use plotly; fix bitrot
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jverzani
@@ -9,6 +9,7 @@ In this section we use the following add-on packages:
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```{julia}
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using SymPy
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using Plots
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plotly()
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```
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```{julia}
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@@ -43,7 +44,7 @@ $$
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#| cache: true
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##{{{ different_poly_graph }}}
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gr()
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anim = @animate for m in 2:2:10
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fn = x -> x^m
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plot(fn, -1, 1, size = fig_size, legend=false, title="graph of x^{$m}", xlims=(-1,1), ylims=(-.1,1))
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@@ -53,6 +54,7 @@ imgfile = tempname() * ".gif"
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gif(anim, imgfile, fps = 1)
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caption = "Polynomials of varying even degrees over ``[-1,1]``."
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plotly()
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ImageFile(imgfile, caption)
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```
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@@ -87,7 +89,7 @@ $$
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#| echo: false
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#| cache: true
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### {{{ lines_m_graph }}}
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gr()
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anim = @animate for m in [-5, -2, -1, 1, 2, 5, 10, 20]
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fn = x -> m * x
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plot(fn, -1, 1, size = fig_size, legend=false, title="m = $m", xlims=(-1,1), ylims=(-20, 20))
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@@ -96,7 +98,7 @@ end
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imgfile = tempname() * ".gif"
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gif(anim, imgfile, fps = 1)
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caption = "Graphs of y = mx for different values of m"
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plotly()
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ImageFile(imgfile, caption)
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```
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@@ -172,17 +174,17 @@ To illustrate, using the just defined `x`, here is how we can create the polynom
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```{julia}
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𝒑 = -16x^2 + 100
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p = -16x^2 + 100
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```
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That is, the expression is created just as you would create it within a function body. But here the result is still a symbolic object. We have assigned this expression to a variable `p`, and have not defined it as a function `p(x)`. Mentally keeping the distinction between symbolic expressions and functions is very important.
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The `typeof` function shows that `𝒑` is of a symbolic type (`Sym`):
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The `typeof` function shows that `p` is of a symbolic type (`Sym`):
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```{julia}
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typeof(𝒑)
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typeof(p)
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```
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We can mix and match symbolic objects. This command creates an arbitrary quadratic polynomial:
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@@ -230,28 +232,28 @@ To illustrate, to do the task above for the polynomial $-16x^2 + 100$ we could h
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```{julia}
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𝒑(x => (x-1)^2)
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p(x => (x-1)^2)
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```
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This "call" notation takes pairs (designated by `a=>b`) where the left-hand side is the variable to substitute for, and the right-hand side the new value. The value to substitute can depend on the variable, as illustrated; be a different variable; or be a numeric value, such as $2$:
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```{julia}
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𝒚 = 𝒑(x=>2)
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y = p(x=>2)
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```
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The result will always be of a symbolic type, even if the answer is just a number:
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```{julia}
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typeof(𝒚)
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typeof(y)
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```
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If there is just one free variable in an expression, the pair notation can be dropped:
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```{julia}
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𝒑(4) # substitutes x=>4
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p(4) # substitutes x=>4
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```
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##### Example
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@@ -311,8 +313,8 @@ Evaluating a symbolic expression and returning a numeric value can be done by co
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```{julia}
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𝐩 = 200 - 16x^2
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N(𝐩(2))
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p = 200 - 16x^2
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N(p(2))
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```
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This approach is direct, but can be slow *if* many such evaluations were needed (such as with a plot). An alternative is to turn the symbolic expression into a `Julia` function and then evaluate that as usual.
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@@ -323,7 +325,7 @@ The `lambdify` function turns a symbolic expression into a `Julia` function
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```{julia}
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#| hold: true
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pp = lambdify(𝐩)
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pp = lambdify(p)
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pp(2)
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```
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@@ -356,7 +358,7 @@ Consider the graph of the polynomial `x^5 - x + 1`:
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plot(x^5 - x + 1, -3/2, 3/2)
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```
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(Plotting symbolic expressions is similar to plotting a function, in that the expression is passed in as the first argument. The expression must have only one free variable, as above, or an error will occur.)
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(Plotting symbolic expressions with `Plots` is similar to plotting a function, in that the expression is passed in as the first argument. The expression must have only one free variable, as above, or an error will occur. This happens, as there is a `Plots` "recipe" for `SymPy` defined.)
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This graph illustrates the key features of polynomial graphs:
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@@ -375,7 +377,7 @@ To investigate this last point, let's consider the case of the monomial $x^n$. W
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#| echo: false
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#| cache: true
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### {{{ poly_growth_graph }}}
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gr()
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anim = @animate for m in 0:2:12
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fn = x -> x^m
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plot(fn, -1.2, 1.2, size = fig_size, legend=false, xlims=(-1.2, 1.2), ylims=(0, 1.2^12), title="x^{$m} over [-1.2, 1.2]")
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@@ -384,7 +386,7 @@ end
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imgfile = tempname() * ".gif"
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gif(anim, imgfile, fps = 1)
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caption = L"Demonstration that $x^{10}$ grows faster than $x^8$, ... and $x^2$ grows faster than $x^0$ (which is constant)."
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plotly()
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ImageFile(imgfile, caption)
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```
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@@ -412,7 +414,7 @@ To see the roots and/or the peaks and valleys of a polynomial requires a judicio
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#| echo: false
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#| cache: true
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### {{{ leading_term_graph }}}
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gr()
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anim = @animate for n in 1:6
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m = [1, .5, -1, -5, -20, -25]
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M = [2, 4, 5, 10, 25, 30]
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@@ -427,7 +429,7 @@ end
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caption = "The previous graph is highlighted in red. Ultimately the leading term (\$x^4\$ here) dominates the graph."
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imgfile = tempname() * ".gif"
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gif(anim, imgfile, fps=1)
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plotly()
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ImageFile(imgfile, caption)
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```
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