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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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||
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|
||
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<h2 id="toc-title">Table of contents</h2>
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<ul>
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<li><a href="#approximate-derivatives" id="toc-approximate-derivatives" class="nav-link active" data-scroll-target="#approximate-derivatives"> <span class="header-section-number">23.0.1</span> Approximate derivatives</a></li>
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<li><a href="#automatic-derivatives" id="toc-automatic-derivatives" class="nav-link" data-scroll-target="#automatic-derivatives"> <span class="header-section-number">23.0.2</span> Automatic derivatives</a></li>
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<li><a href="#recap-on-derivatives-in-julia" id="toc-recap-on-derivatives-in-julia" class="nav-link" data-scroll-target="#recap-on-derivatives-in-julia"> <span class="header-section-number">23.1</span> Recap on derivatives in Julia</a></li>
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<li><a href="#questions" id="toc-questions" class="nav-link" data-scroll-target="#questions"> <span class="header-section-number">23.2</span> Questions</a></li>
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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/numeric_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">23</span> <span class="chapter-title">Numeric derivatives</span></h1>
|
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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>
|
||
<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">ForwardDiff</span></span>
|
||
<span id="cb1-4"><a href="#cb1-4" aria-hidden="true" tabindex="-1"></a><span class="im">using</span> <span class="bu">SymPy</span></span>
|
||
<span id="cb1-5"><a href="#cb1-5" aria-hidden="true" tabindex="-1"></a><span class="im">using</span> <span class="bu">Roots</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||
<hr>
|
||
<p><code>SymPy</code> returns symbolic derivatives. Up to choices of simplification, these answers match those that would be derived by hand. This is useful when comparing with known answers and for seeing the structure of the answer. However, there are times we just want to work with the answer numerically. For that we have other options within <code>Julia</code>. We discuss approximate derivatives and automatic derivatives. The latter will find wide usage in these notes.</p>
|
||
<section id="approximate-derivatives" class="level3" data-number="23.0.1">
|
||
<h3 data-number="23.0.1" class="anchored" data-anchor-id="approximate-derivatives"><span class="header-section-number">23.0.1</span> Approximate derivatives</h3>
|
||
<p>By approximating the limit of the secant line with a value for a small, but positive, <span class="math inline">\(h\)</span>, we get an approximation to the derivative. That is</p>
|
||
<p><span class="math display">\[
|
||
f'(x) \approx \frac{f(x+h) - f(x)}{h}.
|
||
\]</span></p>
|
||
<p>This is the forward-difference approximation. The central difference approximation looks both ways:</p>
|
||
<p><span class="math display">\[
|
||
f'(x) \approx \frac{f(x+h) - f(x-h)}{2h}.
|
||
\]</span></p>
|
||
<p>Though in general they are different, they are both approximations. The central difference is usually more accurate for the same size <span class="math inline">\(h\)</span>. However, both are susceptible to round-off errors. The numerator is a subtraction of like-size numbers - a perfect opportunity to lose precision.</p>
|
||
<p>As such there is a balancing act:</p>
|
||
<ul>
|
||
<li>if <span class="math inline">\(h\)</span> is too small the round-off errors are problematic,</li>
|
||
<li>if <span class="math inline">\(h\)</span> is too big, the approximation to the limit is not good.</li>
|
||
</ul>
|
||
<p>For the forward difference <span class="math inline">\(h\)</span> values around <span class="math inline">\(10^{-8}\)</span> are typically good, for the central difference, values around <span class="math inline">\(10^{-6}\)</span> are typically good.</p>
|
||
<section id="example" class="level5">
|
||
<h5 class="anchored" data-anchor-id="example">Example</h5>
|
||
<p>Let’s verify that the forward difference isn’t too far off.</p>
|
||
<div class="cell" 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">f</span>(x) <span class="op">=</span> <span class="fu">exp</span>(<span class="op">-</span>x<span class="op">^</span><span class="fl">2</span><span class="op">/</span><span class="fl">2</span>)</span>
|
||
<span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a>c <span class="op">=</span> <span class="fl">1</span></span>
|
||
<span id="cb2-3"><a href="#cb2-3" aria-hidden="true" tabindex="-1"></a>h <span class="op">=</span> <span class="fl">1e-8</span></span>
|
||
<span id="cb2-4"><a href="#cb2-4" aria-hidden="true" tabindex="-1"></a>fapprox <span class="op">=</span> (<span class="fu">f</span>(c<span class="op">+</span>h) <span class="op">-</span> <span class="fu">f</span>(c)) <span class="op">/</span> 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="5">
|
||
<pre><code>-0.6065306479285937</code></pre>
|
||
</div>
|
||
</div>
|
||
<p>We can compare to the actual with:</p>
|
||
<div class="cell" data-execution_count="5">
|
||
<div class="sourceCode cell-code" id="cb4"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a><span class="pp">@syms</span> x</span>
|
||
<span id="cb4-2"><a href="#cb4-2" aria-hidden="true" tabindex="-1"></a>df <span class="op">=</span> <span class="fu">diff</span>(<span class="fu">f</span>(x), x)</span>
|
||
<span id="cb4-3"><a href="#cb4-3" aria-hidden="true" tabindex="-1"></a>factual <span class="op">=</span> <span class="fu">N</span>(<span class="fu">df</span>(c))</span>
|
||
<span id="cb4-4"><a href="#cb4-4" aria-hidden="true" tabindex="-1"></a><span class="fu">abs</span>(factual <span class="op">-</span> fapprox)</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>1.178403974334123392219821170344191813548718695568289215873505651941374433299215e-08</code></pre>
|
||
</div>
|
||
</div>
|
||
<p>The error is about <span class="math inline">\(1\)</span> part in <span class="math inline">\(100\)</span> million.</p>
|
||
<p>The central difference is better here:</p>
|
||
<div class="cell" data-hold="true" data-execution_count="6">
|
||
<div class="sourceCode cell-code" id="cb6"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb6-1"><a href="#cb6-1" aria-hidden="true" tabindex="-1"></a>h <span class="op">=</span> <span class="fl">1e-6</span></span>
|
||
<span id="cb6-2"><a href="#cb6-2" aria-hidden="true" tabindex="-1"></a>cdapprox <span class="op">=</span> (<span class="fu">f</span>(c<span class="op">+</span>h) <span class="op">-</span> <span class="fu">f</span>(c<span class="op">-</span>h)) <span class="op">/</span> (<span class="fl">2</span>h)</span>
|
||
<span id="cb6-3"><a href="#cb6-3" aria-hidden="true" tabindex="-1"></a><span class="fu">abs</span>(factual <span class="op">-</span> cdapprox)</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="7">
|
||
<pre><code>1.567568231457459370771602167922282298718695568289215873505651941374433299214873e-11</code></pre>
|
||
</div>
|
||
</div>
|
||
<hr>
|
||
<p>The <a href="https://github.com/JuliaDiff/FiniteDifferences.jl">FiniteDifferences</a> and <a href="https://github.com/JuliaDiff/FiniteDiff.jl">FiniteDiff</a> packages provide performant interfaces for differentiation based on finite differences.</p>
|
||
</section>
|
||
</section>
|
||
<section id="automatic-derivatives" class="level3" data-number="23.0.2">
|
||
<h3 data-number="23.0.2" class="anchored" data-anchor-id="automatic-derivatives"><span class="header-section-number">23.0.2</span> Automatic derivatives</h3>
|
||
<p>There are some other ways to compute derivatives numerically that give much more accuracy at the expense of slightly increased computing time. Automatic differentiation is the general name for a few different approaches. These approaches promise less complexity - in some cases - than symbolic derivatives and more accuracy than approximate derivatives; the accuracy is on the order of machine precision.</p>
|
||
<p>The <code>ForwardDiff</code> package provides one of <a href="https://juliadiff.org/">several</a> ways for <code>Julia</code> to compute automatic derivatives. <code>ForwardDiff</code> is well suited for functions encountered in these notes, which depend on at most a few variables and output no more than a few values at once.</p>
|
||
<p>The <code>ForwardDiff</code> package was loaded in this section; in general its features are available when the <code>CalculusWithJulia</code> package is loaded, as that package provides a more convenient interface. The <code>derivative</code> function is not exported by <code>FiniteDiff</code>, so its usage requires qualification. To illustrate, to find the derivative of <span class="math inline">\(f(x)\)</span> at a <em>point</em> we have this syntax:</p>
|
||
<div class="cell" data-execution_count="7">
|
||
<div class="sourceCode cell-code" id="cb8"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb8-1"><a href="#cb8-1" aria-hidden="true" tabindex="-1"></a>ForwardDiff.<span class="fu">derivative</span>(f, c) <span class="co"># derivative is qualified by a module name</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>-0.6065306597126334</code></pre>
|
||
</div>
|
||
</div>
|
||
<p>The <code>CalculusWithJulia</code> package defines an operator <code>D</code> which goes from finding a derivative at a point with <code>ForwardDiff.derivative</code> to defining a function which evaluates the derivative at each point. It is defined along the lines of <code>D(f) = x -> ForwardDiff.derivative(f,x)</code> in parallel to how the derivative operation for a function is defined mathematically from the definition for its value at a point.</p>
|
||
<p>Here we see the error in estimating <span class="math inline">\(f'(1)\)</span>:</p>
|
||
<div class="cell" data-execution_count="8">
|
||
<div class="sourceCode cell-code" id="cb10"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb10-1"><a href="#cb10-1" aria-hidden="true" tabindex="-1"></a>fauto <span class="op">=</span> <span class="fu">D</span>(f)(c) <span class="co"># D(f) is a function, D(f)(c) is the function called on c</span></span>
|
||
<span id="cb10-2"><a href="#cb10-2" aria-hidden="true" tabindex="-1"></a><span class="fu">abs</span>(factual <span class="op">-</span> fauto)</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>6.593178415491414032952187053331255443171078412649434805862556670078512733295529e-19</code></pre>
|
||
</div>
|
||
</div>
|
||
<p>In this case, it is exact.</p>
|
||
<p>The <code>D</code> operator is defined for most all functions in <code>Julia</code>, though, like the <code>diff</code> operator in <code>SymPy</code> there are some for which it won’t work.</p>
|
||
<section id="example-1" class="level5">
|
||
<h5 class="anchored" data-anchor-id="example-1">Example</h5>
|
||
<p>For <span class="math inline">\(f(x) = \sqrt{1 + \sin(\cos(x))}\)</span> compare the difference between the forward derivative with <span class="math inline">\(h=1e-8\)</span> and that computed by <code>D</code> at <span class="math inline">\(x=\pi/4\)</span>.</p>
|
||
<p>The forward derivative is found with:</p>
|
||
<div class="cell" data-execution_count="9">
|
||
<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><span class="fu">𝒇</span>(x) <span class="op">=</span> <span class="fu">sqrt</span>(<span class="fl">1</span> <span class="op">+</span> <span class="fu">sin</span>(<span class="fu">cos</span>(x)))</span>
|
||
<span id="cb12-2"><a href="#cb12-2" aria-hidden="true" tabindex="-1"></a>𝒄, 𝒉 <span class="op">=</span> <span class="cn">pi</span><span class="op">/</span><span class="fl">4</span>, <span class="fl">1e-8</span></span>
|
||
<span id="cb12-3"><a href="#cb12-3" aria-hidden="true" tabindex="-1"></a>fwd <span class="op">=</span> (<span class="fu">𝒇</span>(𝒄<span class="op">+</span>𝒉) <span class="op">-</span> <span class="fu">𝒇</span>(𝒄))<span class="op">/</span>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="10">
|
||
<pre><code>-0.20927346522370271</code></pre>
|
||
</div>
|
||
</div>
|
||
<p>That given by <code>D</code> is:</p>
|
||
<div class="cell" data-execution_count="10">
|
||
<div class="sourceCode cell-code" id="cb14"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb14-1"><a href="#cb14-1" aria-hidden="true" tabindex="-1"></a>ds_value <span class="op">=</span> <span class="fu">D</span>(𝒇)(𝒄)</span>
|
||
<span id="cb14-2"><a href="#cb14-2" aria-hidden="true" tabindex="-1"></a>ds_value, fwd, ds_value <span class="op">-</span> fwd</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">
|
||
<pre><code>(-0.20927346371432803, -0.20927346522370271, 1.5093746807970376e-9)</code></pre>
|
||
</div>
|
||
</div>
|
||
<p>Finally, <code>SymPy</code> gives an exact value we use to compare:</p>
|
||
<div class="cell" data-execution_count="11">
|
||
<div class="sourceCode cell-code" id="cb16"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb16-1"><a href="#cb16-1" aria-hidden="true" tabindex="-1"></a>𝒇𝒑 <span class="op">=</span> <span class="fu">diff</span>(<span class="fu">𝒇</span>(x), 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="12">
|
||
<span class="math-left-align" style="padding-left: 4px; width:0; float:left;">
|
||
\[
|
||
- \frac{\sin{\left(x \right)} \cos{\left(\cos{\left(x \right)} \right)}}{2 \sqrt{\sin{\left(\cos{\left(x \right)} \right)} + 1}}
|
||
\]
|
||
</span>
|
||
</div>
|
||
</div>
|
||
<div class="cell" data-execution_count="12">
|
||
<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>actual <span class="op">=</span> <span class="fu">N</span>(<span class="fu">𝒇𝒑</span>(PI<span class="op">/</span><span class="fl">4</span>))</span>
|
||
<span id="cb17-2"><a href="#cb17-2" aria-hidden="true" tabindex="-1"></a>actual <span class="op">-</span> ds_value, actual <span class="op">-</span> fwd</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">
|
||
<pre><code>(-5.2941373495493281085179064170722727004081123938542546154188381086149403149753e-17, 1.509374627855664131196135271205690010185476120918876061457453845811618913850597e-09)</code></pre>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="convenient-notation" class="level4">
|
||
<h4 class="anchored" data-anchor-id="convenient-notation">Convenient notation</h4>
|
||
<p><code>Julia</code> allows the possibility of extending functions to different types. Out of the box, the <code>'</code> notation is not employed for functions, but is used for matrices. It is used in postfix position, as with <code>A'</code>. We can define it to do the same thing as <code>D</code> for functions and then, we can evaluate derivatives with the familiar <code>f'(x)</code>. This is done in <code>CalculusWithJulia</code> along the lines of <code>Base.adjoint(f::Function) = D(f)</code>.</p>
|
||
<p>Then, we have, for example:</p>
|
||
<div class="cell" data-hold="true" data-execution_count="13">
|
||
<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>f<span class="op">'</span>(<span class="cn">pi</span>), f<span class="op">''</span>(<span class="cn">pi</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>(-1.0, -1.2246467991473532e-16)</code></pre>
|
||
</div>
|
||
</div>
|
||
<section id="example-2" class="level5">
|
||
<h5 class="anchored" data-anchor-id="example-2">Example</h5>
|
||
<p>Suppose our task is to find a zero of the second derivative of <span class="math inline">\(k(x) = e^{-x^2/2}\)</span> in <span class="math inline">\([0, 10]\)</span>, a known bracket. The <code>D</code> function takes a second argument to indicate the order of the derivative (e.g., <code>D(f,2)</code>), but we use the more familiar notation:</p>
|
||
<div class="cell" data-hold="true" data-execution_count="14">
|
||
<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">k</span>(x) <span class="op">=</span> <span class="fu">exp</span>(<span class="op">-</span>x<span class="op">^</span><span class="fl">2</span><span class="op">/</span><span class="fl">2</span>)</span>
|
||
<span id="cb21-2"><a href="#cb21-2" aria-hidden="true" tabindex="-1"></a><span class="fu">find_zero</span>(k<span class="op">''</span>, <span class="fl">0</span><span class="op">..</span><span class="fl">10</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>1.0</code></pre>
|
||
</div>
|
||
</div>
|
||
<p>We pass in the function object, <code>k''</code>, and not the evaluated function.</p>
|
||
</section>
|
||
</section>
|
||
</section>
|
||
<section id="recap-on-derivatives-in-julia" class="level2" data-number="23.1">
|
||
<h2 data-number="23.1" class="anchored" data-anchor-id="recap-on-derivatives-in-julia"><span class="header-section-number">23.1</span> Recap on derivatives in Julia</h2>
|
||
<p>A quick summary for finding derivatives in <code>Julia</code>, as there are <span class="math inline">\(3\)</span> different manners:</p>
|
||
<ul>
|
||
<li>Symbolic derivatives are found using <code>diff</code> from <code>SymPy</code></li>
|
||
<li>Automatic derivatives are found using the notation <code>f'</code> using <code>ForwardDiff.derivative</code></li>
|
||
<li>approximate derivatives at a point, <code>c</code>, for a given <code>h</code> are found with <code>(f(c+h)-f(c))/h</code>.</li>
|
||
</ul>
|
||
<p>For example, here all three are computed and compared:</p>
|
||
<div class="cell" data-hold="true" data-execution_count="15">
|
||
<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">f</span>(x) <span class="op">=</span> <span class="fu">exp</span>(<span class="op">-</span>x)<span class="fu">*sin</span>(x)</span>
|
||
<span id="cb23-2"><a href="#cb23-2" aria-hidden="true" tabindex="-1"></a></span>
|
||
<span id="cb23-3"><a href="#cb23-3" aria-hidden="true" tabindex="-1"></a>c <span class="op">=</span> <span class="cn">pi</span></span>
|
||
<span id="cb23-4"><a href="#cb23-4" aria-hidden="true" tabindex="-1"></a>h <span class="op">=</span> <span class="fl">1e-8</span></span>
|
||
<span id="cb23-5"><a href="#cb23-5" aria-hidden="true" tabindex="-1"></a></span>
|
||
<span id="cb23-6"><a href="#cb23-6" aria-hidden="true" tabindex="-1"></a>fp <span class="op">=</span> <span class="fu">diff</span>(<span class="fu">f</span>(x),x)</span>
|
||
<span id="cb23-7"><a href="#cb23-7" aria-hidden="true" tabindex="-1"></a></span>
|
||
<span id="cb23-8"><a href="#cb23-8" aria-hidden="true" tabindex="-1"></a>fp, <span class="fu">fp</span>(c), f<span class="op">'</span>(c), (<span class="fu">f</span>(c<span class="op">+</span>h) <span class="op">-</span> <span class="fu">f</span>(c))<span class="op">/</span>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="16">
|
||
<pre><code>(-exp(-x)*sin(x) + exp(-x)*cos(x), -exp(-pi), -0.043213918263772265, -0.04321391756900175)</code></pre>
|
||
</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 use of <code>'</code> to find derivatives provided by <code>CalculusWithJulia</code> is convenient, and used extensively in these notes, but it needs to be noted that it does <strong>not conform</strong> with the generic meaning of <code>'</code> within <code>Julia</code>’s wider package ecosystem and may cause issue with linear algebra operations; the symbol is meant for the adjoint of a matrix.</p>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="questions" class="level2" data-number="23.2">
|
||
<h2 data-number="23.2" class="anchored" data-anchor-id="questions"><span class="header-section-number">23.2</span> Questions</h2>
|
||
<section id="question" class="level5">
|
||
<h5 class="anchored" data-anchor-id="question">Question</h5>
|
||
<p>Find the derivative using a forward difference approximation of <span class="math inline">\(f(x) = x^x\)</span> at the point <span class="math inline">\(x=2\)</span> using <code>h=0.1</code>:</p>
|
||
<div class="cell" data-hold="true" data-execution_count="16">
|
||
<div class="cell-output cell-output-display" data-execution_count="17">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="10811934958227245072" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_10811934958227245072">
|
||
<div style="padding-top: 5px">
|
||
<br>
|
||
<div class="input-group">
|
||
<input id="10811934958227245072" type="number" class="form-control" placeholder="Numeric answer">
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="10811934958227245072_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.getElementById("10811934958227245072").addEventListener("change", function() {
|
||
var correct = (Math.abs(this.value - 7.496380917422423) <= 0.001);
|
||
var msgBox = document.getElementById('10811934958227245072_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_10811934958227245072")
|
||
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_10811934958227245072")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
<p>Using <code>D</code> or <code>f'</code> find the value using automatic differentiation</p>
|
||
<div class="cell" data-hold="true" data-execution_count="17">
|
||
<div class="cell-output cell-output-display" data-execution_count="18">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="17416063512012211261" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_17416063512012211261">
|
||
<div style="padding-top: 5px">
|
||
<br>
|
||
<div class="input-group">
|
||
<input id="17416063512012211261" type="number" class="form-control" placeholder="Numeric answer">
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="17416063512012211261_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.getElementById("17416063512012211261").addEventListener("change", function() {
|
||
var correct = (Math.abs(this.value - 6.772588722239782) <= 0.001);
|
||
var msgBox = document.getElementById('17416063512012211261_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_17416063512012211261")
|
||
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_17416063512012211261")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="question-1" class="level5">
|
||
<h5 class="anchored" data-anchor-id="question-1">Question</h5>
|
||
<p>Mathematically, as the value of <code>h</code> in the forward difference gets smaller the forward difference approximation gets better. On the computer, this is thwarted by floating point representation issues (in particular the error in subtracting two like-sized numbers in forming <span class="math inline">\(f(x+h)-f(x)\)</span>.)</p>
|
||
<p>For <code>1e-16</code> what is the error (in absolute value) in finding the forward difference approximation for the derivative of <span class="math inline">\(\sin(x)\)</span> at <span class="math inline">\(x=0\)</span>?</p>
|
||
<div class="cell" data-hold="true" data-execution_count="18">
|
||
<div class="cell-output cell-output-display" data-execution_count="19">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="17445991177816442810" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_17445991177816442810">
|
||
<div style="padding-top: 5px">
|
||
<br>
|
||
<div class="input-group">
|
||
<input id="17445991177816442810" type="number" class="form-control" placeholder="Numeric answer">
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="17445991177816442810_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.getElementById("17445991177816442810").addEventListener("change", function() {
|
||
var correct = (Math.abs(this.value - 0.0) <= 0.001);
|
||
var msgBox = document.getElementById('17445991177816442810_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_17445991177816442810")
|
||
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_17445991177816442810")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
<p>Repeat for <span class="math inline">\(x=\pi/4\)</span>:</p>
|
||
<div class="cell" data-hold="true" data-execution_count="19">
|
||
<div class="cell-output cell-output-display" data-execution_count="20">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="11448719169103362896" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_11448719169103362896">
|
||
<div style="padding-top: 5px">
|
||
<br>
|
||
<div class="input-group">
|
||
<input id="11448719169103362896" type="number" class="form-control" placeholder="Numeric answer">
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="11448719169103362896_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.getElementById("11448719169103362896").addEventListener("change", function() {
|
||
var correct = (Math.abs(this.value - 0.11022302462515654) <= 0.001);
|
||
var msgBox = document.getElementById('11448719169103362896_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_11448719169103362896")
|
||
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_11448719169103362896")
|
||
if (explanation != null) {
|
||
explanation.style.display = "block";
|
||
}
|
||
}
|
||
|
||
});
|
||
|
||
</script>
|
||
</div>
|
||
</div>
|
||
<section id="question-2" class="level6">
|
||
<h6 class="anchored" data-anchor-id="question-2">Question</h6>
|
||
<p>Let <span class="math inline">\(f(x) = x^x\)</span>. Using <code>D</code>, find <span class="math inline">\(f'(3)\)</span>.</p>
|
||
<div class="cell" data-hold="true" data-execution_count="20">
|
||
<div class="cell-output cell-output-display" data-execution_count="21">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="14612286546881484491" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_14612286546881484491">
|
||
<div style="padding-top: 5px">
|
||
<br>
|
||
<div class="input-group">
|
||
<input id="14612286546881484491" type="number" class="form-control" placeholder="Numeric answer">
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="14612286546881484491_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.getElementById("14612286546881484491").addEventListener("change", function() {
|
||
var correct = (Math.abs(this.value - 56.66253179403897) <= 0.001);
|
||
var msgBox = document.getElementById('14612286546881484491_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_14612286546881484491")
|
||
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_14612286546881484491")
|
||
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>Let <span class="math inline">\(f(x) = \lvert 1 - \sqrt{1 + x}\rvert\)</span>. Using <code>D</code>, find <span class="math inline">\(f'(3)\)</span>.</p>
|
||
<div class="cell" data-hold="true" data-execution_count="21">
|
||
<div class="cell-output cell-output-display" data-execution_count="22">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="6815638655028312497" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_6815638655028312497">
|
||
<div style="padding-top: 5px">
|
||
<br>
|
||
<div class="input-group">
|
||
<input id="6815638655028312497" type="number" class="form-control" placeholder="Numeric answer">
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="6815638655028312497_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.getElementById("6815638655028312497").addEventListener("change", function() {
|
||
var correct = (Math.abs(this.value - 0.25) <= 0.001);
|
||
var msgBox = document.getElementById('6815638655028312497_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_6815638655028312497")
|
||
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_6815638655028312497")
|
||
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>Let <span class="math inline">\(f(x) = e^{\sin(x)}\)</span>. Using <code>D</code>, find <span class="math inline">\(f'(3)\)</span>.</p>
|
||
<div class="cell" data-hold="true" data-execution_count="22">
|
||
<div class="cell-output cell-output-display" data-execution_count="23">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="14559372107848735447" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_14559372107848735447">
|
||
<div style="padding-top: 5px">
|
||
<br>
|
||
<div class="input-group">
|
||
<input id="14559372107848735447" type="number" class="form-control" placeholder="Numeric answer">
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="14559372107848735447_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.getElementById("14559372107848735447").addEventListener("change", function() {
|
||
var correct = (Math.abs(this.value - -1.1400385675133151) <= 0.001);
|
||
var msgBox = document.getElementById('14559372107848735447_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_14559372107848735447")
|
||
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_14559372107848735447")
|
||
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>For <code>Julia</code>’s <code>airyai</code> function find a numeric derivative using the forward difference. For <span class="math inline">\(c=3\)</span> and <span class="math inline">\(h=10^{-8}\)</span> find the forward difference approximation to <span class="math inline">\(f'(3)\)</span> for the <code>airyai</code> function.</p>
|
||
<div class="cell" data-hold="true" data-execution_count="23">
|
||
<div class="cell-output cell-output-display" data-execution_count="24">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="16647551403521855896" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_16647551403521855896">
|
||
<div style="padding-top: 5px">
|
||
<br>
|
||
<div class="input-group">
|
||
<input id="16647551403521855896" type="number" class="form-control" placeholder="Numeric answer">
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="16647551403521855896_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.getElementById("16647551403521855896").addEventListener("change", function() {
|
||
var correct = (Math.abs(this.value - -0.011912976768252426) <= 0.001);
|
||
var msgBox = document.getElementById('16647551403521855896_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_16647551403521855896")
|
||
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_16647551403521855896")
|
||
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>Find the rate of change with respect to time of the function <span class="math inline">\(f(t)= 64 - 16t^2\)</span> at <span class="math inline">\(t=1\)</span>.</p>
|
||
<div class="cell" data-hold="true" data-execution_count="24">
|
||
<div class="cell-output cell-output-display" data-execution_count="25">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="1888790190407988257" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_1888790190407988257">
|
||
<div style="padding-top: 5px">
|
||
<br>
|
||
<div class="input-group">
|
||
<input id="1888790190407988257" type="number" class="form-control" placeholder="Numeric answer">
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="1888790190407988257_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
<script text="text/javascript">
|
||
document.getElementById("1888790190407988257").addEventListener("change", function() {
|
||
var correct = (Math.abs(this.value - -32) <= 0);
|
||
var msgBox = document.getElementById('1888790190407988257_message');
|
||
if(correct) {
|
||
msgBox.innerHTML = "<div class='pluto-output admonition note alert alert-success'><span> 👍 Correct </span></div>";
|
||
var explanation = document.getElementById("explanation_1888790190407988257")
|
||
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_1888790190407988257")
|
||
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>Find the rate of change with respect to height, <span class="math inline">\(h\)</span>, of <span class="math inline">\(f(h) = 32h^3 - 62 h + 12\)</span> at <span class="math inline">\(h=2\)</span>.</p>
|
||
<div class="cell" data-hold="true" data-execution_count="25">
|
||
<div class="cell-output cell-output-display" data-execution_count="26">
|
||
<form class="mx-2 my-3 mw-100" name="WeaveQuestion" data-id="11265879477183451863" data-controltype="">
|
||
<div class="form-group ">
|
||
<div class="controls">
|
||
<div class="form" id="controls_11265879477183451863">
|
||
<div style="padding-top: 5px">
|
||
<br>
|
||
<div class="input-group">
|
||
<input id="11265879477183451863" type="number" class="form-control" placeholder="Numeric answer">
|
||
</div>
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div id="11265879477183451863_message" style="padding-bottom: 15px"></div>
|
||
</div>
|
||
</div>
|
||
</form>
|
||
|
||
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