXn4BNksy90qT+Xhwz7kSglqTa/YjtaQhU7ogmS4GMi8zXhMzSRZUITQR10JwdYIDkeQTtWw0PvEzQxtkA09iA41mVl7D3TohDOwBOoknBK9TvzMTz/czP6MwOkbTH+IBUrwgippUfu80RPNUR1dT/BDzDj8yY0QUDcABASkICM1TxzNURU1CBY9F/R5D27QhC6wBRC0UgQ9z/zU0oHgUv8S9Z/3wIfzS4MDIVPbNFEMTdLRs0Um7U27fNFOMANGoNEuPVD6SdA1TFOCaNCuGMrCAFJN+IJpCFRBLVNCPVP0PFR3S9RzWVSBkIdPkNIq5JlBlSEsRdPd7MImFUG/+glZOAUqeIUCzRMbvVIkTdJLXdMKTaHpEgh8QAVC2AIqlVQ6PVI7rVVTdURUDcXDKIhu6AQwYARYrVJhnVVizVJj9cQ9jcQIqYZYcFQKZbRiOcW92cSnsdVMDZnXkIdUcIM0ANZ9FNU0qtTsLFdkhalLyQZNeIJR6DFbCdezGdedmVdsnRo37dUtYL19hZV+JZt/RUhrZUWBNapLOYdTsIL/QtiHW5WOd+UlWq3WNiy99iyon8gHYaAEIxhOjE1OfkVLCpPM3MxFh91Fep3GjXgHTy0D40JZQ5vUUeXYUvVYhzTHbP0JfriGUwACTqjInNVOcF1ZROO/g5S9gAXZtioMekCFPEgCIt1SDNO/pm3CW5zMl/1Z5gzakP0JcaDYP+hKhO3ayFRLlwU+qfXPYXmNkS3ZMV3b6FJZtwVbuI1amH1Gc+WjS3kHddUCUVJa/ZRWMyVVSwXcfBRcH4sOfrAGo2WEhwU8puXbpw3buH3cvJ1axfwJq8XabbjWzN3bC83IYh1brPoUYrEQgUBbK6CDU9XbhPVaN5Qahi0Zue1R//q5lJHthBrIv9NFM81V3ZZFUWf5XAaV2TT7CXZIBTPQAtGzXdTF3c3VQKiNMN9tzn5cCco12kLosMRNz8Wl1MaV1+bl0MgdTACYh13AWrw9zNtt2+R92+UFEO8t29eMDm44BSkog7I0387dEoXdXe5tL/59XdHdCHxwBkeogVAo0AL22wTU3htUYN1iYOby3+id3iaAB1i1YP1VnNw14azg3Ybp4EixDz7xB2k4BRpQWx371tSt09Xt2P1k2e78XejdCHmQXxwYThumL+TNYeW90xb+l/q6FAAWYGjdWrbNwwzOwg1OLSbGE9y4EADAh2OQYEsAWh+SVcbtWcdt3f+06l8AjQ520IUz6AHTHWNZ21l4VV/k1OI6eYzYBYAYTgUaoAMpdl77rWL87dsUvhz2XVH54+N0IIY8cIFaaNf6xd77TeL8XWJFnuJA5ONsuIUnuIJJpktCzj4ULlQlzOMzIQs3XYZPYIFCYIoSLuP0PeP1TeP4+l4+PodcAAMcoJtY5tpCvuRDzuRbNuL+5WN9iIY/LgNFDeZStmJFNODu1WQ1lSM+lodg2IMUoGBnpmJoNmTOveAFrmZEvaQuJtpbAIImIE1vJuWzjGZNxGLESuWi4WN6UAZI+ADy/cpnhudw3t5pJmdjLkfm0mVcYIMYoF/X/OZ/HmZxRmThKWf/TL0aPsmHZ0gFFjhcJ21oQUHgp1nhhKnn/eVjd/iFPQABSxBkPa1kYR5WHfZZHnbaFnzd2KVcXAACIpDjxHxnjzbleMXjibbgD65aZfiED6hdUOxpPPnonQlpexlp7IldcbgFNEgBU/jO48Xhl1Zi1pXpr2VpPbZojKaBJgBVpW5pcH7ogB5nDhbqf+FjdviFSJAAfvbRG85egNZggXZrgvbhFLVpa8CFJyDilXZdOkZfnqXWmN5O/gxrZ+HjeSgGVPiAAQ5BxEZirsZkr27sHubRieFjbsAFOdAATtAqf/bpeEZFvrYXTyCDOFAHhNiHQBiDOQCHgVgHOwiDOrjt/1RjAzOAhfNVzjyladB2y43IB2f4hRqI47Z257R2aM0m5tw5BJyyAU07CB5QgTPYgQ6IbXAoAVPzgVzChgn4gSxAAFUwj6iGQT6R60iAgEBmbf5Abab+6TveGxh4A36whwmgsoHYBwEgBQDgB7ELuhxQGjYoOwDAMyV17OKW6reMCH+gBlzwghR4BcV1ziPe6mmFacjhBwEIbgD4AWsziBBwPFBAAGYAABGoAzYYgwEHAD8DthBwcM9+bOPjE8lGBRDQgtE67Y62b9UW13lGFmYIAEvMAmgzCE9YgAUggIkDAAPAgDDgAgSwsReIA4HIBKQLCweQARvIARsgBYf7N/8zR/Nx8+Q5gIBCCLhvW7g057d5k3N+i/M6x/NxA7g85/M+JziEUzg/F3R4QzVwCABoIIkoiAKDMAcLYANtUIQFGHAH4AKSqAIfUDa2o7sICLMJmANBAPVg2YdRJ/VSN/VTR/VTr4ZUZ/VWd/VXh/VUx4dlCIYaIGJ9iPVc1/VdL/VV5/VfB/ZgF/ZhB3ZsqAdiR/ZkV/Zlh3V40AZmh/Zoh/WCQYBM8KYzoIlMMADCkAG2O4H9/oc1KLsiyALzmAPsZm+hNRGBYAdfgG84YKQgX+o6aWqceerMQAIh+AdVKIBeAIBNcDxmEAAbYwYHKLEzsIF6gAcY6DhEmABoqIf/F8g4PN3RHJcie1yJGPaFLwCBURAleYfu1NbrK55vb2EGFYgABGCDiHiDBY8DB+iABaiCiFCHHYiACOABiJsCBHAAHYhtil9SCL/44xYIR0aFFCCCcBhG3CNjjd2nU2ZLaFAHL3mRfXBLbFiHqyCHo7CQkDxJwqxKC8kGXAgECdADpQp7EQH7tGd7GoHKtod7FrHJuKfJq3QPiJFLv/7sQ7IQfGiGYEgCF1joWYRIl6vjjV3s53NLCFmlvP9q3Q3coZdwgdhlUYCAH0+sy2Y5py8poOZADdETj1xa2qt4oQdDjI8IfZAGX2gDCbhcgdT8b+H8wvL8z28PCT/um0n3/3PtyCAOhmVwAV/+HH6U/cN/+tpH5bfm+3UHAE82hAZoZqkr/KY3/s7Hb0NV/tO//Yigh2Ughi3QAI9PKabH7A4348S35cf/2EHMfcrPhVjQgCbQWunfcMNPbDuu5aDWe4uPJ9QHCAD5nv1qA+EPgIQKFzJsuPCfw4gSJ1KsCACbxYwaN0Lc6PEjyIzg9oWU+K9jyZQqV4a0Ry4hSpYyZ9KcGTOhBV4OT9a82LPiSZ4AgiZ898sZjhi2ZN78mRKjU5ZNo1KtOLKm0KpatyZ0CZMr2LAgm+bcOVUlVK5BO64FwO8aLkkNxPCzKdZj2rsWz+qlepVm1r6CW74cOviwXv+yOhsGlpm3atuhROkpM7ZFAyOsiCc+3vzQc9W/NvmCLu3VcOnUPRWb7dnZaWSiAM7hiuUCx7TVqhe+Bk16d0jRTH8DF3yaePHkJhuWZYwcr9bYRAf+6gOhDOzkvT0/V26VJODu3rceH29+L/PFDBuz3D5caFujx5qACBVV/F33iPGfZyhcKn/99VSegAV+xlBz6wVIkX4pSccTP9bgsooGTbRD1YJcNThYhv39txJ7BoJFoIgFsuaca7DJ1hY9xhiTRgMIQaZdch2e96FKIZaoFYk7nneigiliJZtkHXFTGw4uLBUdjcXZaB6ODj7po0c9UukdkA9NydCGHK04HUH/j0igxT1qNQnclspFWZKOV/5kpZtOptdaTV3u9SBE7BylhQSZgZXmT3YmFudEa4bUJqEzwZloalkqhGhIgk4UWZFuSbjKbbmFBaiQaDLqkKFjcfrpQouSupmjMI0qqVlZtTVPMcsA0gB2d41KE6ubnrpQqB9BuitIpgIrWKqSdXookW1lg8s1TWgwSl22nrnbrar16tGvw24krLbSIqielsf66ipR+DQTTCkaEFEOsdOqVm1q13rZrUzc0vvnnCjWiSy5PLHjyzNqxHgYvO3VqK28GmV7L0X2Mszkt3TiKmq/APgjjS+rJLkkh+42ijB4oz2cksMj35dvkPt62S9E//MQswwjY5a5n8elFQxawhktbHJDJfOsW8T6TqxwskQtew2ffqJas28gD/nzRz5DbVfQKQ8NFJGVmksMLLclYzPT3Dkd3tTbFnZz2QculGC4KlNEaVvn+BKNHrS+G/bSw+Z8Z9oZSd23gyi3fXVElBb5z8W+dOMsJ9TiTbPeIQ8HeEV/Uz6W4I+uOmnRPMkTzDKhgNADu44Xlyu+kT99uUSWs85R5qqKq3lgyuKSDRwN0HHw6bwDuzfWr0fkuvDoVT34TK+thVK5yhAzTA8pmKIc2k/5vivwb1cfJ/HFvx27sW6rWjtR5+QijSUVXkj94wSPLbL3pZ4dP2Dg76xRXv+Gr0UdO2VAUIh5thep650qe5yjn0K6h0DG2G9z44sJUd4RjGbUYgY4+FoA29cx1ZFtgQpcoNoUwjbNpUh/QXnLLbjxh+sISIAgQZ2ZOAg/BH4QhF85HgnrZMKTUMYY6biCBhrXQg22S4aT8+D8bBg4HMoOMNhoTPlwIY1RkKAH2zCQCz8Cw61ksS8GNEkXd1RDGxbrflh7opZOQh135G53Igoj/ghIqi8WDo4iGiMIy7ilteQvKBJshjZ6QILplciOFtkixH4nOQCBEI8L1COAeAKVtUQIF+LghPp8ZMiKIHJGRmQkEm+oxEM1MEdEwshaejgP/8lIk0Tsyya5Qkf/idHPkQiEJJuy9kSeiOMW1EjGDJTiplhyRo6fmqXQapnEUWKOieEbFxR3ucZCjClaVyKmRDqJofcdkYbLZCa2SglN8j1RT85QR9IShc2IaPNknwTROsdjS/rhkmjsCQo2JCSOV2SKUfFsSDuz805TNvKb4FSYOHWWtcPNYxfGoEfdauXPVw5qoFIqqCgPilBn7sxwh+PGLayxDcad6p9cMiajkGk1ZWZUo8ZbG7hyqL17nsRcwZCHJUZXOlKZlDcoTZRKkRe/ecavnmCEYlD09Ax9wIEBbtxVTxUS0J9EtSZBlSlLUeNSnSW0VU0Jij+okQtxTCNJofDHsKrqEzlZ/zSXGNXqVoHSVQUh9SSwWgY+YkamqbIVOHzVTFuR9Va1TpSjU/HoR3FhDQD4D4BrBZZa/1q/wIpqsHHlqmGdQ9N/2FQewIwBBh8LVYp6S5Gr82ZLL1u4riJ2Lf9aaiEgoAVrirakpBULYWVy1SaiFq6qXW1mH6jZxHXjHuk8aVpvqyvKjsuyv51UKRFbpHS8DB+vSAFuHCLZu/Wur9hbJDyd+1zgwtSrm/3H0QAQ0WxqK7I/JdRun5nV3BJKjyEKytbkUQ76CJGd7VVu6kzbwd7SN06pkm5QzBcNfYxCpwz6b3c9xVxsFdgzRPWeo1p7En1gDB0AaOMhIezX98Ypvv9mhNqFiwek5d0kKJ/D6zQE+YqMbPdjETadgGc43/EuJ2htIR96bwcAS0jAQnFM7o25m+Nu7pjHO0kP3NjCQ+elg7H/gw6SR+zdAoKXoKH0rZNTm5Os8STB5/PHZ0N7ZMgCOIZLBiWBw8zAh4xZs//IRzR8wQ4AxHa2LxTxbmosFW7CuclyfpSWRji+P+bDuH0qiaCblmQbvzm8X67wNR+o6JO8Zaz8uG52IQ1o1UQ6R4S2dJwPjei2sC0oqlSv3dAy6tSUeomV9nKqVX04EaonKOLwBTX4sd8gGizLgSaxm0yMacSkWHhZa44aCeKOfzSYdI6ZdWlqzaZT49rQh17/6JhP4o6j5OMfIE4etkGjbVJOeF6X1rWYefEPflDDkgCIMQlmTDjbThps7SaaeFWtGFgpgx5DLvL69s3TNnOR2xd9N+sWSrtkzllVFpBFSOvS2BNTZB/AaG9Ve0Fbmy07JMwwx2npRw5mpLZsLKZr56A7viWk4h0ASLN8WUKKEIz8UxznigOggaaSg4QHdkh5/N7wg5ZP7eUT/7HEDxSUe2CAF/6IbRmi9XOJ7Lzn/qxq0IeuLaMj3XtKZzrU4APBMhNFurQTigVmcQUJWEKoK+l6WsEudGoR/SNkHzD9zg7mpqsd5lKO+tuDcgEQ4KAX0Hg85CMv+clTvvKY6EDl/zOv+c1zvvOe/zzoQy96aDhAFaM/PepTr/rVez4Hb2A97GMv+9lX/gw+oD3uc695bVSq9w9cXs7TuLwbKIAABjg+8pOv/OUzv/kGAIDxnS/96VO/+ta/Pvazr/3mDwAA2/8++MMv/vFbnwDeJz/606/+9TMf+ux/P/yln0Mg/x74rVrIProM7/3zv/9PaztEACCLbZ3/FaABHiCFfUn92R0CNqADPiA0yZRHESAEVqAFXiAGZqAGbiAHdqAHfiBkIF6UtQkwCEIgRAQvmMEZsNxF8AEZnAEpeN3Q3VfnVA80GAIeRAQ0sAEZwAJDqIId8F4InsUEYlo9eAIeqILXCf9CGPCBQlSCHUShIdQRDQIZBRrIPoACHnjCQk3CGNSBPQRFPeBBGMzBLvVdAEWd/TkdQyiCAYSABTjELCzAFDBB2CkCD3BBFBSAE6ZhFbKd280ELBRACQhAz2HDBPxAFiAAKSgEPKBAAKjCEJ7X2mFaDnTAAqyBQ4RBB4yBCGRBXfjACwiBEIQBFR4WnlxhgTCBBURAFWTNHERAGLwAD5yEOqgADHDBD1QC4sVVAH4VIBpL1NUDACBCHDZEFiwdAOTAGDBEGOTAjwSfAKpdweSfKhhiQ7ABDCTEFCCBQnDBGAiAJG6TxDldIO4GMdqAJjIEORgAKADALBCA0PkAG3z/z2ZViiqWCDEywStmBT9gwAmCQwE04hnIQPjko4oBY+LxVkMYo0OowNEBABvYgDVpgw2EgQyKXcqQWfVcY8/xQDMCgCB0QEKAggnAgzhyUV010TkChzo2RCU4gEJ0wBT6gA5wgR3YwwFt5ACi4Urw4zMNQwCsw1DIwBsAgAq8wRuEASj84rcp5O8xJEM4ZENMACIkBB+IQEL0ggEEwAuoQwtB5TP1Ik14ZEPAwFECwCYgAADYAwpgAgCkZHSs5OG05G68JEPwQQgoxAvEAQCYAReQgQnAgP5JI/P0pMkAZVZ4QgHAhA9gpANYADgiQCOQpS+K5RryBVUyRAcIQkLU/8ELLAQz6MAPZCTfaRZLWqZMmCVDyEA9AsAkRAA/jEEVJIRcDiFdRSVCggVeLoQhYIBCmEAOKgQ5OMBV1pHQZGZi1mZWkEL3DYUOmAEATMAUDAUX5IBdatQ9AqBULsRmLoRrJkQY8MBURkBY6ss0cgVrLgQS1CYAzIEKAAAMFMDxBUABnCBu8uQACkhvKgQoFABR8sMCVAJDnEAdgBF6KufIKCZKaIMAfNw/CKd8nsFQvAEM8INqaudhOiV39uJmwkMYVAMAxIEJqIM2dIATggLKrUMR2MB5ys40ZuhqYmNCnIEPKkIEQIM9wACFLsRtYkjhCSP5FEhvNsJwisBR4tOBBdTFJiSEIRDAOEpMMBaejDIKg4JCWtoAFwAAJhiANvyDhcKDPdhAFvjec70cNabpWTBD8pGnNhSAD9aDDyzAAjABSbCBAUSAAchAL/wI8KlpXe6mQzgA8oXmdDZCQlQBAjhADoDlQiCADwIp1EUlAxYHFyQfopIBeXLpBLQiogJAByxqBESknbEdpdofw7xB8tXBPxxoQqhCCEzAAuDBScipA0RADpzhAS4UAWoDODAENBTmG1UVOaxbcwELPwQrr2wRTQkPNBCjUIADWApqfwQEADs=" class="card-img-top figure-img" alt="A Figure">
|
||
<figcaption class="figure-caption"><div class="markdown"><p>The tangent line is the best linear approximation to the function at the point \((c, f(c))\). As the viewing window zooms in on \((c,f(c))\) we can see how the graph and its tangent line get more similar.</p>
|
||
</div> </figcaption>
|
||
</figure>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>The tangent line is not just a line that intersects the graph in one point, nor does it need only intersect the line in just one point.</p>
|
||
<div class="callout-note callout callout-style-default callout-captioned">
|
||
<div class="callout-header d-flex align-content-center">
|
||
<div class="callout-icon-container">
|
||
<i class="callout-icon"></i>
|
||
</div>
|
||
<div class="callout-caption-container flex-fill">
|
||
Note
|
||
</div>
|
||
</div>
|
||
<div class="callout-body-container callout-body">
|
||
<p>This last point was certainly not obvious at first. <a href="http://www.maa.org/sites/default/files/0746834234133.di020795.02p0640b.pdf">Barrow</a>, who had Newton as a pupil, and was the first to sketch a proof of part of the Fundamental Theorem of Calculus, understood a tangent line to be a line that intersects a curve at only one point.</p>
|
||
</div>
|
||
</div>
|
||
<section id="example" class="level5">
|
||
<h5 class="anchored" data-anchor-id="example">Example</h5>
|
||
<p>What is the slope of the tangent line to <span class="math inline">\(f(x) = \sin(x)\)</span> at <span class="math inline">\(c=0\)</span>?</p>
|
||
<p>We need to compute the limit <span class="math inline">\((\sin(c+h) - \sin(c))/h\)</span> which is the limit as <span class="math inline">\(h\)</span> goes to <span class="math inline">\(0\)</span> of <span class="math inline">\(\sin(h)/h.\)</span> We know this to be <span class="math inline">\(1.\)</span></p>
|
||
<div class="cell" data-hold="true" data-execution_count="18">
|
||
<div class="sourceCode cell-code" id="cb19"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb19-1"><a href="#cb19-1" aria-hidden="true" tabindex="-1"></a><span class="fu">f</span>(x) <span class="op">=</span> <span class="fu">sin</span>(x)</span>
|
||
<span id="cb19-2"><a href="#cb19-2" aria-hidden="true" tabindex="-1"></a>c <span class="op">=</span> <span class="fl">0</span></span>
|
||
<span id="cb19-3"><a href="#cb19-3" aria-hidden="true" tabindex="-1"></a><span class="fu">tl</span>(x) <span class="op">=</span> <span class="fu">f</span>(c) <span class="op">+</span> <span class="fl">1</span> <span class="op">*</span> (x <span class="op">-</span> c)</span>
|
||
<span id="cb19-4"><a href="#cb19-4" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(f, <span class="op">-</span><span class="cn">pi</span><span class="op">/</span><span class="fl">2</span>, <span class="cn">pi</span><span class="op">/</span><span class="fl">2</span>)</span>
|
||
<span id="cb19-5"><a href="#cb19-5" aria-hidden="true" tabindex="-1"></a><span class="fu">plot!</span>(tl, <span class="op">-</span><span class="cn">pi</span><span class="op">/</span><span class="fl">2</span>, <span class="cn">pi</span><span class="op">/</span><span class="fl">2</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="19">
|
||
<p><img src="derivatives_files/figure-html/cell-19-output-1.svg" class="img-fluid"></p>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
</section>
|
||
<section id="the-derivative" class="level2" data-number="22.3">
|
||
<h2 data-number="22.3" class="anchored" data-anchor-id="the-derivative"><span class="header-section-number">22.3</span> The derivative</h2>
|
||
<p>The limit of the slope of the secant line gives an operation: for each <span class="math inline">\(c\)</span> in the domain of <span class="math inline">\(f\)</span> there is a number (the slope of the tangent line) or it does not exist. That is, there is a derived function from <span class="math inline">\(f\)</span>. Call this function the <em>derivative</em> of <span class="math inline">\(f\)</span>.</p>
|
||
<p>There are many notations for the derivative, mostly we use the “prime” notation:</p>
|
||
<p><span class="math display">\[
|
||
f'(x) = \lim_{h \rightarrow 0} \frac{f(x+h) - f(x)}{h}.
|
||
\]</span></p>
|
||
<p>The limit above is identical, only it uses <span class="math inline">\(x\)</span> instead of <span class="math inline">\(c\)</span> to emphasize that we are thinking of a function now, and not just a value at a point.</p>
|
||
<p>The derivative is related to a function, but at times it is more convenient to write only the expression defining the rule of the function. In that case, we use this notation for the derivative <span class="math inline">\([\text{expression}]'\)</span>.</p>
|
||
<section id="some-basic-derivatives" class="level3" data-number="22.3.1">
|
||
<h3 data-number="22.3.1" class="anchored" data-anchor-id="some-basic-derivatives"><span class="header-section-number">22.3.1</span> Some basic derivatives</h3>
|
||
<ul>
|
||
<li><strong>The power rule</strong>. What is the derivative of the monomial <span class="math inline">\(f(x) = x^n\)</span>? We need to look at <span class="math inline">\((x+h)^n - x^n\)</span> for positive, integer-value <span class="math inline">\(n\)</span>. Let’s look at a case, <span class="math inline">\(n=5\)</span></li>
|
||
</ul>
|
||
<div class="cell" data-execution_count="19">
|
||
<div class="sourceCode cell-code" id="cb20"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb20-1"><a href="#cb20-1" aria-hidden="true" tabindex="-1"></a><span class="pp">@syms</span> x<span class="op">::</span><span class="dt">real </span>h<span class="op">::</span><span class="dt">real</span></span>
|
||
<span id="cb20-2"><a href="#cb20-2" aria-hidden="true" tabindex="-1"></a>n <span class="op">=</span> <span class="fl">5</span></span>
|
||
<span id="cb20-3"><a href="#cb20-3" aria-hidden="true" tabindex="-1"></a>ex <span class="op">=</span> <span class="fu">expand</span>((x<span class="op">+</span>h)<span class="op">^</span>n <span class="op">-</span> x<span class="op">^</span>n)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="20">
|
||
<span class="math-left-align" style="padding-left: 4px; width:0; float:left;">
|
||
\[
|
||
h^{5} + 5 h^{4} x + 10 h^{3} x^{2} + 10 h^{2} x^{3} + 5 h x^{4}
|
||
\]
|
||
</span>
|
||
</div>
|
||
</div>
|
||
<p>All terms have an <code>h</code> in them, so we cancel it out:</p>
|
||
<div class="cell" data-execution_count="20">
|
||
<div class="sourceCode cell-code" id="cb21"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb21-1"><a href="#cb21-1" aria-hidden="true" tabindex="-1"></a><span class="fu">cancel</span>(ex<span class="op">/</span>h, h)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="21">
|
||
<span class="math-left-align" style="padding-left: 4px; width:0; float:left;">
|
||
\[
|
||
h^{4} + 5 h^{3} x + 10 h^{2} x^{2} + 10 h x^{3} + 5 x^{4}
|
||
\]
|
||
</span>
|
||
</div>
|
||
</div>
|
||
<p>We see the lone term <code>5x^4</code> without an <span class="math inline">\(h\)</span>, so as we let <span class="math inline">\(h\)</span> go to <span class="math inline">\(0\)</span>, this will be the limit. That is, <span class="math inline">\(f'(x) = 5x^4\)</span>.</p>
|
||
<p>For integer-valued, positive, <span class="math inline">\(n\)</span>, the binomial theorem gives an expansion <span class="math inline">\((x+h)^n = x^n + nx^{n-1}\cdot h^1 + n\cdot(n-1)x^{n-2}\cdot h^2 + \cdots\)</span>. Subtracting <span class="math inline">\(x^n\)</span> then dividing by <span class="math inline">\(h\)</span> leaves just the term <span class="math inline">\(nx^{n-1}\)</span> without a power of <span class="math inline">\(h\)</span>, so the limit, in general, is just this term. That is:</p>
|
||
<p><span class="math display">\[
|
||
[x^n]' = nx^{n-1}.
|
||
\]</span></p>
|
||
<p>It isn’t a special case, but when <span class="math inline">\(n=0\)</span>, we also have the above formula applies, as <span class="math inline">\(x^0\)</span> is the constant <span class="math inline">\(1\)</span>, and all constant functions will have a derivative of <span class="math inline">\(0\)</span> at all <span class="math inline">\(x\)</span>. We will see that in general, the power rule applies for any <span class="math inline">\(n\)</span> where <span class="math inline">\(x^n\)</span> is defined.</p>
|
||
<ul>
|
||
<li>What is the derivative of <span class="math inline">\(f(x) = \sin(x)\)</span>? We know that <span class="math inline">\(f'(0)= 1\)</span> by the earlier example with <span class="math inline">\((\sin(0+h)-\sin(0))/h = \sin(h)/h\)</span>, here we solve in general.</li>
|
||
</ul>
|
||
<p>We need to consider the difference <span class="math inline">\(\sin(x+h) - \sin(x)\)</span>:</p>
|
||
<div class="cell" data-execution_count="21">
|
||
<div class="sourceCode cell-code" id="cb22"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb22-1"><a href="#cb22-1" aria-hidden="true" tabindex="-1"></a>sympy.<span class="fu">expand_trig</span>(<span class="fu">sin</span>(x<span class="op">+</span>h) <span class="op">-</span> <span class="fu">sin</span>(x)) <span class="co"># expand_trig is not exposed in `SymPy`</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="22">
|
||
<span class="math-left-align" style="padding-left: 4px; width:0; float:left;">
|
||
\[
|
||
\sin{\left(h \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(h \right)} - \sin{\left(x \right)}
|
||
\]
|
||
</span>
|
||
</div>
|
||
</div>
|
||
<p>That used the formula <span class="math inline">\(\sin(x+h) = \sin(x)\cos(h) + \sin(h)\cos(x)\)</span>.</p>
|
||
<p>We could then rearrange the secant line slope formula to become:</p>
|
||
<p><span class="math display">\[
|
||
\cos(x) \cdot \frac{\sin(h)}{h} + \sin(x) \cdot \frac{\cos(h) - 1}{h}
|
||
\]</span></p>
|
||
<p>and take a limit. If the answer isn’t clear, we can let <code>SymPy</code> do this work:</p>
|
||
<div class="cell" data-execution_count="22">
|
||
<div class="sourceCode cell-code" id="cb23"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb23-1"><a href="#cb23-1" aria-hidden="true" tabindex="-1"></a><span class="fu">limit</span>((<span class="fu">sin</span>(x<span class="op">+</span>h) <span class="op">-</span> <span class="fu">sin</span>(x))<span class="op">/</span> h, h <span class="op">=></span> <span class="fl">0</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="23">
|
||
<span class="math-left-align" style="padding-left: 4px; width:0; float:left;">
|
||
\[
|
||
\cos{\left(x \right)}
|
||
\]
|
||
</span>
|
||
</div>
|
||
</div>
|
||
<p>From the formula <span class="math inline">\([\sin(x)]' = \cos(x)\)</span> we can easily get the <em>slope</em> of the tangent line to <span class="math inline">\(f(x) = \sin(x)\)</span> at <span class="math inline">\(x=0\)</span> by simply evaluating <span class="math inline">\(\cos(0) = 1\)</span>.</p>
|
||
<ul>
|
||
<li>Let’s see what the derivative of <span class="math inline">\(\ln(x) = \log(x)\)</span> is (using base <span class="math inline">\(e\)</span> for <span class="math inline">\(\log\)</span> unless otherwise indicated). We have</li>
|
||
</ul>
|
||
<p><span class="math display">\[
|
||
\frac{\log(x+h) - \log(x)}{h} = \frac{1}{h}\log(\frac{x+h}{x}) = \log((1+h/x)^{1/h}).
|
||
\]</span></p>
|
||
<p>As noted earlier, Cauchy saw the limit as <span class="math inline">\(u\)</span> goes to <span class="math inline">\(0\)</span> of <span class="math inline">\(f(u) = (1 + u)^{1/u}\)</span> is <span class="math inline">\(e\)</span>. Re-expressing the above we can get <span class="math inline">\(1/h \cdot \log(f(h/x))\)</span>. The limit as <span class="math inline">\(h\)</span> goes to <span class="math inline">\(0\)</span> of this is found from the composition rules for limits: as <span class="math inline">\(\lim_{h \rightarrow 0} f(h/x) = e^{1/x}\)</span>, and since <span class="math inline">\(\log(x)\)</span> is continuous at <span class="math inline">\(e^{1/x}\)</span> we get this expression has a limit of <span class="math inline">\(1/x\)</span>.</p>
|
||
<p>We verify through:</p>
|
||
<div class="cell" data-execution_count="23">
|
||
<div class="sourceCode cell-code" id="cb24"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb24-1"><a href="#cb24-1" aria-hidden="true" tabindex="-1"></a><span class="fu">limit</span>((<span class="fu">log</span>(x<span class="op">+</span>h) <span class="op">-</span> <span class="fu">log</span>(x))<span class="op">/</span>h, h <span class="op">=></span> <span class="fl">0</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="24">
|
||
<span class="math-left-align" style="padding-left: 4px; width:0; float:left;">
|
||
\[
|
||
\frac{1}{x}
|
||
\]
|
||
</span>
|
||
</div>
|
||
</div>
|
||
<ul>
|
||
<li>The derivative of <span class="math inline">\(f(x) = e^x\)</span> can also be done from a limit. We have</li>
|
||
</ul>
|
||
<p><span class="math display">\[
|
||
\frac{e^{x+h} - e^x}{h} = \frac{e^x \cdot(e^h -1)}{h}.
|
||
\]</span></p>
|
||
<p>Earlier, we saw that <span class="math inline">\(\lim_{h \rightarrow 0}(e^h - 1)/h = 1\)</span>. With this, we get <span class="math inline">\([e^x]' = e^x\)</span>, that is it is a function satisfying <span class="math inline">\(f'=f\)</span>.</p>
|
||
<hr>
|
||
<p>There are several different <a href="http://en.wikipedia.org/wiki/Notation_for_differentiation">notations</a> for derivatives. Some are historical, some just add flexibility. We use the prime notation of Lagrange: <span class="math inline">\(f'(x)\)</span>, <span class="math inline">\(u'\)</span> and <span class="math inline">\([\text{expr}]'\)</span>, where the first emphasizes that the derivative is a function with a value at <span class="math inline">\(x\)</span>, the second emphasizes the derivative operates on functions, the last emphasizes that we are taking the derivative of some expression.</p>
|
||
<p>There are many other notations:</p>
|
||
<ul>
|
||
<li>The Leibniz notation uses the infinitesimals: <span class="math inline">\(dy/dx\)</span> to relate to <span class="math inline">\(\Delta y/\Delta x\)</span>. This notation is very common, and especially useful when more than one variable is involved. <code>SymPy</code> uses Leibniz notation in some of its output, expressing somethings such as:</li>
|
||
</ul>
|
||
<p><span class="math display">\[
|
||
f'(x) = \frac{d}{d\xi}(f(\xi)) \big|_{\xi=x}.
|
||
\]</span></p>
|
||
<p>The notation - <span class="math inline">\(\big|\)</span> - on the right-hand side separates the tasks of finding the derivative and evaluating the derivative at a specific value.</p>
|
||
<ul>
|
||
<li>Euler used <code>D</code> for the operator <code>D(f)</code>. This was initially used by <a href="http://jeff560.tripod.com/calculus.html">Argobast</a>. The notation <code>D(f)(c)</code> would be needed to evaluate the derivative at a point.</li>
|
||
<li>Newton used a “dot” above the variable, <span class="math inline">\(\dot{x}(t)\)</span>, which is still widely used in physics to indicate a derivative in time. This inidicates take the derivative and then plug in <span class="math inline">\(t\)</span>.</li>
|
||
<li>The notation <span class="math inline">\([expr]'(c)\)</span> or <span class="math inline">\([expr]'\big|_{x=c}\)</span>would similarly mean, take the derivative of the expression and <strong>then</strong> evaluate at <span class="math inline">\(c\)</span>.</li>
|
||
</ul>
|
||
</section>
|
||
</section>
|
||
<section id="rules-of-derivatives" class="level2" data-number="22.4">
|
||
<h2 data-number="22.4" class="anchored" data-anchor-id="rules-of-derivatives"><span class="header-section-number">22.4</span> Rules of derivatives</h2>
|
||
<p>We could proceed in a similar manner – using limits to find other derivatives, but let’s not. If we have a function <span class="math inline">\(f(x) = x^5 \sin(x)\)</span>, it would be nice to leverage our previous work on the derivatives of <span class="math inline">\(f(x) =x^5\)</span> and <span class="math inline">\(g(x) = \sin(x)\)</span>, rather than derive an answer from scratch.</p>
|
||
<p>As with limits and continuity, it proves very useful to consider rules that make the process of finding derivatives of combinations of functions a matter of combining derivatives of the individual functions in some manner.</p>
|
||
<p>We already have one such rule:</p>
|
||
<section id="power-rule" class="level3" data-number="22.4.1">
|
||
<h3 data-number="22.4.1" class="anchored" data-anchor-id="power-rule"><span class="header-section-number">22.4.1</span> Power rule</h3>
|
||
<p>We have seen for integer <span class="math inline">\(n \geq 0\)</span> the formula:</p>
|
||
<p><span class="math display">\[
|
||
[x^n]' = n x^{n-1}.
|
||
\]</span></p>
|
||
<p>This will be shown true for all real exponents.</p>
|
||
</section>
|
||
<section id="sum-rule" class="level3" data-number="22.4.2">
|
||
<h3 data-number="22.4.2" class="anchored" data-anchor-id="sum-rule"><span class="header-section-number">22.4.2</span> Sum rule</h3>
|
||
<p>Let’s consider <span class="math inline">\(k(x) = a\cdot f(x) + b\cdot g(x)\)</span>, what is its derivative? That is, in terms of <span class="math inline">\(f\)</span>, <span class="math inline">\(g\)</span> and their derivatives, can we express <span class="math inline">\(k'(x)\)</span>?</p>
|
||
<p>We can rearrange <span class="math inline">\((k(x+h) - k(x))\)</span> as follows:</p>
|
||
<p><span class="math display">\[
|
||
(a\cdot f(x+h) + b\cdot g(x+h)) - (a\cdot f(x) + b \cdot g(x)) =
|
||
a\cdot (f(x+h) - f(x)) + b \cdot (g(x+h) - g(x)).
|
||
\]</span></p>
|
||
<p>Dividing by <span class="math inline">\(h\)</span>, we see that this becomes</p>
|
||
<p><span class="math display">\[
|
||
a\cdot \frac{f(x+h) - f(x)}{h} + b \cdot \frac{g(x+h) - g(x)}{h} \rightarrow a\cdot f'(x) + b\cdot g'(x).
|
||
\]</span></p>
|
||
<p>That is <span class="math inline">\([a\cdot f(x) + b \cdot g(x)]' = a\cdot f'(x) + b\cdot g'(x)\)</span>.</p>
|
||
<p>This holds two rules: the derivative of a constant times a function is the constant times the derivative of the function; and the derivative of a sum of functions is the sum of the derivative of the functions.</p>
|
||
<p>This example shows a useful template:</p>
|
||
<p><span class="math display">\[
|
||
\begin{align*}
|
||
[2x^2 - \frac{x}{3} + 3e^x]' & = 2[\square]' - \frac{[\square]]}{3} + 3[\square]'\\
|
||
&= 2[x^2]' - \frac{[x]'}{3} + 3[e^x]'\\
|
||
&= 2(2x) - \frac{1}{3} + 3e^x\\
|
||
&= 4x - \frac{1}{3} + 3e^x
|
||
\end{align*}
|
||
\]</span></p>
|
||
</section>
|
||
<section id="product-rule" class="level3" data-number="22.4.3">
|
||
<h3 data-number="22.4.3" class="anchored" data-anchor-id="product-rule"><span class="header-section-number">22.4.3</span> Product rule</h3>
|
||
<p>Other rules can be similarly derived. <code>SymPy</code> can give us them as well. Here we define two symbolic functions <code>u</code> and <code>v</code> and let <code>SymPy</code> derive a formula for the derivative of a product of functions:</p>
|
||
<div class="cell" data-hold="true" data-execution_count="24">
|
||
<div class="sourceCode cell-code" id="cb25"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb25-1"><a href="#cb25-1" aria-hidden="true" tabindex="-1"></a><span class="pp">@syms</span> <span class="fu">u</span>() <span class="fu">v</span>()</span>
|
||
<span id="cb25-2"><a href="#cb25-2" aria-hidden="true" tabindex="-1"></a><span class="fu">f</span>(x) <span class="op">=</span> <span class="fu">u</span>(x) <span class="op">*</span> <span class="fu">v</span>(x)</span>
|
||
<span id="cb25-3"><a href="#cb25-3" aria-hidden="true" tabindex="-1"></a><span class="fu">limit</span>((<span class="fu">f</span>(x<span class="op">+</span>h) <span class="op">-</span> <span class="fu">f</span>(x))<span class="op">/</span>h, h <span class="op">=></span> <span class="fl">0</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="25">
|
||
<span class="math-left-align" style="padding-left: 4px; width:0; float:left;">
|
||
\[
|
||
u{\left(x \right)} \left. \frac{d}{d \xi_{1}} v{\left(\xi_{1} \right)} \right|_{\substack{ \xi_{1}=x }} + v{\left(x \right)} \left. \frac{d}{d \xi_{1}} u{\left(\xi_{1} \right)} \right|_{\substack{ \xi_{1}=x }}
|
||
\]
|
||
</span>
|
||
</div>
|
||
</div>
|
||
<p>The output uses the Leibniz notation to represent that the derivative of <span class="math inline">\(u(x) \cdot v(x)\)</span> is the <span class="math inline">\(u\)</span> times the derivative of <span class="math inline">\(v\)</span> evaluated at <span class="math inline">\(x\)</span> plus <span class="math inline">\(v\)</span> times the derivative of <span class="math inline">\(u\)</span> evaluated at <span class="math inline">\(x\)</span>. A common shorthand is <span class="math inline">\([uv]' = u'v + uv'\)</span>.</p>
|
||
<p>This example shows a useful template for the product rule:</p>
|
||
<p><span class="math display">\[
|
||
\begin{align*}
|
||
[(x^2+1)\cdot e^x]' &= [\square]' \cdot (\square) + (\square) \cdot [\square]'\\
|
||
&= [x^2 + 1]' \cdot (e^x) + (x^2+1) \cdot [e^x]'\\
|
||
&= (2x)\cdot e^x + (x^2+1)\cdot e^x
|
||
\end{align*}
|
||
\]</span></p>
|
||
</section>
|
||
<section id="quotient-rule" class="level3" data-number="22.4.4">
|
||
<h3 data-number="22.4.4" class="anchored" data-anchor-id="quotient-rule"><span class="header-section-number">22.4.4</span> Quotient rule</h3>
|
||
<p>The derivative of <span class="math inline">\(f(x) = u(x)/v(x)\)</span> - a ratio of functions - can be similarly computed. The result will be <span class="math inline">\([u/v]' = (u'v - uv')/u^2\)</span>:</p>
|
||
<div class="cell" data-hold="true" data-execution_count="25">
|
||
<div class="sourceCode cell-code" id="cb26"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb26-1"><a href="#cb26-1" aria-hidden="true" tabindex="-1"></a><span class="pp">@syms</span> <span class="fu">u</span>() <span class="fu">v</span>()</span>
|
||
<span id="cb26-2"><a href="#cb26-2" aria-hidden="true" tabindex="-1"></a><span class="fu">f</span>(x) <span class="op">=</span> <span class="fu">u</span>(x) <span class="op">/</span> <span class="fu">v</span>(x)</span>
|
||
<span id="cb26-3"><a href="#cb26-3" aria-hidden="true" tabindex="-1"></a><span class="fu">limit</span>((<span class="fu">f</span>(x<span class="op">+</span>h) <span class="op">-</span> <span class="fu">f</span>(x))<span class="op">/</span>h, h <span class="op">=></span> <span class="fl">0</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="26">
|
||
<span class="math-left-align" style="padding-left: 4px; width:0; float:left;">
|
||
\[
|
||
\frac{- u{\left(x \right)} \left. \frac{d}{d \xi_{1}} v{\left(\xi_{1} \right)} \right|_{\substack{ \xi_{1}=x }} + v{\left(x \right)} \left. \frac{d}{d \xi_{1}} u{\left(\xi_{1} \right)} \right|_{\substack{ \xi_{1}=x }}}{v^{2}{\left(x \right)}}
|
||
\]
|
||
</span>
|
||
</div>
|
||
</div>
|
||
<p>This example shows a useful template for the quotient rule:</p>
|
||
<p><span class="math display">\[
|
||
\begin{align*}
|
||
[\frac{x^2+1}{e^x}]' &= \frac{[\square]' \cdot (\square) - (\square) \cdot [\square]'}{(\square)^2}\\
|
||
&= \frac{[x^2 + 1]' \cdot (e^x) - (x^2+1) \cdot [e^x]'}{(e^x)^2}\\
|
||
&= \frac{(2x)\cdot e^x - (x^2+1)\cdot e^x}{e^{2x}}
|
||
\end{align*}
|
||
\]</span></p>
|
||
<section id="examples" class="level5">
|
||
<h5 class="anchored" data-anchor-id="examples">Examples</h5>
|
||
<p>Compute the derivative of <span class="math inline">\(f(x) = (1 + \sin(x)) + (1 + x^2)\)</span>.</p>
|
||
<p>As written we can identify <span class="math inline">\(f(x) = u(x) + v(x)\)</span> with <span class="math inline">\(u=(1 + \sin(x))\)</span>, <span class="math inline">\(v=(1 + x^2)\)</span>. The sum rule immediately applies to give:</p>
|
||
<p><span class="math display">\[
|
||
f'(x) = (\cos(x)) + (2x).
|
||
\]</span></p>
|
||
<hr>
|
||
<p>Compute the derivative of <span class="math inline">\(f(x) = (1 + \sin(x)) \cdot (1 + x^2)\)</span>.</p>
|
||
<p>The same <span class="math inline">\(u\)</span> and <span class="math inline">\(v\)</span> my be identified. The product rule readily applies to yield:</p>
|
||
<p><span class="math display">\[
|
||
f'(x) = u'v + uv' = \cos(x) \cdot (1 + x^2) + (1 + \sin(x)) \cdot (2x).
|
||
\]</span></p>
|
||
<hr>
|
||
<p>Compute the derivative of <span class="math inline">\(f(x) = (1 + \sin(x)) / (1 + x^2)\)</span>.</p>
|
||
<p>The same <span class="math inline">\(u\)</span> and <span class="math inline">\(v\)</span> my be identified. The quotient rule readily applies to yield:</p>
|
||
<p><span class="math display">\[
|
||
f'(x) = u'v - uv' = \frac{\cos(x) \cdot (1 + x^2) - (1 + \sin(x)) \cdot (2x)}{(1+x^2)^2}.
|
||
\]</span></p>
|
||
<hr>
|
||
<p>Compute the derivative of <span class="math inline">\(f(x) = (x-1) \cdot (x-2)\)</span>.</p>
|
||
<p>This can be done using the product rule <em>or</em> by expanding the polynomial and using the power and sum rule. As this polynomial is easy to expand, we do both and compare:</p>
|
||
<p><span class="math display">\[
|
||
[(x-1)(x-2)]' = [x^2 - 3x + 2]' = 2x -3.
|
||
\]</span></p>
|
||
<p>Whereas the product rule gives:</p>
|
||
<p><span class="math display">\[
|
||
[(x-1)(x-2)]' = 1\cdot (x-2) + (x-1)\cdot 1 = 2x - 3.
|
||
\]</span></p>
|
||
<hr>
|
||
<p>Find the derivative of <span class="math inline">\(f(x) = (x-1)(x-2)(x-3)(x-4)(x-5)\)</span>.</p>
|
||
<p>We could expand this, as above, but without computer assistance the potential for error is high. Instead we will use the product rule on the product of <span class="math inline">\(5\)</span> terms.</p>
|
||
<p>Let’s first treat the case of <span class="math inline">\(3\)</span> products:</p>
|
||
<p><span class="math display">\[
|
||
[u\cdot v\cdot w]' =[ u \cdot (vw)]' = u' (vw) + u [vw]' = u'(vw) + u[v' w + v w'] =
|
||
u' vw + u v' w + uvw'.
|
||
\]</span></p>
|
||
<p>This pattern generalizes, clearly, to:</p>
|
||
<p><span class="math display">\[
|
||
[f_1\cdot f_2 \cdots f_n]' = f_1' f_2 \cdots f_n + f_1 \cdot f_2' \cdot (f_3 \cdots f_n) + \dots +
|
||
f_1 \cdots f_{n-1} \cdot f_n'.
|
||
\]</span></p>
|
||
<p>There are <span class="math inline">\(n\)</span> terms, each where one of the <span class="math inline">\(f_i\)</span>s have a derivative. Were we to multiply top and bottom by <span class="math inline">\(f_i\)</span>, we would get each term looks like: <span class="math inline">\(f \cdot f_i'/f_i\)</span>.</p>
|
||
<p>With this, we can proceed. Each term <span class="math inline">\(x-i\)</span> has derivative <span class="math inline">\(1\)</span>, so the answer to <span class="math inline">\(f'(x)\)</span>, with <span class="math inline">\(f\)</span> as above, is <span class="math inline">\(f'(x) = f(x)/(x-1) + f(x)/(x-2) + f(x)/(x-3) + f(x)/(x-4) + f(x)/(x-5)\)</span>, that is:</p>
|
||
<p><span class="math display">\[
|
||
f'(x) = (x-2)(x-3)(x-4)(x-5) + (x-1)(x-3)(x-4)(x-5) + (x-1)(x-2)(x-4)(x-5) + (x-1)(x-2)(x-3)(x-5) + (x-1)(x-2)(x-3)(x-4).
|
||
\]</span></p>
|
||
<hr>
|
||
<p>Find the derivative of <span class="math inline">\(x\sin(x)\)</span> evaluated at <span class="math inline">\(\pi\)</span>.</p>
|
||
<p><span class="math display">\[
|
||
[x\sin(x)]'\big|_{x=\pi} = (1\sin(x) + x\cos(x))\big|_{x=\pi} = (\sin(\pi) + \pi \cdot \cos(\pi)) = -\pi.
|
||
\]</span></p>
|
||
</section>
|
||
</section>
|
||
<section id="chain-rule" class="level3" data-number="22.4.5">
|
||
<h3 data-number="22.4.5" class="anchored" data-anchor-id="chain-rule"><span class="header-section-number">22.4.5</span> Chain rule</h3>
|
||
<p>Finally, the derivative of a composition of functions can be computed using pieces of each function. This gives a rule called the <em>chain rule</em>. Before deriving, let’s give a slight motivation.</p>
|
||
<p>Consider the output of a factory for some widget. It depends on two steps: an initial manufacturing step and a finishing step. The number of employees is important in how much is initially manufactured. Suppose <span class="math inline">\(x\)</span> is the number of employees and <span class="math inline">\(g(x)\)</span> is the amount initially manufactured. Adding more employees increases the amount made by the made-up rule <span class="math inline">\(g(x) = \sqrt{x}\)</span>. The finishing step depends on how much is made by the employees. If <span class="math inline">\(y\)</span> is the amount made, then <span class="math inline">\(f(y)\)</span> is the number of widgets finished. Suppose for some reason that <span class="math inline">\(f(y) = y^2.\)</span></p>
|
||
<p>How many widgets are made as a function of employees? The composition <span class="math inline">\(u(x) = f(g(x))\)</span> would provide that. Changes in the initial manufacturing step lead to changes in how much is initially made; changes in the initial amount made leads to changes in the finished products. Each change contributes to the overall change.</p>
|
||
<p>What is the effect of adding employees on the rate of output of widgets? In this specific case we know the answer, as <span class="math inline">\((f \circ g)(x) = x\)</span>, so the answer is just the rate is <span class="math inline">\(1\)</span>.</p>
|
||
<p>In general, we want to express <span class="math inline">\(\Delta f / \Delta x\)</span> in a form so that we can take a limit.</p>
|
||
<p>But what do we know? We know <span class="math inline">\(\Delta g / \Delta x\)</span> and <span class="math inline">\(\Delta f/\Delta y\)</span>. Using <span class="math inline">\(y=g(x)\)</span>, this suggests that we might have luck with the right side of this equation:</p>
|
||
<p><span class="math display">\[
|
||
\frac{\Delta f}{\Delta x} = \frac{\Delta f}{\Delta y} \cdot \frac{\Delta y}{\Delta x}.
|
||
\]</span></p>
|
||
<p>Interpreting this, we get the <em>average</em> rate of change in the composition can be thought of as a product: The <em>average</em> rate of change of the initial step (<span class="math inline">\(\Delta y/ \Delta x\)</span>) times the <em>average</em> rate of the change of the second step evaluated not at <span class="math inline">\(x\)</span>, but at <span class="math inline">\(y\)</span>, <span class="math inline">\(\Delta f/ \Delta y\)</span>.</p>
|
||
<p>Re-expressing using derivative notation with <span class="math inline">\(h\)</span> would be:</p>
|
||
<p><span class="math display">\[
|
||
\frac{f(g(x+h)) - f(g(x))}{h} = \frac{f(g(x+h)) - f(g(x))}{g(x+h) - g(x)} \cdot \frac{g(x+h) - g(x)}{h}.
|
||
\]</span></p>
|
||
<p>The left hand side will converge to the derivative of <span class="math inline">\(u(x)\)</span> or <span class="math inline">\([f(g(x))]'\)</span>.</p>
|
||
<p>The right most part of the right side would have a limit <span class="math inline">\(g'(x)\)</span>, were we to let <span class="math inline">\(h\)</span> go to <span class="math inline">\(0\)</span>.</p>
|
||
<p>It isn’t obvious, but the left part of the right side has the limit <span class="math inline">\(f'(g(x))\)</span>. This would be clear if <em>only</em> <span class="math inline">\(g(x+h) = g(x) + h\)</span>, for then the expression would be exactly the limit expression with <span class="math inline">\(c=g(x)\)</span>. But, alas, except to some hopeful students and some special cases, it is definitely not the case in general that <span class="math inline">\(g(x+h) = g(x) + h\)</span> - that right parentheses actually means something. However, it is <em>nearly</em> the case that <span class="math inline">\(g(x+h) = g(x) + kh\)</span> for some <span class="math inline">\(k\)</span> and this can be used to formulate a proof (one of the two detailed <a href="http://en.wikipedia.org/wiki/Chain_rule#Proofs">here</a> and <a href="http://kruel.co/math/chainrule.pdf">here</a>).</p>
|
||
<p>Combined, we would end up with:</p>
|
||
<blockquote class="blockquote">
|
||
<p><em>The chain rule</em>: <span class="math inline">\([f(g(x))]' = f'(g(x)) \cdot g'(x)\)</span>. That is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.</p>
|
||
</blockquote>
|
||
<p>To see that this works in our specific case, we assume the general power rule that <span class="math inline">\([x^n]' = n x^{n-1}\)</span> to get:</p>
|
||
<p><span class="math display">\[
|
||
\begin{align*}
|
||
f(x) &= x^2 & g(x) &= \sqrt{x}\\
|
||
f'(\square) &= 2(\square) & g'(x) &= \frac{1}{2}x^{-1/2}
|
||
\end{align*}
|
||
\]</span></p>
|
||
<p>We use <span class="math inline">\(\square\)</span> for the argument of <code>f'</code> to emphasize that <span class="math inline">\(g(x)\)</span> is the needed value, not just <span class="math inline">\(x\)</span>:</p>
|
||
<p><span class="math display">\[
|
||
\begin{align*}
|
||
[(\sqrt{x})^2]' &= [f(g(x)]'\\
|
||
&= f'(g(x)) \cdot g'(x) \\
|
||
&= 2(\sqrt{x}) \cdot \frac{1}{2}x^{-1/2}\\
|
||
&= \frac{2\sqrt{x}}{2\sqrt{x}}\\
|
||
&=1
|
||
\end{align*}
|
||
\]</span></p>
|
||
<p>This is the same as the derivative of <span class="math inline">\(x\)</span> found by first evaluating the composition. For this problem, the chain rule is not necessary, but typically it is a needed rule to fully differentiate a function.</p>
|
||
<section id="examples-1" class="level5">
|
||
<h5 class="anchored" data-anchor-id="examples-1">Examples</h5>
|
||
<p>Find the derivative of <span class="math inline">\(f(x) = \sqrt{1 - x^2}\)</span>. We identify the composition of <span class="math inline">\(\sqrt{x}\)</span> and <span class="math inline">\((1-x^2)\)</span>. We set the functions and their derivatives into a pattern to emphasize the pieces in the chain-rule formula:</p>
|
||
<p><span class="math display">\[
|
||
\begin{align*}
|
||
f(x) &=\sqrt{x} = x^{1/2} & g(x) &= 1 - x^2 \\
|
||
f'(\square) &=(1/2)(\square)^{-1/2} & g'(x) &= -2x
|
||
\end{align*}
|
||
\]</span></p>
|
||
<p>Then:</p>
|
||
<p><span class="math display">\[
|
||
[f(g(x))]' = (1/2)(1-x^2)^{-1/2} \cdot (-2x).
|
||
\]</span></p>
|
||
<hr>
|
||
<p>Find the derivative of <span class="math inline">\(\log(2 + \sin(x))\)</span>. This is a composition <span class="math inline">\(\log(x)\)</span> – with derivative <span class="math inline">\(1/x\)</span> and <span class="math inline">\(2 + \sin(x)\)</span> – with derivative <span class="math inline">\(\cos(x)\)</span>. We get <span class="math inline">\((1/\sin(x)) \cos(x)\)</span>.</p>
|
||
<p>In general,</p>
|
||
<p><span class="math display">\[
|
||
[\log(f(x))]' \frac{f'(x)}{f(x)}.
|
||
\]</span></p>
|
||
<hr>
|
||
<p>Find the derivative of <span class="math inline">\(e^{f(x)}\)</span>. The inner function has derivative <span class="math inline">\(f'(x)\)</span>, the outer function has derivative <span class="math inline">\(e^x\)</span> (the same as the outer function itself). We get for a derivative</p>
|
||
<p><span class="math display">\[
|
||
[e^{f(x)}]' = e^{f(x)} \cdot f'(x).
|
||
\]</span></p>
|
||
<p>This is a useful rule to remember for expressions involving exponentials.</p>
|
||
<hr>
|
||
<p>Find the derivative of <span class="math inline">\(\sin(x)\cos(2x)\)</span> at <span class="math inline">\(x=\pi\)</span>.</p>
|
||
<p><span class="math display">\[
|
||
[\sin(x)\cos(2x)]'\big|_{x=\pi} =
|
||
(\cos(x)\cos(2x) + \sin(x)(-\sin(2x)\cdot 2))\big|_{x=\pi} =
|
||
((-1)(1) + (0)(-0)(2)) = -1.
|
||
\]</span></p>
|
||
</section>
|
||
<section id="proof-of-the-chain-rule" class="level5">
|
||
<h5 class="anchored" data-anchor-id="proof-of-the-chain-rule">Proof of the Chain Rule</h5>
|
||
<p>A function is <em>differentiable</em> at <span class="math inline">\(a\)</span> if the following limit exists <span class="math inline">\(\lim_{h \rightarrow 0}(f(a+h)-f(a))/h\)</span>. Reexpressing this as: <span class="math inline">\(f(a+h) - f(a) - f'(a)h = \epsilon_f(h) h\)</span> where as <span class="math inline">\(h\rightarrow 0\)</span>, <span class="math inline">\(\epsilon_f(h) \rightarrow 0\)</span>. Then, we have:</p>
|
||
<p><span class="math display">\[
|
||
g(a+h) = g(a) + g'(a)h + \epsilon_g(h) h = g(a) + h',
|
||
\]</span></p>
|
||
<p>Where <span class="math inline">\(h' = (g'(a) + \epsilon_g(h))h \rightarrow 0\)</span> as <span class="math inline">\(h \rightarrow 0\)</span> will be used to simplify the following:</p>
|
||
<p><span class="math display">\[
|
||
\begin{align}
|
||
f(g(a+h)) - f(g(a)) &=
|
||
f(g(a) + g'(a)h + \epsilon_g(h)h) - f(g(a)) \\
|
||
&= f(g(a)) + f'(g(a)) (g'(a)h + \epsilon_g(h)h) + \epsilon_f(h')(h') - f(g(a))\\
|
||
&= f'(g(a)) g'(a)h + f'(g(a))(\epsilon_g(h)h) + \epsilon_f(h')(h').
|
||
\end{align}
|
||
\]</span></p>
|
||
<p>Rearranging:</p>
|
||
<p><span class="math display">\[
|
||
f(g(a+h)) - f(g(a)) - f'(g(a)) g'(a) h = f'(g(a))\epsilon_g(h))h + \epsilon_f(h')(h') =
|
||
(f'(g(a)) \epsilon_g(h) + \epsilon_f(h')( (g'(a) + \epsilon_g(h))))h =
|
||
\epsilon(h)h,
|
||
\]</span></p>
|
||
<p>where <span class="math inline">\(\epsilon(h)\)</span> combines the above terms which go to zero as <span class="math inline">\(h\rightarrow 0\)</span> into one. This is the alternative definition of the derivative, showing <span class="math inline">\((f\circ g)'(a) = f'(g(a)) g'(a)\)</span> when <span class="math inline">\(g\)</span> is differentiable at <span class="math inline">\(a\)</span> and <span class="math inline">\(f\)</span> is differentiable at <span class="math inline">\(g(a)\)</span>.</p>
|
||
</section>
|
||
<section id="the-chain-rule" class="level5">
|
||
<h5 class="anchored" data-anchor-id="the-chain-rule">The “chain” rule</h5>
|
||
<p>The chain rule name could also be simply the “composition rule,” as that is the operation the rule works for. However, in practice, there are usually <em>multiple</em> compositions, and the “chain” rule is used to chain together the different pieces. To get a sense, consider a triple composition <span class="math inline">\(u(v(w(x())))\)</span>. This will have derivative:</p>
|
||
<p><span class="math display">\[
|
||
\begin{align*}
|
||
[u(v(w(x)))]' &= u'(v(w(x))) \cdot [v(w(x))]' \\
|
||
&= u'(v(w(x))) \cdot v'(w(x)) \cdot w'(x)
|
||
\end{align*}
|
||
\]</span></p>
|
||
<p>The answer can be viewed as a repeated peeling off of the outer function, a view with immediate application to many compositions. To see that in action with an expression, consider this derivative problem, shown in steps:</p>
|
||
<p><span class="math display">\[
|
||
\begin{align*}
|
||
[\sin(e^{\cos(x^2-x)})]'
|
||
&= \cos(e^{\cos(x^2-x)}) \cdot [e^{\cos(x^2-x)}]'\\
|
||
&= \cos(e^{\cos(x^2-x)}) \cdot e^{\cos(x^2-x)} \cdot [\cos(x^2-x)]'\\
|
||
&= \cos(e^{\cos(x^2-x)}) \cdot e^{\cos(x^2-x)} \cdot (-\sin(x^2-x)) \cdot [x^2-x]'\\
|
||
&= \cos(e^{\cos(x^2-x)}) \cdot e^{\cos(x^2-x)} \cdot (-\sin(x^2-x)) \cdot (2x-1)\\
|
||
\end{align*}
|
||
\]</span></p>
|
||
</section>
|
||
<section id="more-examples-of-differentiation" class="level5">
|
||
<h5 class="anchored" data-anchor-id="more-examples-of-differentiation">More examples of differentiation</h5>
|
||
<p>Find the derivative of <span class="math inline">\(x^5 \cdot \sin(x)\)</span>.</p>
|
||
<p>This is a product of functions, using <span class="math inline">\([u\cdot v]' = u'v + uv'\)</span> we get:</p>
|
||
<p><span class="math display">\[
|
||
5x^4 \cdot \sin(x) + x^5 \cdot \cos(x)
|
||
\]</span></p>
|
||
<hr>
|
||
<p>Find the derivative of <span class="math inline">\(x^5 / \sin(x)\)</span>.</p>
|
||
<p>This is a quotient of functions. Using <span class="math inline">\([u/v]' = (u'v - uv')/v^2\)</span> we get</p>
|
||
<p><span class="math display">\[
|
||
(5x^4 \cdot \sin(x) - x^5 \cdot \cos(x)) / (\sin(x))^2.
|
||
\]</span></p>
|
||
<hr>
|
||
<p>Find the derivative of <span class="math inline">\(\sin(x^5)\)</span>. This is a composition of functions <span class="math inline">\(u(v(x))\)</span> with <span class="math inline">\(v(x) = x^5\)</span>. The chain rule says find the derivative of <span class="math inline">\(u\)</span> (<span class="math inline">\(\cos(x)\)</span>) and evaluate at <span class="math inline">\(v(x)\)</span> (<span class="math inline">\(\cos(x^5)\)</span>) then multiply by the derivative of <span class="math inline">\(v\)</span>:</p>
|
||
<p><span class="math display">\[
|
||
\cos(x^5) \cdot 5x^4.
|
||
\]</span></p>
|
||
<hr>
|
||
<p>Similarly, but differently, find the derivative of <span class="math inline">\(\sin(x)^5\)</span>. Now <span class="math inline">\(v(x) = \sin(x)\)</span>, so the derivative of <span class="math inline">\(u(x)\)</span> (<span class="math inline">\(5x^4\)</span>) evaluated at <span class="math inline">\(v(x)\)</span> is <span class="math inline">\(5(\sin(x))^4\)</span> so multiplying by <span class="math inline">\(v'\)</span> gives:</p>
|
||
<p><span class="math display">\[
|
||
5(\sin(x))^4 \cdot \cos(x)
|
||
\]</span></p>
|
||
<hr>
|
||
<p>We can verify these with <code>SymPy</code>. Rather than take a limit, we will use <code>SymPy</code>’s <code>diff</code> function to compute derivatives.</p>
|
||
<div class="cell" data-execution_count="26">
|
||
<div class="sourceCode cell-code" id="cb27"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb27-1"><a href="#cb27-1" aria-hidden="true" tabindex="-1"></a><span class="fu">diff</span>(x<span class="op">^</span><span class="fl">5</span> <span class="op">*</span> <span class="fu">sin</span>(x))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="27">
|
||
<span class="math-left-align" style="padding-left: 4px; width:0; float:left;">
|
||
\[
|
||
x^{5} \cos{\left(x \right)} + 5 x^{4} \sin{\left(x \right)}
|
||
\]
|
||
</span>
|
||
</div>
|
||
</div>
|
||
<div class="cell" data-execution_count="27">
|
||
<div class="sourceCode cell-code" id="cb28"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb28-1"><a href="#cb28-1" aria-hidden="true" tabindex="-1"></a><span class="fu">diff</span>(x<span class="op">^</span><span class="fl">5</span><span class="op">/</span><span class="fu">sin</span>(x))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="28">
|
||
<span class="math-left-align" style="padding-left: 4px; width:0; float:left;">
|
||
\[
|
||
- \frac{x^{5} \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{5 x^{4}}{\sin{\left(x \right)}}
|
||
\]
|
||
</span>
|
||
</div>
|
||
</div>
|
||
<div class="cell" data-execution_count="28">
|
||
<div class="sourceCode cell-code" id="cb29"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb29-1"><a href="#cb29-1" aria-hidden="true" tabindex="-1"></a><span class="fu">diff</span>(<span class="fu">sin</span>(x<span class="op">^</span><span class="fl">5</span>))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="29">
|
||
<span class="math-left-align" style="padding-left: 4px; width:0; float:left;">
|
||
\[
|
||
5 x^{4} \cos{\left(x^{5} \right)}
|
||
\]
|
||
</span>
|
||
</div>
|
||
</div>
|
||
<p>and finally,</p>
|
||
<div class="cell" data-execution_count="29">
|
||
<div class="sourceCode cell-code" id="cb30"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb30-1"><a href="#cb30-1" aria-hidden="true" tabindex="-1"></a><span class="fu">diff</span>(<span class="fu">sin</span>(x)<span class="op">^</span><span class="fl">5</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="30">
|
||
<span class="math-left-align" style="padding-left: 4px; width:0; float:left;">
|
||
\[
|
||
5 \sin^{4}{\left(x \right)} \cos{\left(x \right)}
|
||
\]
|
||
</span>
|
||
</div>
|
||
</div>
|
||
<div class="callout-note callout callout-style-default callout-captioned">
|
||
<div class="callout-header d-flex align-content-center">
|
||
<div class="callout-icon-container">
|
||
<i class="callout-icon"></i>
|
||
</div>
|
||
<div class="callout-caption-container flex-fill">
|
||
Note
|
||
</div>
|
||
</div>
|
||
<div class="callout-body-container callout-body">
|
||
<p>The <code>diff</code> function can be called as <code>diff(ex)</code> when there is just one free variable, as in the above examples; as <code>diff(ex, var)</code> when there are parameters in the expression.</p>
|
||
</div>
|
||
</div>
|
||
<hr>
|
||
<p>The general product rule: For any <span class="math inline">\(n\)</span> - not just integer values - we can re-express <span class="math inline">\(x^n\)</span> using <span class="math inline">\(e\)</span>: <span class="math inline">\(x^n = e^{n \log(x)}\)</span>. Now the chain rule can be applied:</p>
|
||
<p><span class="math display">\[
|
||
[x^n]' = [e^{n\log(x)}]' = e^{n\log(x)} \cdot (n \frac{1}{x}) = n x^n \cdot \frac{1}{x} = n x^{n-1}.
|
||
\]</span></p>
|
||
<hr>
|
||
<p>Find the derivative of <span class="math inline">\(f(x) = x^3 (1-x)^2\)</span> using either the power rule or the sum rule.</p>
|
||
<p>The power rule expresses <span class="math inline">\(f=u\cdot v\)</span>. With <span class="math inline">\(u(x)=x^3\)</span> and <span class="math inline">\(v(x)=(1-x)^2\)</span> we get:</p>
|
||
<p><span class="math display">\[
|
||
u'(x) = 3x^2, \quad v'(x) = 2 \cdot (1-x)^1 \cdot (-1),
|
||
\]</span></p>
|
||
<p>the last by the chain rule. Combining with <span class="math inline">\(u' v + u v'\)</span> we get: <span class="math inline">\(f'(x) = (3x^2)\cdot (1-x)^2 + x^3 \cdot (-2) \cdot (1-x)\)</span>.</p>
|
||
<p>Otherwise, the polynomial can be expanded to give <span class="math inline">\(f(x)=x^5-2x^4+x^3\)</span> which has derivative <span class="math inline">\(f'(x) = 5x^4 - 8x^3 + 3x^2\)</span>.</p>
|
||
<hr>
|
||
<p>Find the derivative of <span class="math inline">\(f(x) = x \cdot e^{-x^2}\)</span>.</p>
|
||
<p>Using the product rule and then the chain rule, we have:</p>
|
||
<p><span class="math display">\[
|
||
\begin{align}
|
||
f'(x) &= [x \cdot e^{-x^2}]'\\
|
||
&= [x]' \cdot e^{-x^2} + x \cdot [e^{-x^2}]'\\
|
||
&= 1 \cdot e^{-x^2} + x \cdot (e^{-x^2}) \cdot [-x^2]'\\
|
||
&= e^{-x^2} + x \cdot e^{-x^2} \cdot (-2x)\\
|
||
&= e^{-x^2} (1 - 2x^2).
|
||
\end{align}
|
||
\]</span></p>
|
||
<hr>
|
||
<p>Find the derivative of <span class="math inline">\(f(x) = e^{-ax} \cdot \sin(x)\)</span>.</p>
|
||
<p>Using the product rule and then the chain rule, we have:</p>
|
||
<p><span class="math display">\[
|
||
\begin{align}
|
||
f'(x) &= [e^{-ax} \cdot \sin(x)]'\\
|
||
&= [e^{-ax}]' \cdot \sin(x) + e^{-ax} \cdot [\sin(x)]'\\
|
||
&= e^{-ax} \cdot [-ax]' \cdot \sin(x) + e^{-ax} \cdot \cos(x)\\
|
||
&= e^{-ax} \cdot (-a) \cdot \sin(x) + e^{-ax} \cos(x)\\
|
||
&= e^{-ax}(\cos(x) - a\sin(x)).
|
||
\end{align}
|
||
\]</span></p>
|
||
<hr>
|
||
<p>Find the derivative of <span class="math inline">\(e^{-x^2/2}\)</span> at <span class="math inline">\(x=1\)</span>.</p>
|
||
<p><span class="math display">\[
|
||
[e^{-x^2/2}]'\big|_{x=1} =
|
||
(e^{-x^2/2} \cdot \frac{-2x}{2}) \big|_{x=1} =
|
||
e^{-1/2} \cdot (-1) = -e^{-1/2}.
|
||
\]</span></p>
|
||
</section>
|
||
<section id="example-derivative-of-inverse-functions" class="level5">
|
||
<h5 class="anchored" data-anchor-id="example-derivative-of-inverse-functions">Example: derivative of inverse functions</h5>
|
||
<p>Suppose we knew that <span class="math inline">\(\log(x)\)</span> had derivative of <span class="math inline">\(1/x\)</span>, but didn’t know the derivative of <span class="math inline">\(e^x\)</span>. From their inverse relation, we have: <span class="math inline">\(x=\log(e^x)\)</span>, so taking derivatives of both sides would yield:</p>
|
||
<p><span class="math display">\[
|
||
1 = (\frac{1}{e^x}) \cdot [e^x]'.
|
||
\]</span></p>
|
||
<p>Or solving, <span class="math inline">\([e^x]' = e^x\)</span>. This is a general strategy to find the derivative of an <em>inverse</em> function.</p>
|
||
<p>The graph of an inverse function is related to the graph of the function through the symmetry <span class="math inline">\(y=x\)</span>.</p>
|
||
<p>For example, the graph of <span class="math inline">\(e^x\)</span> and <span class="math inline">\(\log(x)\)</span> have this symmetry, emphasized below:</p>
|
||
<div class="cell" data-hold="true" data-execution_count="30">
|
||
<div class="cell-output cell-output-display" data-execution_count="31">
|
||
<p><img src="derivatives_files/figure-html/cell-31-output-1.svg" class="img-fluid"></p>
|
||
</div>
|
||
</div>
|
||
<p>The point <span class="math inline">\((1, e)\)</span> on the graph of <span class="math inline">\(e^x\)</span> matches the point <span class="math inline">\((e, 1)\)</span> on the graph of the inverse function, <span class="math inline">\(\log(x)\)</span>. The slope of the tangent line at <span class="math inline">\(x=1\)</span> to <span class="math inline">\(e^x\)</span> is given by <span class="math inline">\(e\)</span> as well. What is the slope of the tangent line to <span class="math inline">\(\log(x)\)</span> at <span class="math inline">\(x=e\)</span>?</p>
|
||
<p>As seen, the value can be computed, but how?</p>
|
||
<p>Finding the derivative of the inverse function can be achieved from the chain rule using the identify <span class="math inline">\(f^{-1}(f(x)) = x\)</span> for all <span class="math inline">\(x\)</span> in the domain of <span class="math inline">\(f\)</span>.</p>
|
||
<p>The chain rule applied to both sides, yields:</p>
|
||
<p><span class="math display">\[
|
||
1 = [f^{-1}]'(f(x)) \cdot f'(x)
|
||
\]</span></p>
|
||
<p>Solving, we see that <span class="math inline">\([f^{-1}]'(f(x)) = 1/f'(x)\)</span>. To emphasize the evaluation of the derivative of the inverse function at <span class="math inline">\(f(x)\)</span> we might write:</p>
|
||
<p><span class="math display">\[
|
||
\frac{d}{du} (f^{-1}(u)) \big|_{u=f(x)} = \frac{1}{f'(x)}
|
||
\]</span></p>
|
||
<p>So the reciprocal of the slope of the tangent line of <span class="math inline">\(f\)</span> at the mirror image point. In the above, we see if the slope of the tangent line at <span class="math inline">\((1,e)\)</span> to <span class="math inline">\(f\)</span> is <span class="math inline">\(e\)</span>, then the slope of the tangent line to <span class="math inline">\(f^{-1}(x)\)</span> at <span class="math inline">\((e,1)\)</span> would be <span class="math inline">\(1/e\)</span>.</p>
|
||
</section>
|
||
<section id="rules-of-derivatives-and-some-sample-functions" class="level4">
|
||
<h4 class="anchored" data-anchor-id="rules-of-derivatives-and-some-sample-functions">Rules of derivatives and some sample functions</h4>
|
||
<p>This table summarizes the rules of derivatives that allow derivatives of more complicated expressions to be computed with the derivatives of their pieces.</p>
|
||
<div class="cell" data-hold="true" data-execution_count="31">
|
||
<div class="cell-output cell-output-display" data-execution_count="32">
|
||
|
||
|
||
<div class="table-responsive">
|
||
<table class="table table-hover">
|
||
<tbody><tr><th>Name</th><th>Rule</th></tr>
|
||
|
||
<tr><td><div class="markdown"><p>Power rule</p>
|
||
</div></td><td><div class="markdown">\([x^n]' = n\cdot x^{n-1}\)
|
||
</div></td></tr>
|
||
<tr><td><div class="markdown"><p>constant</p>
|
||
</div></td><td><div class="markdown">\([cf(x)]' = c \cdot f'(x)\)
|
||
</div></td></tr>
|
||
<tr><td><div class="markdown"><p>sum/difference</p>
|
||
</div></td><td><div class="markdown">\([f(x) \pm g(x)]' = f'(x) \pm g'(x)\)
|
||
</div></td></tr>
|
||
<tr><td><div class="markdown"><p>product</p>
|
||
</div></td><td><div class="markdown">\([f(x) \cdot g(x)]' = f'(x)\cdot g(x) + f(x) \cdot g'(x)\)
|
||
</div></td></tr>
|
||
<tr><td><div class="markdown"><p>quotient</p>
|
||
</div></td><td><div class="markdown">\([f(x)/g(x)]' = (f'(x) \cdot g(x) - f(x) \cdot g'(x)) / g(x)^2\)
|
||
</div></td></tr>
|
||
<tr><td><div class="markdown"><p>chain</p>
|
||
</div></td><td><div class="markdown">\([f(g(x))]' = f'(g(x)) \cdot g'(x)\)
|
||
</div></td></tr>
|
||
|
||
</tbody></table>
|
||
</div>
|
||
|
||
</div>
|
||
</div>
|
||
<p>This table gives some useful derivatives:</p>
|
||
<div class="cell" data-hold="true" data-execution_count="32">
|
||
<div class="cell-output cell-output-display" data-execution_count="33">
|
||
|
||
|
||
<div class="table-responsive">
|
||
<table class="table table-hover">
|
||
<tbody><tr><th>Function</th><th>Derivative</th></tr>
|
||
|
||
<tr><td><div class="markdown">\(x^n (\text{ all } n)\)
|
||
</div></td><td><div class="markdown">\(nx^{n-1}\)
|
||
</div></td></tr>
|
||
<tr><td><div class="markdown">\(e^x\)
|
||
</div></td><td><div class="markdown">\(e^x\)
|
||
</div></td></tr>
|
||
<tr><td><div class="markdown">\(\log(x)\)
|
||
</div></td><td><div class="markdown">\(1/x\)
|
||
</div></td></tr>
|
||
<tr><td><div class="markdown">\(\sin(x)\)
|
||
</div></td><td><div class="markdown">\(\cos(x)\)
|
||
</div></td></tr>
|
||
<tr><td><div class="markdown">\(\cos(x)\)
|
||
</div></td><td><div class="markdown">\(-\sin(x)\)
|
||
</div></td></tr>
|
||
|
||
</tbody></table>
|
||
</div>
|
||
|
||
</div>
|
||
</div>
|
||
</section>
|
||
</section>
|
||
</section>
|
||
<section id="higher-order-derivatives" class="level2" data-number="22.5">
|
||
<h2 data-number="22.5" class="anchored" data-anchor-id="higher-order-derivatives"><span class="header-section-number">22.5</span> Higher-order derivatives</h2>
|
||
<p>The derivative of a function is an operator, it takes a function and returns a new, derived, function. We could repeat this operation. The result is called a higher-order derivative. The Lagrange notation uses additional “primes” to indicate how many. So <span class="math inline">\(f''(x)\)</span> is the second derivative and <span class="math inline">\(f'''(x)\)</span> the third. For even higher orders, sometimes the notation is <span class="math inline">\(f^{(n)}(x)\)</span> to indicate an <span class="math inline">\(n\)</span>th derivative.</p>
|
||
<section id="examples-2" class="level5">
|
||
<h5 class="anchored" data-anchor-id="examples-2">Examples</h5>
|
||
<p>Find the first <span class="math inline">\(3\)</span> derivatives of <span class="math inline">\(f(x) = ax^3 + bx^2 + cx + d\)</span>.</p>
|
||
<p>Differentiating a polynomial is done with the sum rule, here we repeat three times:</p>
|
||
<p><span class="math display">\[
|
||
\begin{align}
|
||
f(x) &= ax^3 + bx^2 + cx + d\\
|
||
f'(x) &= 3ax^2 + 2bx + c \\
|
||
f''(x) &= 3\cdot 2 a x + 2b \\
|
||
f'''(x) &= 6a
|
||
\end{align}
|
||
\]</span></p>
|
||
<p>We can see, the fourth derivative – and all higher order ones – would be identically <span class="math inline">\(0\)</span>. This is part of a general phenomenon: an <span class="math inline">\(n\)</span>th degree polynomial has only <span class="math inline">\(n\)</span> non-zero derivatives.</p>
|
||
<hr>
|
||
<p>Find the first <span class="math inline">\(5\)</span> derivatives of <span class="math inline">\(\sin(x)\)</span>.</p>
|
||
<p><span class="math display">\[
|
||
\begin{align}
|
||
f(x) &= \sin(x) \\
|
||
f'(x) &= \cos(x) \\
|
||
f''(x) &= -\sin(x) \\
|
||
f'''(x) &= -\cos(x) \\
|
||
f^{(4)} &= \sin(x) \\
|
||
f^{(5)} &= \cos(x)
|
||
\end{align}
|
||
\]</span></p>
|
||
<p>We see the derivatives repeat themselves. (We also see alternative notation for higher order derivatives.)</p>
|
||
<hr>
|
||
<p>Find the second derivative of <span class="math inline">\(e^{-x^2}\)</span>.</p>
|
||
<p>We need the chain rule <em>and</em> the product rule:</p>
|
||
<p><span class="math display">\[
|
||
[e^{-x^2}]'' = [e^{-x^2} \cdot (-2x)]' = \left(e^{-x^2} \cdot (-2x)\right) \cdot(-2x) + e^{-x^2} \cdot (-2) =
|
||
e^{-x^2}(4x^2 - 2).
|
||
\]</span></p>
|
||
<p>This can be verified:</p>
|
||
<div class="cell" data-execution_count="33">
|
||
<div class="sourceCode cell-code" id="cb31"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb31-1"><a href="#cb31-1" aria-hidden="true" tabindex="-1"></a><span class="fu">diff</span>(<span class="fu">diff</span>(<span class="fu">exp</span>(<span class="op">-</span>x<span class="op">^</span><span class="fl">2</span>))) <span class="op">|></span> simplify</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="34">
|
||
<span class="math-left-align" style="padding-left: 4px; width:0; float:left;">
|
||
\[
|
||
2 \cdot \left(2 x^{2} - 1\right) e^{- x^{2}}
|
||
\]
|
||
</span>
|
||
</div>
|
||
</div>
|
||
<p>Having to iterate the use of <code>diff</code> is cumbersome. An alternate notation is either specifying the variable twice: <code>diff(ex, x, x)</code> or using a number after the variable: <code>diff(ex, x, 2)</code>:</p>
|
||
<div class="cell" data-execution_count="34">
|
||
<div class="sourceCode cell-code" id="cb32"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb32-1"><a href="#cb32-1" aria-hidden="true" tabindex="-1"></a><span class="fu">diff</span>(<span class="fu">exp</span>(<span class="op">-</span>x<span class="op">^</span><span class="fl">2</span>), x, x) <span class="op">|></span> simplify</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="35">
|
||
<span class="math-left-align" style="padding-left: 4px; width:0; float:left;">
|
||
\[
|
||
2 \cdot \left(2 x^{2} - 1\right) e^{- x^{2}}
|
||
\]
|
||
</span>
|
||
</div>
|
||
</div>
|
||
<p>Higher-order derivatives can become involved when the product or quotient rules becomes involved.</p>
|
||
</section>
|
||
</section>
|
||
<section id="questions" class="level2" data-number="22.6">
|
||
<h2 data-number="22.6" class="anchored" data-anchor-id="questions"><span class="header-section-number">22.6</span> Questions</h2>
|
||
<section id="question" class="level6">
|
||
<h6 class="anchored" data-anchor-id="question">Question</h6>
|
||
<p>The derivative at <span class="math inline">\(c\)</span> is the slope of the tangent line at <span class="math inline">\(x=c\)</span>. Answer the following based on this graph:</p>
|
||
<div class="cell" data-execution_count="35">
|
||
<div class="sourceCode cell-code" id="cb33"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb33-1"><a href="#cb33-1" aria-hidden="true" tabindex="-1"></a>fn <span class="op">=</span> x <span class="op">-></span> <span class="fu">-x*exp</span>(x)<span class="fu">*sin</span>(<span class="cn">pi</span><span class="op">*</span>x)</span>
|
||
<span id="cb33-2"><a href="#cb33-2" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(fn, <span class="fl">0</span>, <span class="fl">2</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<div class="cell-output cell-output-display" data-execution_count="36">
|
||
<p><img src="derivatives_files/figure-html/cell-36-output-1.svg" class="img-fluid"></p>
|
||
</div>
|
||
</div>
|
||
<p>At which of these points <span class="math inline">\(c= 1/2, 1, 3/2\)</span> is the derivative negative?</p>
|
||
<div class="cell" data-hold="true" data-execution_count="36">
|
||
<div class="cell-output cell-output-display" data-execution_count="37">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="18070623817803723201" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_18070623817803723201">
|
||
<div style="padding-top: 5px">
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_18070623817803723201_1">
|
||
<input class="form-check-input" type="radio" name="radio_18070623817803723201" id="radio_18070623817803723201_1" value="1">
|
||
|
||
<span class="label-body px-1">
|
||
\(1/2\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_18070623817803723201_2">
|
||
<input class="form-check-input" type="radio" name="radio_18070623817803723201" id="radio_18070623817803723201_2" value="2">
|
||
|
||
<span class="label-body px-1">
|
||
\(1\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_18070623817803723201_3">
|
||
<input class="form-check-input" type="radio" name="radio_18070623817803723201" id="radio_18070623817803723201_3" value="3">
|
||
|
||
<span class="label-body px-1">
|
||
\(3/2\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="18070623817803723201_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.querySelectorAll('input[name="radio_18070623817803723201"]').forEach(function(rb) {
|
||
rb.addEventListener("change", function() {
|
||
var correct = rb.value == 1;
|
||
var msgBox = document.getElementById('18070623817803723201_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_18070623817803723201")
|
||
if (explanation != null) {
|
||
explanation.style.display = "none";
|
||
}
|
||
} else {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
|
||
var explanation = document.getElementById("explanation_18070623817803723201")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
})});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
<p>Which value looks bigger from reading the graph:</p>
|
||
<div class="cell" data-hold="true" data-execution_count="37">
|
||
<div class="cell-output cell-output-display" data-execution_count="38">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="8514362285684083836" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_8514362285684083836">
|
||
<div style="padding-top: 5px">
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_8514362285684083836_1">
|
||
<input class="form-check-input" type="radio" name="radio_8514362285684083836" id="radio_8514362285684083836_1" value="1">
|
||
|
||
<span class="label-body px-1">
|
||
\(f(1)\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_8514362285684083836_2">
|
||
<input class="form-check-input" type="radio" name="radio_8514362285684083836" id="radio_8514362285684083836_2" value="2">
|
||
|
||
<span class="label-body px-1">
|
||
\(f(3/2)\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="8514362285684083836_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.querySelectorAll('input[name="radio_8514362285684083836"]').forEach(function(rb) {
|
||
rb.addEventListener("change", function() {
|
||
var correct = rb.value == 2;
|
||
var msgBox = document.getElementById('8514362285684083836_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_8514362285684083836")
|
||
if (explanation != null) {
|
||
explanation.style.display = "none";
|
||
}
|
||
} else {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
|
||
var explanation = document.getElementById("explanation_8514362285684083836")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
})});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
<p>At <span class="math inline">\(0.708 \dots\)</span> and <span class="math inline">\(1.65\dots\)</span> the derivative has a common value. What is it?</p>
|
||
<div class="cell" data-hold="true" data-execution_count="38">
|
||
<div class="cell-output cell-output-display" data-execution_count="39">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="17836212321002218750" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_17836212321002218750">
|
||
<div style="padding-top: 5px">
|
||
<br>
|
||
<div class="input-group">
|
||
<input id="17836212321002218750" type="number" class="form-control" placeholder="Numeric answer">
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="17836212321002218750_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.getElementById("17836212321002218750").addEventListener("change", function() {
|
||
var correct = (Math.abs(this.value - 0) <= 0.01);
|
||
var msgBox = document.getElementById('17836212321002218750_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_17836212321002218750")
|
||
if (explanation != null) {
|
||
explanation.style.display = "none";
|
||
}
|
||
} else {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
|
||
var explanation = document.getElementById("explanation_17836212321002218750")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="question-1" class="level6">
|
||
<h6 class="anchored" data-anchor-id="question-1">Question</h6>
|
||
<p>Consider the graph of the <code>airyai</code> function (from <code>SpecialFunctions</code>) over <span class="math inline">\([-5, 5]\)</span>.</p>
|
||
<div class="cell" data-hold="true" data-execution_count="39">
|
||
<div class="cell-output cell-output-display" data-execution_count="40">
|
||
<p><img src="derivatives_files/figure-html/cell-40-output-1.svg" class="img-fluid"></p>
|
||
</div>
|
||
</div>
|
||
<p>At <span class="math inline">\(x = -2.5\)</span> the derivative is postive or negative?</p>
|
||
<div class="cell" data-hold="true" data-execution_count="40">
|
||
<div class="cell-output cell-output-display" data-execution_count="41">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="11772867052490330990" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_11772867052490330990">
|
||
<div style="padding-top: 5px">
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_11772867052490330990_1">
|
||
<input class="form-check-input" type="radio" name="radio_11772867052490330990" id="radio_11772867052490330990_1" value="1">
|
||
|
||
<span class="label-body px-1">
|
||
positive
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_11772867052490330990_2">
|
||
<input class="form-check-input" type="radio" name="radio_11772867052490330990" id="radio_11772867052490330990_2" value="2">
|
||
|
||
<span class="label-body px-1">
|
||
negative
|
||
</span>
|
||
</label>
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="11772867052490330990_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.querySelectorAll('input[name="radio_11772867052490330990"]').forEach(function(rb) {
|
||
rb.addEventListener("change", function() {
|
||
var correct = rb.value == 1;
|
||
var msgBox = document.getElementById('11772867052490330990_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_11772867052490330990")
|
||
if (explanation != null) {
|
||
explanation.style.display = "none";
|
||
}
|
||
} else {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
|
||
var explanation = document.getElementById("explanation_11772867052490330990")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
})});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
<p>At <span class="math inline">\(x=0\)</span> the derivative is postive or negative?</p>
|
||
<div class="cell" data-hold="true" data-execution_count="41">
|
||
<div class="cell-output cell-output-display" data-execution_count="42">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="4867065852861829581" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_4867065852861829581">
|
||
<div style="padding-top: 5px">
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_4867065852861829581_1">
|
||
<input class="form-check-input" type="radio" name="radio_4867065852861829581" id="radio_4867065852861829581_1" value="1">
|
||
|
||
<span class="label-body px-1">
|
||
positive
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_4867065852861829581_2">
|
||
<input class="form-check-input" type="radio" name="radio_4867065852861829581" id="radio_4867065852861829581_2" value="2">
|
||
|
||
<span class="label-body px-1">
|
||
negative
|
||
</span>
|
||
</label>
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="4867065852861829581_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.querySelectorAll('input[name="radio_4867065852861829581"]').forEach(function(rb) {
|
||
rb.addEventListener("change", function() {
|
||
var correct = rb.value == 2;
|
||
var msgBox = document.getElementById('4867065852861829581_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_4867065852861829581")
|
||
if (explanation != null) {
|
||
explanation.style.display = "none";
|
||
}
|
||
} else {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
|
||
var explanation = document.getElementById("explanation_4867065852861829581")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
})});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
<p>At <span class="math inline">\(x = 2.5\)</span> the derivative is postive or negative?</p>
|
||
<div class="cell" data-hold="true" data-execution_count="42">
|
||
<div class="cell-output cell-output-display" data-execution_count="43">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="5199065362316066675" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_5199065362316066675">
|
||
<div style="padding-top: 5px">
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_5199065362316066675_1">
|
||
<input class="form-check-input" type="radio" name="radio_5199065362316066675" id="radio_5199065362316066675_1" value="1">
|
||
|
||
<span class="label-body px-1">
|
||
positive
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_5199065362316066675_2">
|
||
<input class="form-check-input" type="radio" name="radio_5199065362316066675" id="radio_5199065362316066675_2" value="2">
|
||
|
||
<span class="label-body px-1">
|
||
negative
|
||
</span>
|
||
</label>
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="5199065362316066675_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.querySelectorAll('input[name="radio_5199065362316066675"]').forEach(function(rb) {
|
||
rb.addEventListener("change", function() {
|
||
var correct = rb.value == 2;
|
||
var msgBox = document.getElementById('5199065362316066675_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_5199065362316066675")
|
||
if (explanation != null) {
|
||
explanation.style.display = "none";
|
||
}
|
||
} else {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
|
||
var explanation = document.getElementById("explanation_5199065362316066675")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
})});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="question-2" class="level6">
|
||
<h6 class="anchored" data-anchor-id="question-2">Question</h6>
|
||
<p>Compute the derivative of <span class="math inline">\(e^x\)</span> using <code>limit</code>. What do you get?</p>
|
||
<div class="cell" data-hold="true" data-execution_count="43">
|
||
<div class="cell-output cell-output-display" data-execution_count="44">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="8450758211151318864" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_8450758211151318864">
|
||
<div style="padding-top: 5px">
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_8450758211151318864_1">
|
||
<input class="form-check-input" type="radio" name="radio_8450758211151318864" id="radio_8450758211151318864_1" value="1">
|
||
|
||
<span class="label-body px-1">
|
||
\(e^x\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_8450758211151318864_2">
|
||
<input class="form-check-input" type="radio" name="radio_8450758211151318864" id="radio_8450758211151318864_2" value="2">
|
||
|
||
<span class="label-body px-1">
|
||
\(x^e\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_8450758211151318864_3">
|
||
<input class="form-check-input" type="radio" name="radio_8450758211151318864" id="radio_8450758211151318864_3" value="3">
|
||
|
||
<span class="label-body px-1">
|
||
\((e-1)x^e\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_8450758211151318864_4">
|
||
<input class="form-check-input" type="radio" name="radio_8450758211151318864" id="radio_8450758211151318864_4" value="4">
|
||
|
||
<span class="label-body px-1">
|
||
\(e x^{(e-1)}\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_8450758211151318864_5">
|
||
<input class="form-check-input" type="radio" name="radio_8450758211151318864" id="radio_8450758211151318864_5" value="5">
|
||
|
||
<span class="label-body px-1">
|
||
something else
|
||
</span>
|
||
</label>
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="8450758211151318864_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.querySelectorAll('input[name="radio_8450758211151318864"]').forEach(function(rb) {
|
||
rb.addEventListener("change", function() {
|
||
var correct = rb.value == 1;
|
||
var msgBox = document.getElementById('8450758211151318864_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_8450758211151318864")
|
||
if (explanation != null) {
|
||
explanation.style.display = "none";
|
||
}
|
||
} else {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
|
||
var explanation = document.getElementById("explanation_8450758211151318864")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
})});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="question-3" class="level6">
|
||
<h6 class="anchored" data-anchor-id="question-3">Question</h6>
|
||
<p>Compute the derivative of <span class="math inline">\(x^e\)</span> using <code>limit</code>. What do you get?</p>
|
||
<div class="cell" data-hold="true" data-execution_count="44">
|
||
<div class="cell-output cell-output-display" data-execution_count="45">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="15667146029355605918" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_15667146029355605918">
|
||
<div style="padding-top: 5px">
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_15667146029355605918_1">
|
||
<input class="form-check-input" type="radio" name="radio_15667146029355605918" id="radio_15667146029355605918_1" value="1">
|
||
|
||
<span class="label-body px-1">
|
||
\(e^x\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_15667146029355605918_2">
|
||
<input class="form-check-input" type="radio" name="radio_15667146029355605918" id="radio_15667146029355605918_2" value="2">
|
||
|
||
<span class="label-body px-1">
|
||
\(x^e\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_15667146029355605918_3">
|
||
<input class="form-check-input" type="radio" name="radio_15667146029355605918" id="radio_15667146029355605918_3" value="3">
|
||
|
||
<span class="label-body px-1">
|
||
\((e-1)x^e\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_15667146029355605918_4">
|
||
<input class="form-check-input" type="radio" name="radio_15667146029355605918" id="radio_15667146029355605918_4" value="4">
|
||
|
||
<span class="label-body px-1">
|
||
\(e x^{(e-1)}\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_15667146029355605918_5">
|
||
<input class="form-check-input" type="radio" name="radio_15667146029355605918" id="radio_15667146029355605918_5" value="5">
|
||
|
||
<span class="label-body px-1">
|
||
something else
|
||
</span>
|
||
</label>
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="15667146029355605918_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.querySelectorAll('input[name="radio_15667146029355605918"]').forEach(function(rb) {
|
||
rb.addEventListener("change", function() {
|
||
var correct = rb.value == 5;
|
||
var msgBox = document.getElementById('15667146029355605918_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_15667146029355605918")
|
||
if (explanation != null) {
|
||
explanation.style.display = "none";
|
||
}
|
||
} else {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
|
||
var explanation = document.getElementById("explanation_15667146029355605918")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
})});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="question-4" class="level6">
|
||
<h6 class="anchored" data-anchor-id="question-4">Question</h6>
|
||
<p>Compute the derivative of <span class="math inline">\(e^{e\cdot x}\)</span> using <code>limit</code>. What do you get?</p>
|
||
<div class="cell" data-hold="true" data-execution_count="45">
|
||
<div class="cell-output cell-output-display" data-execution_count="46">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="15910140022522252323" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_15910140022522252323">
|
||
<div style="padding-top: 5px">
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_15910140022522252323_1">
|
||
<input class="form-check-input" type="radio" name="radio_15910140022522252323" id="radio_15910140022522252323_1" value="1">
|
||
|
||
<span class="label-body px-1">
|
||
\(e^x\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_15910140022522252323_2">
|
||
<input class="form-check-input" type="radio" name="radio_15910140022522252323" id="radio_15910140022522252323_2" value="2">
|
||
|
||
<span class="label-body px-1">
|
||
\(x^e\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_15910140022522252323_3">
|
||
<input class="form-check-input" type="radio" name="radio_15910140022522252323" id="radio_15910140022522252323_3" value="3">
|
||
|
||
<span class="label-body px-1">
|
||
\((e-1)x^e\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_15910140022522252323_4">
|
||
<input class="form-check-input" type="radio" name="radio_15910140022522252323" id="radio_15910140022522252323_4" value="4">
|
||
|
||
<span class="label-body px-1">
|
||
\(e x^{(e-1)}\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_15910140022522252323_5">
|
||
<input class="form-check-input" type="radio" name="radio_15910140022522252323" id="radio_15910140022522252323_5" value="5">
|
||
|
||
<span class="label-body px-1">
|
||
\(e \cdot e^{e\cdot x}\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_15910140022522252323_6">
|
||
<input class="form-check-input" type="radio" name="radio_15910140022522252323" id="radio_15910140022522252323_6" value="6">
|
||
|
||
<span class="label-body px-1">
|
||
something else
|
||
</span>
|
||
</label>
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="15910140022522252323_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.querySelectorAll('input[name="radio_15910140022522252323"]').forEach(function(rb) {
|
||
rb.addEventListener("change", function() {
|
||
var correct = rb.value == 5;
|
||
var msgBox = document.getElementById('15910140022522252323_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_15910140022522252323")
|
||
if (explanation != null) {
|
||
explanation.style.display = "none";
|
||
}
|
||
} else {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
|
||
var explanation = document.getElementById("explanation_15910140022522252323")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
})});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="question-5" class="level6">
|
||
<h6 class="anchored" data-anchor-id="question-5">Question</h6>
|
||
<p>In the derivation of the derivative of <span class="math inline">\(\sin(x)\)</span>, the following limit is needed:</p>
|
||
<p><span class="math display">\[
|
||
L = \lim_{h \rightarrow 0} \frac{\cos(h) - 1}{h}.
|
||
\]</span></p>
|
||
<p>This is</p>
|
||
<div class="cell" data-hold="true" data-execution_count="46">
|
||
<div class="cell-output cell-output-display" data-execution_count="47">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="12570137480795112399" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_12570137480795112399">
|
||
<div style="padding-top: 5px">
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_12570137480795112399_1">
|
||
<input class="form-check-input" type="radio" name="radio_12570137480795112399" id="radio_12570137480795112399_1" value="1">
|
||
|
||
<span class="label-body px-1">
|
||
Does not exist. The answer is \(0/0\) which is undefined
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_12570137480795112399_2">
|
||
<input class="form-check-input" type="radio" name="radio_12570137480795112399" id="radio_12570137480795112399_2" value="2">
|
||
|
||
<span class="label-body px-1">
|
||
\(0\), as this expression is the derivative of cosine at \(0\). The answer follows, as cosine clearly has a tangent line with slope \(0\) at \(x=0\).
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_12570137480795112399_3">
|
||
<input class="form-check-input" type="radio" name="radio_12570137480795112399" id="radio_12570137480795112399_3" value="3">
|
||
|
||
<span class="label-body px-1">
|
||
\(1\), as this is clearly the analog of the limit of \(\sin(h)/h\).
|
||
</span>
|
||
</label>
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="12570137480795112399_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.querySelectorAll('input[name="radio_12570137480795112399"]').forEach(function(rb) {
|
||
rb.addEventListener("change", function() {
|
||
var correct = rb.value == 2;
|
||
var msgBox = document.getElementById('12570137480795112399_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_12570137480795112399")
|
||
if (explanation != null) {
|
||
explanation.style.display = "none";
|
||
}
|
||
} else {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
|
||
var explanation = document.getElementById("explanation_12570137480795112399")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
})});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="question-6" class="level6">
|
||
<h6 class="anchored" data-anchor-id="question-6">Question</h6>
|
||
<p>Let <span class="math inline">\(f(x) = (e^x + e^{-x})/2\)</span> and <span class="math inline">\(g(x) = (e^x - e^{-x})/2\)</span>. Which is true?</p>
|
||
<div class="cell" data-hold="true" data-execution_count="47">
|
||
<div class="cell-output cell-output-display" data-execution_count="48">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="5658218585561973155" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_5658218585561973155">
|
||
<div style="padding-top: 5px">
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_5658218585561973155_1">
|
||
<input class="form-check-input" type="radio" name="radio_5658218585561973155" id="radio_5658218585561973155_1" value="1">
|
||
|
||
<span class="label-body px-1">
|
||
\(f'(x) = -g(x)\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_5658218585561973155_2">
|
||
<input class="form-check-input" type="radio" name="radio_5658218585561973155" id="radio_5658218585561973155_2" value="2">
|
||
|
||
<span class="label-body px-1">
|
||
\(f'(x) = -f(x)\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_5658218585561973155_3">
|
||
<input class="form-check-input" type="radio" name="radio_5658218585561973155" id="radio_5658218585561973155_3" value="3">
|
||
|
||
<span class="label-body px-1">
|
||
\(f'(x) = g(x)\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_5658218585561973155_4">
|
||
<input class="form-check-input" type="radio" name="radio_5658218585561973155" id="radio_5658218585561973155_4" value="4">
|
||
|
||
<span class="label-body px-1">
|
||
\(f'(x) = f(x)\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="5658218585561973155_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.querySelectorAll('input[name="radio_5658218585561973155"]').forEach(function(rb) {
|
||
rb.addEventListener("change", function() {
|
||
var correct = rb.value == 3;
|
||
var msgBox = document.getElementById('5658218585561973155_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_5658218585561973155")
|
||
if (explanation != null) {
|
||
explanation.style.display = "none";
|
||
}
|
||
} else {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
|
||
var explanation = document.getElementById("explanation_5658218585561973155")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
})});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="question-7" class="level6">
|
||
<h6 class="anchored" data-anchor-id="question-7">Question</h6>
|
||
<p>Let <span class="math inline">\(f(x) = (e^x + e^{-x})/2\)</span> and <span class="math inline">\(g(x) = (e^x - e^{-x})/2\)</span>. Which is true?</p>
|
||
<div class="cell" data-hold="true" data-execution_count="48">
|
||
<div class="cell-output cell-output-display" data-execution_count="49">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="10652710322969307913" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_10652710322969307913">
|
||
<div style="padding-top: 5px">
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_10652710322969307913_1">
|
||
<input class="form-check-input" type="radio" name="radio_10652710322969307913" id="radio_10652710322969307913_1" value="1">
|
||
|
||
<span class="label-body px-1">
|
||
\(f''(x) = f(x)\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_10652710322969307913_2">
|
||
<input class="form-check-input" type="radio" name="radio_10652710322969307913" id="radio_10652710322969307913_2" value="2">
|
||
|
||
<span class="label-body px-1">
|
||
\(f''(x) = -f(x)\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_10652710322969307913_3">
|
||
<input class="form-check-input" type="radio" name="radio_10652710322969307913" id="radio_10652710322969307913_3" value="3">
|
||
|
||
<span class="label-body px-1">
|
||
\(f''(x) = -g(x)\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_10652710322969307913_4">
|
||
<input class="form-check-input" type="radio" name="radio_10652710322969307913" id="radio_10652710322969307913_4" value="4">
|
||
|
||
<span class="label-body px-1">
|
||
\(f''(x) = g(x)\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="10652710322969307913_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.querySelectorAll('input[name="radio_10652710322969307913"]').forEach(function(rb) {
|
||
rb.addEventListener("change", function() {
|
||
var correct = rb.value == 1;
|
||
var msgBox = document.getElementById('10652710322969307913_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_10652710322969307913")
|
||
if (explanation != null) {
|
||
explanation.style.display = "none";
|
||
}
|
||
} else {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
|
||
var explanation = document.getElementById("explanation_10652710322969307913")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
})});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="question-8" class="level6">
|
||
<h6 class="anchored" data-anchor-id="question-8">Question</h6>
|
||
<p>Consider the function <span class="math inline">\(f\)</span> and its transformation <span class="math inline">\(g(x) = a + f(x)\)</span> (shift up by <span class="math inline">\(a\)</span>). Do <span class="math inline">\(f\)</span> and <span class="math inline">\(g\)</span> have the same derivative?</p>
|
||
<div class="cell" data-hold="true" data-execution_count="49">
|
||
<div class="cell-output cell-output-display" data-execution_count="50">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="9137945322402582354" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_9137945322402582354">
|
||
<div style="padding-top: 5px">
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_9137945322402582354_1">
|
||
<input class="form-check-input" type="radio" name="radio_9137945322402582354" id="radio_9137945322402582354_1" value="1">
|
||
|
||
<span class="label-body px-1">
|
||
Yes
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_9137945322402582354_2">
|
||
<input class="form-check-input" type="radio" name="radio_9137945322402582354" id="radio_9137945322402582354_2" value="2">
|
||
|
||
<span class="label-body px-1">
|
||
No
|
||
</span>
|
||
</label>
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="9137945322402582354_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.querySelectorAll('input[name="radio_9137945322402582354"]').forEach(function(rb) {
|
||
rb.addEventListener("change", function() {
|
||
var correct = rb.value == 1;
|
||
var msgBox = document.getElementById('9137945322402582354_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_9137945322402582354")
|
||
if (explanation != null) {
|
||
explanation.style.display = "none";
|
||
}
|
||
} else {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
|
||
var explanation = document.getElementById("explanation_9137945322402582354")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
})});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
<p>Consider the function <span class="math inline">\(f\)</span> and its transformation <span class="math inline">\(g(x) = f(x - a)\)</span> (shift right by <span class="math inline">\(a\)</span>). Do <span class="math inline">\(f\)</span> and <span class="math inline">\(g\)</span> have the same derivative?</p>
|
||
<div class="cell" data-hold="true" data-execution_count="50">
|
||
<div class="cell-output cell-output-display" data-execution_count="51">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="13613035363754329792" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_13613035363754329792">
|
||
<div style="padding-top: 5px">
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_13613035363754329792_1">
|
||
<input class="form-check-input" type="radio" name="radio_13613035363754329792" id="radio_13613035363754329792_1" value="1">
|
||
|
||
<span class="label-body px-1">
|
||
Yes
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_13613035363754329792_2">
|
||
<input class="form-check-input" type="radio" name="radio_13613035363754329792" id="radio_13613035363754329792_2" value="2">
|
||
|
||
<span class="label-body px-1">
|
||
No
|
||
</span>
|
||
</label>
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="13613035363754329792_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.querySelectorAll('input[name="radio_13613035363754329792"]').forEach(function(rb) {
|
||
rb.addEventListener("change", function() {
|
||
var correct = rb.value == 2;
|
||
var msgBox = document.getElementById('13613035363754329792_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_13613035363754329792")
|
||
if (explanation != null) {
|
||
explanation.style.display = "none";
|
||
}
|
||
} else {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
|
||
var explanation = document.getElementById("explanation_13613035363754329792")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
})});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
<p>Consider the function <span class="math inline">\(f\)</span> and its transformation <span class="math inline">\(g(x) = f(x - a)\)</span> (shift right by <span class="math inline">\(a\)</span>). Is <span class="math inline">\(f'\)</span> at <span class="math inline">\(x\)</span> equal to <span class="math inline">\(g'\)</span> at <span class="math inline">\(x-a\)</span>?</p>
|
||
<div class="cell" data-hold="true" data-execution_count="51">
|
||
<div class="cell-output cell-output-display" data-execution_count="52">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="11913924926352511074" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_11913924926352511074">
|
||
<div style="padding-top: 5px">
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_11913924926352511074_1">
|
||
<input class="form-check-input" type="radio" name="radio_11913924926352511074" id="radio_11913924926352511074_1" value="1">
|
||
|
||
<span class="label-body px-1">
|
||
Yes
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_11913924926352511074_2">
|
||
<input class="form-check-input" type="radio" name="radio_11913924926352511074" id="radio_11913924926352511074_2" value="2">
|
||
|
||
<span class="label-body px-1">
|
||
No
|
||
</span>
|
||
</label>
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="11913924926352511074_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.querySelectorAll('input[name="radio_11913924926352511074"]').forEach(function(rb) {
|
||
rb.addEventListener("change", function() {
|
||
var correct = rb.value == 1;
|
||
var msgBox = document.getElementById('11913924926352511074_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_11913924926352511074")
|
||
if (explanation != null) {
|
||
explanation.style.display = "none";
|
||
}
|
||
} else {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
|
||
var explanation = document.getElementById("explanation_11913924926352511074")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
})});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
<p>Consider the function <span class="math inline">\(f\)</span> and its transformation <span class="math inline">\(g(x) = c f(x)\)</span>, <span class="math inline">\(c > 1\)</span>. Do <span class="math inline">\(f\)</span> and <span class="math inline">\(g\)</span> have the same derivative?</p>
|
||
<div class="cell" data-hold="true" data-execution_count="52">
|
||
<div class="cell-output cell-output-display" data-execution_count="53">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="3372556018929962169" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_3372556018929962169">
|
||
<div style="padding-top: 5px">
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_3372556018929962169_1">
|
||
<input class="form-check-input" type="radio" name="radio_3372556018929962169" id="radio_3372556018929962169_1" value="1">
|
||
|
||
<span class="label-body px-1">
|
||
Yes
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_3372556018929962169_2">
|
||
<input class="form-check-input" type="radio" name="radio_3372556018929962169" id="radio_3372556018929962169_2" value="2">
|
||
|
||
<span class="label-body px-1">
|
||
No
|
||
</span>
|
||
</label>
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="3372556018929962169_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.querySelectorAll('input[name="radio_3372556018929962169"]').forEach(function(rb) {
|
||
rb.addEventListener("change", function() {
|
||
var correct = rb.value == 2;
|
||
var msgBox = document.getElementById('3372556018929962169_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_3372556018929962169")
|
||
if (explanation != null) {
|
||
explanation.style.display = "none";
|
||
}
|
||
} else {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
|
||
var explanation = document.getElementById("explanation_3372556018929962169")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
})});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
<p>Consider the function <span class="math inline">\(f\)</span> and its transformation <span class="math inline">\(g(x) = f(x/c)\)</span>, <span class="math inline">\(c > 1\)</span>. Do <span class="math inline">\(f\)</span> and <span class="math inline">\(g\)</span> have the same derivative?</p>
|
||
<div class="cell" data-hold="true" data-execution_count="53">
|
||
<div class="cell-output cell-output-display" data-execution_count="54">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="10394193485362826314" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_10394193485362826314">
|
||
<div style="padding-top: 5px">
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_10394193485362826314_1">
|
||
<input class="form-check-input" type="radio" name="radio_10394193485362826314" id="radio_10394193485362826314_1" value="1">
|
||
|
||
<span class="label-body px-1">
|
||
Yes
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_10394193485362826314_2">
|
||
<input class="form-check-input" type="radio" name="radio_10394193485362826314" id="radio_10394193485362826314_2" value="2">
|
||
|
||
<span class="label-body px-1">
|
||
No
|
||
</span>
|
||
</label>
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="10394193485362826314_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.querySelectorAll('input[name="radio_10394193485362826314"]').forEach(function(rb) {
|
||
rb.addEventListener("change", function() {
|
||
var correct = rb.value == 2;
|
||
var msgBox = document.getElementById('10394193485362826314_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_10394193485362826314")
|
||
if (explanation != null) {
|
||
explanation.style.display = "none";
|
||
}
|
||
} else {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
|
||
var explanation = document.getElementById("explanation_10394193485362826314")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
})});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
<p>Which of the following is true?</p>
|
||
<div class="cell" data-hold="true" data-execution_count="54">
|
||
<div class="cell-output cell-output-display" data-execution_count="55">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="15431856474718980528" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_15431856474718980528">
|
||
<div style="padding-top: 5px">
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_15431856474718980528_1">
|
||
<input class="form-check-input" type="radio" name="radio_15431856474718980528" id="radio_15431856474718980528_1" value="1">
|
||
|
||
<span class="label-body px-1">
|
||
If the graphs of \(f\) and \(g\) are rescalings of each other through \(g(x)=cf(x)\), \(c > 1\). Then the tangent line for corresponding points is the same.
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_15431856474718980528_2">
|
||
<input class="form-check-input" type="radio" name="radio_15431856474718980528" id="radio_15431856474718980528_2" value="2">
|
||
|
||
<span class="label-body px-1">
|
||
If the graphs of \(f\) and \(g\) are translations up and down, the tangent line at corresponding points is unchanged.
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_15431856474718980528_3">
|
||
<input class="form-check-input" type="radio" name="radio_15431856474718980528" id="radio_15431856474718980528_3" value="3">
|
||
|
||
<span class="label-body px-1">
|
||
If the graphs of \(f\) and \(g\) are rescalings of each other through \(g(x)=f(x/c)\), \(c > 1\). Then the tangent line for corresponding points is the same.
|
||
</span>
|
||
</label>
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="15431856474718980528_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.querySelectorAll('input[name="radio_15431856474718980528"]').forEach(function(rb) {
|
||
rb.addEventListener("change", function() {
|
||
var correct = rb.value == 2;
|
||
var msgBox = document.getElementById('15431856474718980528_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_15431856474718980528")
|
||
if (explanation != null) {
|
||
explanation.style.display = "none";
|
||
}
|
||
} else {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
|
||
var explanation = document.getElementById("explanation_15431856474718980528")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
})});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="question-9" class="level6">
|
||
<h6 class="anchored" data-anchor-id="question-9">Question</h6>
|
||
<p>The rate of change of volume with respect to height is <span class="math inline">\(3h\)</span>. The rate of change of height with respect to time is <span class="math inline">\(2t\)</span>. At at <span class="math inline">\(t=3\)</span> the height is <span class="math inline">\(h=14\)</span> what is the rate of change of volume with respect to time when <span class="math inline">\(t=3\)</span>?</p>
|
||
<div class="cell" data-hold="true" data-execution_count="55">
|
||
<div class="cell-output cell-output-display" data-execution_count="56">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="4090216136358313545" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_4090216136358313545">
|
||
<div style="padding-top: 5px">
|
||
<br>
|
||
<div class="input-group">
|
||
<input id="4090216136358313545" type="number" class="form-control" placeholder="Numeric answer">
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="4090216136358313545_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.getElementById("4090216136358313545").addEventListener("change", function() {
|
||
var correct = (Math.abs(this.value - 252) <= 0);
|
||
var msgBox = document.getElementById('4090216136358313545_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_4090216136358313545")
|
||
if (explanation != null) {
|
||
explanation.style.display = "none";
|
||
}
|
||
} else {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
|
||
var explanation = document.getElementById("explanation_4090216136358313545")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="question-10" class="level6">
|
||
<h6 class="anchored" data-anchor-id="question-10">Question</h6>
|
||
<p>Which equation below is <span class="math inline">\(f(x) = \sin(k\cdot x)\)</span> a solution of (<span class="math inline">\(k > 1\)</span>)?</p>
|
||
<div class="cell" data-hold="true" data-execution_count="56">
|
||
<div class="cell-output cell-output-display" data-execution_count="57">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="15523345641145931290" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_15523345641145931290">
|
||
<div style="padding-top: 5px">
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_15523345641145931290_1">
|
||
<input class="form-check-input" type="radio" name="radio_15523345641145931290" id="radio_15523345641145931290_1" value="1">
|
||
|
||
<span class="label-body px-1">
|
||
\(f'(x) = k^2 \cdot f(x)\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_15523345641145931290_2">
|
||
<input class="form-check-input" type="radio" name="radio_15523345641145931290" id="radio_15523345641145931290_2" value="2">
|
||
|
||
<span class="label-body px-1">
|
||
\(f'(x) = -k^2 \cdot f(x)\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_15523345641145931290_3">
|
||
<input class="form-check-input" type="radio" name="radio_15523345641145931290" id="radio_15523345641145931290_3" value="3">
|
||
|
||
<span class="label-body px-1">
|
||
\(f''(x) = -k^2 \cdot f(x)\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_15523345641145931290_4">
|
||
<input class="form-check-input" type="radio" name="radio_15523345641145931290" id="radio_15523345641145931290_4" value="4">
|
||
|
||
<span class="label-body px-1">
|
||
\(f''(x) = k^2 \cdot f(x)\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="15523345641145931290_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.querySelectorAll('input[name="radio_15523345641145931290"]').forEach(function(rb) {
|
||
rb.addEventListener("change", function() {
|
||
var correct = rb.value == 3;
|
||
var msgBox = document.getElementById('15523345641145931290_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_15523345641145931290")
|
||
if (explanation != null) {
|
||
explanation.style.display = "none";
|
||
}
|
||
} else {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
|
||
var explanation = document.getElementById("explanation_15523345641145931290")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
})});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="question-11" class="level6">
|
||
<h6 class="anchored" data-anchor-id="question-11">Question</h6>
|
||
<p>Let <span class="math inline">\(f(x) = e^{k\cdot x}\)</span>, <span class="math inline">\(k > 1\)</span>. Which equation below is <span class="math inline">\(f(x)\)</span> a solution of?</p>
|
||
<div class="cell" data-hold="true" data-execution_count="57">
|
||
<div class="cell-output cell-output-display" data-execution_count="58">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="8017076594990075220" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_8017076594990075220">
|
||
<div style="padding-top: 5px">
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_8017076594990075220_1">
|
||
<input class="form-check-input" type="radio" name="radio_8017076594990075220" id="radio_8017076594990075220_1" value="1">
|
||
|
||
<span class="label-body px-1">
|
||
\(f''(x) = -k^2 \cdot f(x)\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_8017076594990075220_2">
|
||
<input class="form-check-input" type="radio" name="radio_8017076594990075220" id="radio_8017076594990075220_2" value="2">
|
||
|
||
<span class="label-body px-1">
|
||
\(f'(x) = k^2 \cdot f(x)\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_8017076594990075220_3">
|
||
<input class="form-check-input" type="radio" name="radio_8017076594990075220" id="radio_8017076594990075220_3" value="3">
|
||
|
||
<span class="label-body px-1">
|
||
\(f'(x) = -k^2 \cdot f(x)\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_8017076594990075220_4">
|
||
<input class="form-check-input" type="radio" name="radio_8017076594990075220" id="radio_8017076594990075220_4" value="4">
|
||
|
||
<span class="label-body px-1">
|
||
\(f''(x) = k^2 \cdot f(x)\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="8017076594990075220_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.querySelectorAll('input[name="radio_8017076594990075220"]').forEach(function(rb) {
|
||
rb.addEventListener("change", function() {
|
||
var correct = rb.value == 4;
|
||
var msgBox = document.getElementById('8017076594990075220_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_8017076594990075220")
|
||
if (explanation != null) {
|
||
explanation.style.display = "none";
|
||
}
|
||
} else {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
|
||
var explanation = document.getElementById("explanation_8017076594990075220")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
})});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="question-12" class="level5">
|
||
<h5 class="anchored" data-anchor-id="question-12">Question</h5>
|
||
<p>Their are <span class="math inline">\(6\)</span> trig functions. The derivatives of <span class="math inline">\(\sin(x)\)</span> and <span class="math inline">\(\cos(x)\)</span> should be memorized. The others can be derived if not memorized using the quotient rule or chain rule.</p>
|
||
<p>What is <span class="math inline">\([\tan(x)]'\)</span>? (Use <span class="math inline">\(\tan(x) = \sin(x)/\cos(x)\)</span>.)</p>
|
||
<div class="cell" data-execution_count="58">
|
||
<div class="cell-output cell-output-display" data-execution_count="59">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="10550654213121411664" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_10550654213121411664">
|
||
<div style="padding-top: 5px">
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_10550654213121411664_1">
|
||
<input class="form-check-input" type="radio" name="radio_10550654213121411664" id="radio_10550654213121411664_1" value="1">
|
||
|
||
<span class="label-body px-1">
|
||
\(\sec^2(x)\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_10550654213121411664_2">
|
||
<input class="form-check-input" type="radio" name="radio_10550654213121411664" id="radio_10550654213121411664_2" value="2">
|
||
|
||
<span class="label-body px-1">
|
||
\(-\csc(x)\cot(x)\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_10550654213121411664_3">
|
||
<input class="form-check-input" type="radio" name="radio_10550654213121411664" id="radio_10550654213121411664_3" value="3">
|
||
|
||
<span class="label-body px-1">
|
||
\(-\csc^2(x)\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_10550654213121411664_4">
|
||
<input class="form-check-input" type="radio" name="radio_10550654213121411664" id="radio_10550654213121411664_4" value="4">
|
||
|
||
<span class="label-body px-1">
|
||
\(\sec(x)\tan(x)\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="10550654213121411664_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.querySelectorAll('input[name="radio_10550654213121411664"]').forEach(function(rb) {
|
||
rb.addEventListener("change", function() {
|
||
var correct = rb.value == 1;
|
||
var msgBox = document.getElementById('10550654213121411664_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_10550654213121411664")
|
||
if (explanation != null) {
|
||
explanation.style.display = "none";
|
||
}
|
||
} else {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
|
||
var explanation = document.getElementById("explanation_10550654213121411664")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
})});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
<p>What is <span class="math inline">\([\cot(x)]'\)</span>? (Use <span class="math inline">\(\tan(x) = \cos(x)/\sin(x)\)</span>.)</p>
|
||
<div class="cell" data-execution_count="59">
|
||
<div class="cell-output cell-output-display" data-execution_count="60">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="3463334550781411206" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_3463334550781411206">
|
||
<div style="padding-top: 5px">
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_3463334550781411206_1">
|
||
<input class="form-check-input" type="radio" name="radio_3463334550781411206" id="radio_3463334550781411206_1" value="1">
|
||
|
||
<span class="label-body px-1">
|
||
\(-\csc(x)\cot(x)\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_3463334550781411206_2">
|
||
<input class="form-check-input" type="radio" name="radio_3463334550781411206" id="radio_3463334550781411206_2" value="2">
|
||
|
||
<span class="label-body px-1">
|
||
\(\sec(x)\tan(x)\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_3463334550781411206_3">
|
||
<input class="form-check-input" type="radio" name="radio_3463334550781411206" id="radio_3463334550781411206_3" value="3">
|
||
|
||
<span class="label-body px-1">
|
||
\(-\csc^2(x)\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_3463334550781411206_4">
|
||
<input class="form-check-input" type="radio" name="radio_3463334550781411206" id="radio_3463334550781411206_4" value="4">
|
||
|
||
<span class="label-body px-1">
|
||
\(\sec^2(x)\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="3463334550781411206_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.querySelectorAll('input[name="radio_3463334550781411206"]').forEach(function(rb) {
|
||
rb.addEventListener("change", function() {
|
||
var correct = rb.value == 3;
|
||
var msgBox = document.getElementById('3463334550781411206_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_3463334550781411206")
|
||
if (explanation != null) {
|
||
explanation.style.display = "none";
|
||
}
|
||
} else {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
|
||
var explanation = document.getElementById("explanation_3463334550781411206")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
})});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
<p>What is <span class="math inline">\([\sec(x)]'\)</span>? (Use <span class="math inline">\(\sec(x) = 1/\cos(x)\)</span>.)</p>
|
||
<div class="cell" data-execution_count="60">
|
||
<div class="cell-output cell-output-display" data-execution_count="61">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="17709900094297008015" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_17709900094297008015">
|
||
<div style="padding-top: 5px">
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_17709900094297008015_1">
|
||
<input class="form-check-input" type="radio" name="radio_17709900094297008015" id="radio_17709900094297008015_1" value="1">
|
||
|
||
<span class="label-body px-1">
|
||
\(-\csc^2(x)\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_17709900094297008015_2">
|
||
<input class="form-check-input" type="radio" name="radio_17709900094297008015" id="radio_17709900094297008015_2" value="2">
|
||
|
||
<span class="label-body px-1">
|
||
\(\sec(x)\tan(x)\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_17709900094297008015_3">
|
||
<input class="form-check-input" type="radio" name="radio_17709900094297008015" id="radio_17709900094297008015_3" value="3">
|
||
|
||
<span class="label-body px-1">
|
||
\(\sec^2(x)\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_17709900094297008015_4">
|
||
<input class="form-check-input" type="radio" name="radio_17709900094297008015" id="radio_17709900094297008015_4" value="4">
|
||
|
||
<span class="label-body px-1">
|
||
\(-\csc(x)\cot(x)\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="17709900094297008015_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.querySelectorAll('input[name="radio_17709900094297008015"]').forEach(function(rb) {
|
||
rb.addEventListener("change", function() {
|
||
var correct = rb.value == 2;
|
||
var msgBox = document.getElementById('17709900094297008015_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_17709900094297008015")
|
||
if (explanation != null) {
|
||
explanation.style.display = "none";
|
||
}
|
||
} else {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
|
||
var explanation = document.getElementById("explanation_17709900094297008015")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
})});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
<p>What is <span class="math inline">\([\csc(x)]'\)</span>? (Use <span class="math inline">\(\csc(x) = 1/\sin(x)\)</span>.)</p>
|
||
<div class="cell" data-execution_count="61">
|
||
<div class="cell-output cell-output-display" data-execution_count="62">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="8899624616288527206" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_8899624616288527206">
|
||
<div style="padding-top: 5px">
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_8899624616288527206_1">
|
||
<input class="form-check-input" type="radio" name="radio_8899624616288527206" id="radio_8899624616288527206_1" value="1">
|
||
|
||
<span class="label-body px-1">
|
||
\(\sec^2(x)\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_8899624616288527206_2">
|
||
<input class="form-check-input" type="radio" name="radio_8899624616288527206" id="radio_8899624616288527206_2" value="2">
|
||
|
||
<span class="label-body px-1">
|
||
\(\sec(x)\tan(x)\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_8899624616288527206_3">
|
||
<input class="form-check-input" type="radio" name="radio_8899624616288527206" id="radio_8899624616288527206_3" value="3">
|
||
|
||
<span class="label-body px-1">
|
||
\(-\csc(x)\cot(x)\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
<div class="form-check">
|
||
<label class="form-check-label" for="radio_8899624616288527206_4">
|
||
<input class="form-check-input" type="radio" name="radio_8899624616288527206" id="radio_8899624616288527206_4" value="4">
|
||
|
||
<span class="label-body px-1">
|
||
\(-\csc^2(x)\)
|
||
</span>
|
||
</label>
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="8899624616288527206_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.querySelectorAll('input[name="radio_8899624616288527206"]').forEach(function(rb) {
|
||
rb.addEventListener("change", function() {
|
||
var correct = rb.value == 3;
|
||
var msgBox = document.getElementById('8899624616288527206_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_8899624616288527206")
|
||
if (explanation != null) {
|
||
explanation.style.display = "none";
|
||
}
|
||
} else {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition alert alert-danger'><span>👎 Incorrect </span></div>";
|
||
var explanation = document.getElementById("explanation_8899624616288527206")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
})});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="question-13" class="level5">
|
||
<h5 class="anchored" data-anchor-id="question-13">Question</h5>
|
||
<p>Consider this picture of composition:</p>
|
||
<div class="cell" data-hold="true" data-execution_count="62">
|
||
<div class="cell-output cell-output-display" data-execution_count="63">
|
||
<p><img src="derivatives_files/figure-html/cell-63-output-1.svg" class="img-fluid"></p>
|
||
</div>
|
||
</div>
|
||
<p>The right graph is of <span class="math inline">\(g(x) = \exp(x)\)</span> at <span class="math inline">\(x=1\)</span>, the left graph of <span class="math inline">\(f(x) = \sin(x)\)</span> <em>rotated</em> <span class="math inline">\(90\)</span> degrees counter-clockwise. Chasing the arrows shows graphically how <span class="math inline">\(f(g(1))\)</span> can be computed. The nearby values <span class="math inline">\(f(g(1+h))\)</span> are – using the tangent line of <span class="math inline">\(g\)</span> at <span class="math inline">\(x-1\)</span> – approximated by <span class="math inline">\(f(g(1) + g'(1)\cdot h)\)</span>, as shown in the graph segment on the left.</p>
|
||
<p>Assuming the approximation gets better for <span class="math inline">\(h\)</span> close to <span class="math inline">\(0\)</span>, as it visually does, the derivative at <span class="math inline">\(1\)</span> for <span class="math inline">\(f(g(x))\)</span> should be given by this limit:</p>
|
||
<p><span class="math display">\[
|
||
\begin{align*}
|
||
\frac{d(f\circ g)}{dx}\mid_{x=1}
|
||
&= \lim_{h\rightarrow 0} \frac{f(g(1) + g'(1)h)-f(g(1))}{h}\\
|
||
&= \lim_{h\rightarrow 0} \frac{f(g(1) + g'(1)h)-f(g(1))}{h}\\
|
||
&= \lim_{h\rightarrow 0} \frac{f(g(1) + g'(1)h)-f(g(1))}{g'(1)h} \cdot g'(1)\\
|
||
&= \lim_{h\rightarrow 0} (f\circ g)'(1) \cdot g'(1).
|
||
\end{align*}
|
||
\]</span></p>
|
||
<p>What limit law, described below assuming all limits exist. allows the last equals sign?</p>
|
||
<div class="cell" data-hold="true" data-execution_count="63">
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|
||
<span class="label-body px-1">
|
||
The limit of a sum is the sum of the limits: \(\lim_{x\rightarrow c}(au(x)+bv(x)) = a\lim_{x\rightarrow c}u(x) + b\lim_{x\rightarrow c}v(x)\)
|
||
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<span class="label-body px-1">
|
||
The limit of a product is the product of the limits: \(\lim_{x\rightarrow c}(u(x)\cdot v(x)) = \lim_{x\rightarrow c}u(x) \cdot \lim_{x\rightarrow c}v(x)\)
|
||
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<span class="label-body px-1">
|
||
The limit of a composition (under assumptions on \(v\)): \(\lim_{x \rightarrow c}u(v(x)) = \lim_{w \rightarrow \lim_{x \rightarrow c}v(x)} u(w)\).
|
||
</span>
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<i class="bi bi-arrow-left-short"></i> <span class="nav-page-text"><span class="chapter-number">21</span> <span class="chapter-title">Implications of continuity</span></span>
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