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<html> <html>
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@@ -16,43 +16,43 @@ var ptx_lunr_docs = [
"type": "Worksheet", "type": "Worksheet",
"number": "", "number": "",
"title": "Calculus Review", "title": "Calculus Review",
"body": " Calculus Review In Calculus I, you learned about derivatives, antiderivatives, and definite integrals. All of these will make an appearance in the context of probability density functions, cumulative distribution functions, and probability calculations in continuous sample spaces, so it will be useful to refresh your memory and practice some calculations. The Power Rule for derivatives states: The Power Rule for antiderivatives states: Practice with the following derivatives: Practice with the following antiderivatives: Fundamental Theorem of Calculus If is an antiderivative of i.e., if then: If is a probability density function (don't worry about what that means for now), then integrals such as are used to calculate certain probabilities, and so are function values of the form . Practice with the following integrals: " "body": " Calculus Review In Calculus I, you learned about derivatives, antiderivatives, and definite integrals. All of these will make an appearance in the context of probability density functions, cumulative distribution functions, and probability calculations in continuous sample spaces, so it will be useful to refresh your memory and practice some calculations. The Power Rule for derivatives states: The Power Rule for antiderivatives states: Practice with the following derivatives: Practice with the following antiderivatives: Fundamental Theorem of Calculus If is an antiderivative of i.e., if then: If is a probability density function (don't worry about what that means for now), then integrals such as are used to calculate certain probabilities, as are values of the form . Practice with the following integrals: "
}, },
{ {
"id": "recitation-calculus-review-3", "id": "recitation-calculus-review-3-1",
"level": "2", "level": "2",
"url": "recitation-calculus-review.html#recitation-calculus-review-3", "url": "recitation-calculus-review.html#recitation-calculus-review-3-1",
"type": "Worksheet Exercise", "type": "Worksheet Exercise",
"number": "1", "number": "1",
"title": "", "title": "",
"body": " The Power Rule for derivatives states: " "body": " The Power Rule for derivatives states: "
}, },
{ {
"id": "recitation-calculus-review-4", "id": "recitation-calculus-review-3-2",
"level": "2", "level": "2",
"url": "recitation-calculus-review.html#recitation-calculus-review-4", "url": "recitation-calculus-review.html#recitation-calculus-review-3-2",
"type": "Worksheet Exercise", "type": "Worksheet Exercise",
"number": "2", "number": "2",
"title": "", "title": "",
"body": " The Power Rule for antiderivatives states: " "body": " The Power Rule for antiderivatives states: "
}, },
{ {
"id": "recitation-calculus-review-5", "id": "recitation-calculus-review-3-3",
"level": "2", "level": "2",
"url": "recitation-calculus-review.html#recitation-calculus-review-5", "url": "recitation-calculus-review.html#recitation-calculus-review-3-3",
"type": "Worksheet Exercise", "type": "Worksheet Exercise",
"number": "3", "number": "3",
"title": "", "title": "",
"body": " Practice with the following derivatives: " "body": " Practice with the following derivatives: "
}, },
{ {
"id": "recitation-calculus-review-6", "id": "recitation-calculus-review-3-4",
"level": "2", "level": "2",
"url": "recitation-calculus-review.html#recitation-calculus-review-6", "url": "recitation-calculus-review.html#recitation-calculus-review-3-4",
"type": "Worksheet Exercise", "type": "Worksheet Exercise",
"number": "4", "number": "4",
"title": "", "title": "",
"body": " Practice with the following antiderivatives: " "body": " Practice with the following antiderivatives: "
}, },
{ {
"id": "thm-FTC", "id": "thm-FTC",
@@ -61,16 +61,16 @@ var ptx_lunr_docs = [
"type": "Theorem", "type": "Theorem",
"number": "1", "number": "1",
"title": "Fundamental Theorem of Calculus.", "title": "Fundamental Theorem of Calculus.",
"body": " Fundamental Theorem of Calculus If is an antiderivative of i.e., if then: If is a probability density function (don't worry about what that means for now), then integrals such as are used to calculate certain probabilities, and so are function values of the form . " "body": " Fundamental Theorem of Calculus If is an antiderivative of i.e., if then: "
}, },
{ {
"id": "recitation-calculus-review-8", "id": "recitation-calculus-review-4-3",
"level": "2", "level": "2",
"url": "recitation-calculus-review.html#recitation-calculus-review-8", "url": "recitation-calculus-review.html#recitation-calculus-review-4-3",
"type": "Worksheet Exercise", "type": "Worksheet Exercise",
"number": "5", "number": "5",
"title": "", "title": "",
"body": " Practice with the following integrals: " "body": " Practice with the following integrals: "
} }
] ]
@@ -185,49 +185,53 @@ eBookConfig.enable_chatcodes = false;
<span class="type">Worksheet</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber"></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Calculus Review</span><div class="print-links"><a href="recitation-calculus-review-printable.html" class="print-link" title="Print preview"><span class="icon material-symbols-outlined" aria-hidden="true">&#xe8ad;</span></a></div> <span class="type">Worksheet</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber"></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Calculus Review</span><div class="print-links"><a href="recitation-calculus-review-printable.html" class="print-link" title="Print preview"><span class="icon material-symbols-outlined" aria-hidden="true">&#xe8ad;</span></a></div>
</h2> </h2>
<section class="introduction" id="recitation-calculus-review-2"><div class="para" id="recitation-calculus-review-2-1">In Calculus I, you learned about derivatives, antiderivatives, and definite integrals. All of these will make an appearance in the context of probability density functions, cumulative distribution functions, and probability calculations in continuous sample spaces, so it will be useful to refresh your memory and practice some calculations.<div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div> <section class="introduction" id="recitation-calculus-review-2"><div class="para" id="recitation-calculus-review-2-1">In Calculus I, you learned about derivatives, antiderivatives, and definite integrals. All of these will make an appearance in the context of probability density functions, cumulative distribution functions, and probability calculations in continuous sample spaces, so it will be useful to refresh your memory and practice some calculations.<div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div></section><article class="exercise exercise-like" id="recitation-calculus-review-3"><h3 class="heading"><span class="codenumber">1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3> </div></section><section class="onepage" id="recitation-calculus-review-3"><article class="exercise exercise-like" id="recitation-calculus-review-3-1"><h3 class="heading"><span class="codenumber">1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
<div class="para logical" id="recitation-calculus-review-3-1-1"> <div class="para logical" id="recitation-calculus-review-3-1-1-1">
<div class="para">The Power Rule for derivatives states:</div> <div class="para">The Power Rule for derivatives states:</div>
<div class="displaymath process-math" id="recitation-calculus-review-3-1-1-1"> <div class="displaymath process-math" id="recitation-calculus-review-3-1-1-1-1">
\begin{gather*} \begin{gather*}
\frac{d}{dx}\left( x^n \right) = \frac{d}{dx}\left( x^n \right) =
\end{gather*} \end{gather*}
</div> </div>
<div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div> <div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-3-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> </div>
<div class="autopermalink" data-description="Worksheet Exercise 1"><a href="#recitation-calculus-review-3" title="Copy heading and permalink for Worksheet Exercise 1" aria-label="Copy heading and permalink for Worksheet Exercise 1">🔗</a></div></article><article class="exercise exercise-like" id="recitation-calculus-review-4"><h3 class="heading"><span class="codenumber">2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3> <div class="autopermalink" data-description="Worksheet Exercise 1"><a href="#recitation-calculus-review-3-1" title="Copy heading and permalink for Worksheet Exercise 1" aria-label="Copy heading and permalink for Worksheet Exercise 1">🔗</a></div></article><article class="exercise exercise-like" id="recitation-calculus-review-3-2"><h3 class="heading"><span class="codenumber">2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
<div class="para logical" id="recitation-calculus-review-4-1-1"> <div class="para logical" id="recitation-calculus-review-3-2-1-1">
<div class="para">The Power Rule for antiderivatives states:</div> <div class="para">The Power Rule for antiderivatives states:</div>
<div class="displaymath process-math" id="recitation-calculus-review-4-1-1-1"> <div class="displaymath process-math" id="recitation-calculus-review-3-2-1-1-1">
\begin{gather*} \begin{gather*}
\int\left( x^n \right)\ dx = \int\left( x^n \right)\ dx =
\end{gather*} \end{gather*}
</div> </div>
<div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div> <div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-3-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> </div>
<div class="autopermalink" data-description="Worksheet Exercise 2"><a href="#recitation-calculus-review-4" title="Copy heading and permalink for Worksheet Exercise 2" aria-label="Copy heading and permalink for Worksheet Exercise 2">🔗</a></div></article><article class="exercise exercise-like" id="recitation-calculus-review-5"><h3 class="heading"><span class="codenumber">3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3> <div class="autopermalink" data-description="Worksheet Exercise 2"><a href="#recitation-calculus-review-3-2" title="Copy heading and permalink for Worksheet Exercise 2" aria-label="Copy heading and permalink for Worksheet Exercise 2">🔗</a></div></article><article class="exercise exercise-like" id="recitation-calculus-review-3-3"><h3 class="heading"><span class="codenumber">3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
<div class="para logical" id="recitation-calculus-review-5-1-1"> <div class="introduction" id="recitation-calculus-review-3-3-1"><div class="para" id="recitation-calculus-review-3-3-1-1">Practice with the following derivatives:<div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-3-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
<div class="para">Practice with the following derivatives:</div> </div></div>
<div class="displaymath process-math" id="recitation-calculus-review-5-1-1-1"> <article class="task exercise-like" id="recitation-calculus-review-3-3-2"><h4 class="heading"><span class="codenumber">(a)</span></h4>
\begin{align*} <div class="para" id="recitation-calculus-review-3-3-2-1">
\frac{d}{dx}\left( 2x^2 + x^{-3} + 4\sqrt{x} \right) \amp = \\ <span class="process-math">\(\displaystyle \frac{d}{dx}\left( 2x^2 + x^{-3} + 4\sqrt{x} \right) = \)</span><div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-3-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
\frac{d}{dx}\left( \frac{1}{x} + 3e^{2x} + x^e + \pi^e \right) \amp =
\end{align*}
</div> </div>
<div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-5-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div> <div class="workspace" data-space="2cm"></div>
<div class="autopermalink" data-description="Task 3.a"><a href="#recitation-calculus-review-3-3-2" title="Copy heading and permalink for Task 3.a" aria-label="Copy heading and permalink for Task 3.a">🔗</a></div></article><article class="task exercise-like" id="recitation-calculus-review-3-3-3"><h4 class="heading"><span class="codenumber">(b)</span></h4>
<div class="para" id="recitation-calculus-review-3-3-3-1">
<span class="process-math">\(\displaystyle \frac{d}{dx}\left( \frac{1}{x} + 3e^{2x} + x^e + \pi^e \right) = \)</span><div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-3-3-3-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> </div>
<div class="autopermalink" data-description="Worksheet Exercise 3"><a href="#recitation-calculus-review-5" title="Copy heading and permalink for Worksheet Exercise 3" aria-label="Copy heading and permalink for Worksheet Exercise 3">🔗</a></div></article><article class="exercise exercise-like" id="recitation-calculus-review-6"><h3 class="heading"><span class="codenumber">4<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3> <div class="workspace" data-space="2cm"></div>
<div class="para logical" id="recitation-calculus-review-6-1-1"> <div class="autopermalink" data-description="Task 3.b"><a href="#recitation-calculus-review-3-3-3" title="Copy heading and permalink for Task 3.b" aria-label="Copy heading and permalink for Task 3.b">🔗</a></div></article><div class="autopermalink" data-description="Worksheet Exercise 3"><a href="#recitation-calculus-review-3-3" title="Copy heading and permalink for Worksheet Exercise 3" aria-label="Copy heading and permalink for Worksheet Exercise 3">🔗</a></div></article><article class="exercise exercise-like" id="recitation-calculus-review-3-4"><h3 class="heading"><span class="codenumber">4<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
<div class="para">Practice with the following antiderivatives:</div> <div class="introduction" id="recitation-calculus-review-3-4-1"><div class="para" id="recitation-calculus-review-3-4-1-1">Practice with the following antiderivatives:<div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-3-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
<div class="displaymath process-math" id="recitation-calculus-review-6-1-1-1"> </div></div>
\begin{align*} <article class="task exercise-like" id="recitation-calculus-review-3-4-2"><h4 class="heading"><span class="codenumber">(a)</span></h4>
\int 2x^2 + x^{-3} + 4\sqrt{x} \ dx \amp = \\ <div class="para" id="recitation-calculus-review-3-4-2-1">
\int \frac{1}{x} + 3e^{2x} + x^e + \pi^e \ dx \amp = <span class="process-math">\(\displaystyle \int 2x^2 + x^{-3} + 4\sqrt{x} \ dx = \)</span><div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-3-4-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
\end{align*}
</div> </div>
<div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-6-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div> <div class="workspace" data-space="2cm"></div>
<div class="autopermalink" data-description="Task 4.a"><a href="#recitation-calculus-review-3-4-2" title="Copy heading and permalink for Task 4.a" aria-label="Copy heading and permalink for Task 4.a">🔗</a></div></article><article class="task exercise-like" id="recitation-calculus-review-3-4-3"><h4 class="heading"><span class="codenumber">(b)</span></h4>
<div class="para" id="recitation-calculus-review-3-4-3-1">
<span class="process-math">\(\displaystyle \int \frac{1}{x} + 3e^{2x} + x^e + \pi^e \ dx = \)</span><div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-3-4-3-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> </div>
<div class="autopermalink" data-description="Worksheet Exercise 4"><a href="#recitation-calculus-review-6" title="Copy heading and permalink for Worksheet Exercise 4" aria-label="Copy heading and permalink for Worksheet Exercise 4">🔗</a></div></article><article class="theorem theorem-like" id="thm-FTC"><h3 class="heading"> <div class="workspace" data-space="2cm"></div>
<div class="autopermalink" data-description="Task 4.b"><a href="#recitation-calculus-review-3-4-3" title="Copy heading and permalink for Task 4.b" aria-label="Copy heading and permalink for Task 4.b">🔗</a></div></article><div class="autopermalink" data-description="Worksheet Exercise 4"><a href="#recitation-calculus-review-3-4" title="Copy heading and permalink for Worksheet Exercise 4" aria-label="Copy heading and permalink for Worksheet Exercise 4">🔗</a></div></article></section><section class="onepage" id="recitation-calculus-review-4"><article class="theorem theorem-like" id="thm-FTC"><h3 class="heading">
<span class="type">Theorem</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1</span><span class="period heading-divison-mark heading-divison-mark__period">.</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Fundamental Theorem of Calculus.</span> <span class="type">Theorem</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1</span><span class="period heading-divison-mark heading-divison-mark__period">.</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Fundamental Theorem of Calculus.</span>
</h3> </h3>
<div class="para logical" id="thm-FTC-2-1"> <div class="para logical" id="thm-FTC-2-1">
@@ -237,22 +241,24 @@ eBookConfig.enable_chatcodes = false;
\int_a^b f(x)\ dx = F(x)\bigg|_a^b = F(b) - F(a). \int_a^b f(x)\ dx = F(x)\bigg|_a^b = F(b) - F(a).
\end{gather*} \end{gather*}
</div> </div>
<div class="para">If <span class="process-math">\(f(x)\)</span> is a probability density function (don’t worry about what that means for now), then integrals such as <span class="process-math">\(\int_a^b f(x)\ dx\)</span> are used to calculate certain probabilities, and so are function values of the form <span class="process-math">\(F(x)\text{.}\)</span>
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#thm-FTC-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div> <div class="autopermalink" data-description="Paragraph"><a href="#thm-FTC-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> </div>
<div class="autopermalink" data-description="Theorem 1: Fundamental Theorem of Calculus"><a href="#thm-FTC" title="Copy heading and permalink for Theorem 1: Fundamental Theorem of Calculus" aria-label="Copy heading and permalink for Theorem 1: Fundamental Theorem of Calculus">🔗</a></div></article><article class="exercise exercise-like" id="recitation-calculus-review-8"><h3 class="heading"><span class="codenumber">5<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3> <div class="autopermalink" data-description="Theorem 1: Fundamental Theorem of Calculus"><a href="#thm-FTC" title="Copy heading and permalink for Theorem 1: Fundamental Theorem of Calculus" aria-label="Copy heading and permalink for Theorem 1: Fundamental Theorem of Calculus">🔗</a></div></article><div class="para" id="recitation-calculus-review-4-2">If <span class="process-math">\(f(x)\)</span> is a probability density function (don’t worry about what that means for now), then integrals such as <span class="process-math">\(\int_a^b f(x)\ dx\)</span> are used to calculate certain probabilities, as are values of the form <span class="process-math">\(F(x)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-4-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
<div class="para logical" id="recitation-calculus-review-8-1-1">
<div class="para">Practice with the following integrals:</div>
<div class="displaymath process-math" id="recitation-calculus-review-8-1-1-1">
\begin{align*}
\int_0^1 \frac{3}{2} x^{3/2} \ dx \amp = \\
\int_0^x 3e^{-3t} \ dt \amp =
\end{align*}
</div> </div>
<div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-8-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div> <article class="exercise exercise-like" id="recitation-calculus-review-4-3"><h3 class="heading"><span class="codenumber">5<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
<div class="introduction" id="recitation-calculus-review-4-3-1"><div class="para" id="recitation-calculus-review-4-3-1-1">Practice with the following integrals:<div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-4-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div></div>
<article class="task exercise-like" id="recitation-calculus-review-4-3-2"><h4 class="heading"><span class="codenumber">(a)</span></h4>
<div class="para" id="recitation-calculus-review-4-3-2-1">
<span class="process-math">\(\displaystyle \int_0^1 \frac{3}{2} x^{3/2} \ dx = \)</span><div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-4-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> </div>
<div class="autopermalink" data-description="Worksheet Exercise 5"><a href="#recitation-calculus-review-8" title="Copy heading and permalink for Worksheet Exercise 5" aria-label="Copy heading and permalink for Worksheet Exercise 5">🔗</a></div></article><div class="autopermalink" data-description="Worksheet: Calculus Review"><a href="#recitation-calculus-review" title="Copy heading and permalink for Worksheet: Calculus Review" aria-label="Copy heading and permalink for Worksheet: Calculus Review">🔗</a></div></section> <div class="workspace" data-space="2cm"></div>
<div class="autopermalink" data-description="Task 5.a"><a href="#recitation-calculus-review-4-3-2" title="Copy heading and permalink for Task 5.a" aria-label="Copy heading and permalink for Task 5.a">🔗</a></div></article><article class="task exercise-like" id="recitation-calculus-review-4-3-3"><h4 class="heading"><span class="codenumber">(b)</span></h4>
<div class="para" id="recitation-calculus-review-4-3-3-1">
<span class="process-math">\(\displaystyle \int_0^x 3e^{-3t} \ dt = \)</span><div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-4-3-3-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="workspace" data-space="2cm"></div>
<div class="autopermalink" data-description="Task 5.b"><a href="#recitation-calculus-review-4-3-3" title="Copy heading and permalink for Task 5.b" aria-label="Copy heading and permalink for Task 5.b">🔗</a></div></article><div class="autopermalink" data-description="Worksheet Exercise 5"><a href="#recitation-calculus-review-4-3" title="Copy heading and permalink for Worksheet Exercise 5" aria-label="Copy heading and permalink for Worksheet Exercise 5">🔗</a></div></article></section><div class="autopermalink" data-description="Worksheet: Calculus Review"><a href="#recitation-calculus-review" title="Copy heading and permalink for Worksheet: Calculus Review" aria-label="Copy heading and permalink for Worksheet: Calculus Review">🔗</a></div></section>
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@@ -170,49 +170,53 @@ eBookConfig.enable_chatcodes = false;
<span class="type">Worksheet</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber"></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Calculus Review</span><div class="print-links"><a href="recitation-calculus-review-printable.html" class="print-link" title="Print preview"><span class="icon material-symbols-outlined" aria-hidden="true">&#xe8ad;</span></a></div> <span class="type">Worksheet</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber"></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Calculus Review</span><div class="print-links"><a href="recitation-calculus-review-printable.html" class="print-link" title="Print preview"><span class="icon material-symbols-outlined" aria-hidden="true">&#xe8ad;</span></a></div>
</h2> </h2>
<section class="introduction" id="recitation-calculus-review-2"><div class="para" id="recitation-calculus-review-2-1">In Calculus I, you learned about derivatives, antiderivatives, and definite integrals. All of these will make an appearance in the context of probability density functions, cumulative distribution functions, and probability calculations in continuous sample spaces, so it will be useful to refresh your memory and practice some calculations.<div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div> <section class="introduction" id="recitation-calculus-review-2"><div class="para" id="recitation-calculus-review-2-1">In Calculus I, you learned about derivatives, antiderivatives, and definite integrals. All of these will make an appearance in the context of probability density functions, cumulative distribution functions, and probability calculations in continuous sample spaces, so it will be useful to refresh your memory and practice some calculations.<div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div></section><article class="exercise exercise-like" id="recitation-calculus-review-3"><h3 class="heading"><span class="codenumber">1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3> </div></section><section class="onepage" id="recitation-calculus-review-3"><article class="exercise exercise-like" id="recitation-calculus-review-3-1"><h3 class="heading"><span class="codenumber">1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
<div class="para logical" id="recitation-calculus-review-3-1-1"> <div class="para logical" id="recitation-calculus-review-3-1-1-1">
<div class="para">The Power Rule for derivatives states:</div> <div class="para">The Power Rule for derivatives states:</div>
<div class="displaymath process-math" id="recitation-calculus-review-3-1-1-1"> <div class="displaymath process-math" id="recitation-calculus-review-3-1-1-1-1">
\begin{gather*} \begin{gather*}
\frac{d}{dx}\left( x^n \right) = \frac{d}{dx}\left( x^n \right) =
\end{gather*} \end{gather*}
</div> </div>
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<div class="autopermalink" data-description="Worksheet Exercise 1"><a href="#recitation-calculus-review-3" title="Copy heading and permalink for Worksheet Exercise 1" aria-label="Copy heading and permalink for Worksheet Exercise 1">🔗</a></div></article><article class="exercise exercise-like" id="recitation-calculus-review-4"><h3 class="heading"><span class="codenumber">2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3> <div class="autopermalink" data-description="Worksheet Exercise 1"><a href="#recitation-calculus-review-3-1" title="Copy heading and permalink for Worksheet Exercise 1" aria-label="Copy heading and permalink for Worksheet Exercise 1">🔗</a></div></article><article class="exercise exercise-like" id="recitation-calculus-review-3-2"><h3 class="heading"><span class="codenumber">2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
<div class="para logical" id="recitation-calculus-review-4-1-1"> <div class="para logical" id="recitation-calculus-review-3-2-1-1">
<div class="para">The Power Rule for antiderivatives states:</div> <div class="para">The Power Rule for antiderivatives states:</div>
<div class="displaymath process-math" id="recitation-calculus-review-4-1-1-1"> <div class="displaymath process-math" id="recitation-calculus-review-3-2-1-1-1">
\begin{gather*} \begin{gather*}
\int\left( x^n \right)\ dx = \int\left( x^n \right)\ dx =
\end{gather*} \end{gather*}
</div> </div>
<div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div> <div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-3-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Worksheet Exercise 2"><a href="#recitation-calculus-review-4" title="Copy heading and permalink for Worksheet Exercise 2" aria-label="Copy heading and permalink for Worksheet Exercise 2">🔗</a></div></article><article class="exercise exercise-like" id="recitation-calculus-review-5"><h3 class="heading"><span class="codenumber">3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3> <div class="autopermalink" data-description="Worksheet Exercise 2"><a href="#recitation-calculus-review-3-2" title="Copy heading and permalink for Worksheet Exercise 2" aria-label="Copy heading and permalink for Worksheet Exercise 2">🔗</a></div></article><article class="exercise exercise-like" id="recitation-calculus-review-3-3"><h3 class="heading"><span class="codenumber">3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
<div class="para logical" id="recitation-calculus-review-5-1-1"> <div class="introduction" id="recitation-calculus-review-3-3-1"><div class="para" id="recitation-calculus-review-3-3-1-1">Practice with the following derivatives:<div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-3-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
<div class="para">Practice with the following derivatives:</div> </div></div>
<div class="displaymath process-math" id="recitation-calculus-review-5-1-1-1"> <article class="task exercise-like" id="recitation-calculus-review-3-3-2"><h4 class="heading"><span class="codenumber">(a)</span></h4>
\begin{align*} <div class="para" id="recitation-calculus-review-3-3-2-1">
\frac{d}{dx}\left( 2x^2 + x^{-3} + 4\sqrt{x} \right) \amp = \\ <span class="process-math">\(\displaystyle \frac{d}{dx}\left( 2x^2 + x^{-3} + 4\sqrt{x} \right) = \)</span><div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-3-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
\frac{d}{dx}\left( \frac{1}{x} + 3e^{2x} + x^e + \pi^e \right) \amp =
\end{align*}
</div> </div>
<div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-5-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div> <div class="workspace" data-space="2cm"></div>
<div class="autopermalink" data-description="Task 3.a"><a href="#recitation-calculus-review-3-3-2" title="Copy heading and permalink for Task 3.a" aria-label="Copy heading and permalink for Task 3.a">🔗</a></div></article><article class="task exercise-like" id="recitation-calculus-review-3-3-3"><h4 class="heading"><span class="codenumber">(b)</span></h4>
<div class="para" id="recitation-calculus-review-3-3-3-1">
<span class="process-math">\(\displaystyle \frac{d}{dx}\left( \frac{1}{x} + 3e^{2x} + x^e + \pi^e \right) = \)</span><div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-3-3-3-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> </div>
<div class="autopermalink" data-description="Worksheet Exercise 3"><a href="#recitation-calculus-review-5" title="Copy heading and permalink for Worksheet Exercise 3" aria-label="Copy heading and permalink for Worksheet Exercise 3">🔗</a></div></article><article class="exercise exercise-like" id="recitation-calculus-review-6"><h3 class="heading"><span class="codenumber">4<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3> <div class="workspace" data-space="2cm"></div>
<div class="para logical" id="recitation-calculus-review-6-1-1"> <div class="autopermalink" data-description="Task 3.b"><a href="#recitation-calculus-review-3-3-3" title="Copy heading and permalink for Task 3.b" aria-label="Copy heading and permalink for Task 3.b">🔗</a></div></article><div class="autopermalink" data-description="Worksheet Exercise 3"><a href="#recitation-calculus-review-3-3" title="Copy heading and permalink for Worksheet Exercise 3" aria-label="Copy heading and permalink for Worksheet Exercise 3">🔗</a></div></article><article class="exercise exercise-like" id="recitation-calculus-review-3-4"><h3 class="heading"><span class="codenumber">4<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
<div class="para">Practice with the following antiderivatives:</div> <div class="introduction" id="recitation-calculus-review-3-4-1"><div class="para" id="recitation-calculus-review-3-4-1-1">Practice with the following antiderivatives:<div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-3-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
<div class="displaymath process-math" id="recitation-calculus-review-6-1-1-1"> </div></div>
\begin{align*} <article class="task exercise-like" id="recitation-calculus-review-3-4-2"><h4 class="heading"><span class="codenumber">(a)</span></h4>
\int 2x^2 + x^{-3} + 4\sqrt{x} \ dx \amp = \\ <div class="para" id="recitation-calculus-review-3-4-2-1">
\int \frac{1}{x} + 3e^{2x} + x^e + \pi^e \ dx \amp = <span class="process-math">\(\displaystyle \int 2x^2 + x^{-3} + 4\sqrt{x} \ dx = \)</span><div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-3-4-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
\end{align*}
</div> </div>
<div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-6-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div> <div class="workspace" data-space="2cm"></div>
<div class="autopermalink" data-description="Task 4.a"><a href="#recitation-calculus-review-3-4-2" title="Copy heading and permalink for Task 4.a" aria-label="Copy heading and permalink for Task 4.a">🔗</a></div></article><article class="task exercise-like" id="recitation-calculus-review-3-4-3"><h4 class="heading"><span class="codenumber">(b)</span></h4>
<div class="para" id="recitation-calculus-review-3-4-3-1">
<span class="process-math">\(\displaystyle \int \frac{1}{x} + 3e^{2x} + x^e + \pi^e \ dx = \)</span><div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-3-4-3-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> </div>
<div class="autopermalink" data-description="Worksheet Exercise 4"><a href="#recitation-calculus-review-6" title="Copy heading and permalink for Worksheet Exercise 4" aria-label="Copy heading and permalink for Worksheet Exercise 4">🔗</a></div></article><article class="theorem theorem-like" id="thm-FTC"><h3 class="heading"> <div class="workspace" data-space="2cm"></div>
<div class="autopermalink" data-description="Task 4.b"><a href="#recitation-calculus-review-3-4-3" title="Copy heading and permalink for Task 4.b" aria-label="Copy heading and permalink for Task 4.b">🔗</a></div></article><div class="autopermalink" data-description="Worksheet Exercise 4"><a href="#recitation-calculus-review-3-4" title="Copy heading and permalink for Worksheet Exercise 4" aria-label="Copy heading and permalink for Worksheet Exercise 4">🔗</a></div></article></section><section class="onepage" id="recitation-calculus-review-4"><article class="theorem theorem-like" id="thm-FTC"><h3 class="heading">
<span class="type">Theorem</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1</span><span class="period heading-divison-mark heading-divison-mark__period">.</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Fundamental Theorem of Calculus.</span> <span class="type">Theorem</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">1</span><span class="period heading-divison-mark heading-divison-mark__period">.</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Fundamental Theorem of Calculus.</span>
</h3> </h3>
<div class="para logical" id="thm-FTC-2-1"> <div class="para logical" id="thm-FTC-2-1">
@@ -222,22 +226,24 @@ eBookConfig.enable_chatcodes = false;
\int_a^b f(x)\ dx = F(x)\bigg|_a^b = F(b) - F(a). \int_a^b f(x)\ dx = F(x)\bigg|_a^b = F(b) - F(a).
\end{gather*} \end{gather*}
</div> </div>
<div class="para">If <span class="process-math">\(f(x)\)</span> is a probability density function (don’t worry about what that means for now), then integrals such as <span class="process-math">\(\int_a^b f(x)\ dx\)</span> are used to calculate certain probabilities, and so are function values of the form <span class="process-math">\(F(x)\text{.}\)</span>
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#thm-FTC-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div> <div class="autopermalink" data-description="Paragraph"><a href="#thm-FTC-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> </div>
<div class="autopermalink" data-description="Theorem 1: Fundamental Theorem of Calculus"><a href="#thm-FTC" title="Copy heading and permalink for Theorem 1: Fundamental Theorem of Calculus" aria-label="Copy heading and permalink for Theorem 1: Fundamental Theorem of Calculus">🔗</a></div></article><article class="exercise exercise-like" id="recitation-calculus-review-8"><h3 class="heading"><span class="codenumber">5<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3> <div class="autopermalink" data-description="Theorem 1: Fundamental Theorem of Calculus"><a href="#thm-FTC" title="Copy heading and permalink for Theorem 1: Fundamental Theorem of Calculus" aria-label="Copy heading and permalink for Theorem 1: Fundamental Theorem of Calculus">🔗</a></div></article><div class="para" id="recitation-calculus-review-4-2">If <span class="process-math">\(f(x)\)</span> is a probability density function (don’t worry about what that means for now), then integrals such as <span class="process-math">\(\int_a^b f(x)\ dx\)</span> are used to calculate certain probabilities, as are values of the form <span class="process-math">\(F(x)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-4-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
<div class="para logical" id="recitation-calculus-review-8-1-1">
<div class="para">Practice with the following integrals:</div>
<div class="displaymath process-math" id="recitation-calculus-review-8-1-1-1">
\begin{align*}
\int_0^1 \frac{3}{2} x^{3/2} \ dx \amp = \\
\int_0^x 3e^{-3t} \ dt \amp =
\end{align*}
</div> </div>
<div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-8-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div> <article class="exercise exercise-like" id="recitation-calculus-review-4-3"><h3 class="heading"><span class="codenumber">5<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
<div class="introduction" id="recitation-calculus-review-4-3-1"><div class="para" id="recitation-calculus-review-4-3-1-1">Practice with the following integrals:<div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-4-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div></div>
<article class="task exercise-like" id="recitation-calculus-review-4-3-2"><h4 class="heading"><span class="codenumber">(a)</span></h4>
<div class="para" id="recitation-calculus-review-4-3-2-1">
<span class="process-math">\(\displaystyle \int_0^1 \frac{3}{2} x^{3/2} \ dx = \)</span><div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-4-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> </div>
<div class="autopermalink" data-description="Worksheet Exercise 5"><a href="#recitation-calculus-review-8" title="Copy heading and permalink for Worksheet Exercise 5" aria-label="Copy heading and permalink for Worksheet Exercise 5">🔗</a></div></article><div class="autopermalink" data-description="Worksheet: Calculus Review"><a href="#recitation-calculus-review" title="Copy heading and permalink for Worksheet: Calculus Review" aria-label="Copy heading and permalink for Worksheet: Calculus Review">🔗</a></div></section></div> <div class="workspace" data-space="2cm"></div>
<div class="autopermalink" data-description="Task 5.a"><a href="#recitation-calculus-review-4-3-2" title="Copy heading and permalink for Task 5.a" aria-label="Copy heading and permalink for Task 5.a">🔗</a></div></article><article class="task exercise-like" id="recitation-calculus-review-4-3-3"><h4 class="heading"><span class="codenumber">(b)</span></h4>
<div class="para" id="recitation-calculus-review-4-3-3-1">
<span class="process-math">\(\displaystyle \int_0^x 3e^{-3t} \ dt = \)</span><div class="autopermalink" data-description="Paragraph"><a href="#recitation-calculus-review-4-3-3-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="workspace" data-space="2cm"></div>
<div class="autopermalink" data-description="Task 5.b"><a href="#recitation-calculus-review-4-3-3" title="Copy heading and permalink for Task 5.b" aria-label="Copy heading and permalink for Task 5.b">🔗</a></div></article><div class="autopermalink" data-description="Worksheet Exercise 5"><a href="#recitation-calculus-review-4-3" title="Copy heading and permalink for Worksheet Exercise 5" aria-label="Copy heading and permalink for Worksheet Exercise 5">🔗</a></div></article></section><div class="autopermalink" data-description="Worksheet: Calculus Review"><a href="#recitation-calculus-review" title="Copy heading and permalink for Worksheet: Calculus Review" aria-label="Copy heading and permalink for Worksheet: Calculus Review">🔗</a></div></section></div>
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