use quarto, not Pluto to render pages
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@@ -14,7 +14,7 @@ using Roots
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using CalculusWithJulia.WeaveSupport
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fig_size = (600, 400)
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fig_size = (800, 600)
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using Markdown, Mustache
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const frontmatter = (
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@@ -284,17 +284,13 @@ first $n$ natural numbers.
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With this expression, it is readily seen that as $n$ gets large this value gets close to $2/6 = 1/3$.
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```julia; echo=false
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note("""
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!!! note
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The above approach, like Archimedes', ends with a limit being
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taken. The answer comes from using a limit to add a big number of
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small values. As with all limit questions, worrying about whether a
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limit exists is fundamental. For this problem, we will see that for
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the general statement there is a stretching of the formal concept of a limit.
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The above approach, like Archimedes', ends with a limit being
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taken. The answer comes from using a limit to add a big number of
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small values. As with all limit questions, worrying about whether a
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limit exists is fundamental. For this problem, we will see that for
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the general statement there is a stretching of the formal concept of a limit.
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""")
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```
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----
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@@ -897,17 +893,12 @@ derivative over $[a,b]$. This is significant, the error in $10$ steps
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of Simpson's rule is on the scale of the error of $10,000$ steps of
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the Riemann sum for well-behaved functions.
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```julia; echo=false
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note(L"""
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The Wikipedia article mentions that Kepler used a similar formula $100$
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years prior to Simpson, or about $200$ years before Riemann published
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his work. Again, the value in Riemann's work is not the computation of
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the answer, but the framework it provides in determining if a function
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is Riemann integrable or not.
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""")
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```
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!!! note
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The Wikipedia article mentions that Kepler used a similar formula $100$
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years prior to Simpson, or about $200$ years before Riemann published
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his work. Again, the value in Riemann's work is not the computation of
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the answer, but the framework it provides in determining if a function
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is Riemann integrable or not.
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## Gauss quadrature
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@@ -1289,8 +1280,8 @@ choices = [
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"``p``",
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"``1-p``",
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"``p^2``"]
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ans = 3
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radioq(choices, ans)
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answ = 3
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radioq(choices, answ)
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```
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###### Question
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@@ -1303,8 +1294,8 @@ choices = [
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"``2^5/5 - 0^5/5``",
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"``2^4/4 - 0^4/4``",
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"``3\\cdot 2^3 - 3 \\cdot 0^3``"]
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ans = 2
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radioq(choices, ans)
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answ = 2
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radioq(choices, answ)
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```
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@@ -1344,8 +1335,8 @@ L"The area between $c$ and $b$ must be positive, so $F(c) < F(b)$.",
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"``F(b) - F(c) = F(a).``",
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L" $F(x)$ is continuous, so between $a$ and $b$ has an extreme value, which must be at $c$. So $F(c) \geq F(b)$."
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]
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ans = 1
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radioq(choices, ans)
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answ = 1
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radioq(choices, answ)
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```
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@@ -1359,8 +1350,8 @@ choices = [
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"``10/100``",
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"``(10 - 0) \\cdot e^{10} / 100^4``"
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]
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ans = 1
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radioq(choices, ans)
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answ = 1
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radioq(choices, answ)
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```
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@@ -1440,9 +1431,11 @@ The area under a curve approximated by a Riemann sum.
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#CalculusWithJulia.WeaveSupport.JSXGraph(:integrals, url, caption)
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# This is just wrong...
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url = "https://raw.githubusercontent.com/jverzani/CalculusWithJulia.jl/master/CwJ/integrals/riemann.js"
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url = "./riemann.js"
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CalculusWithJulia.WeaveSupport.JSXGraph(url, caption)
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```
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The interactive graphic shows the area of a right-Riemann sum for different partitions. The function is
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```math
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@@ -1502,8 +1495,8 @@ L"around $10^{-2}$",
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L"around $10^{-4}$",
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L"around $10^{-6}$",
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L"around $10^{-8}$"]
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ans = 4
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radioq(choices, ans, keep_order=true)
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answ = 4
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radioq(choices, answ, keep_order=true)
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```
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###### Question
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