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@@ -170,49 +170,53 @@ eBookConfig.enable_chatcodes = false;
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<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"></span></a></div>
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</h2>
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<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>
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</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>
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<div class="para logical" id="recitation-calculus-review-3-1-1">
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</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>
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<div class="para logical" id="recitation-calculus-review-3-1-1-1">
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<div class="para">The Power Rule for derivatives states:</div>
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<div class="displaymath process-math" id="recitation-calculus-review-3-1-1-1">
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<div class="displaymath process-math" id="recitation-calculus-review-3-1-1-1-1">
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\begin{gather*}
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\frac{d}{dx}\left( x^n \right) =
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\end{gather*}
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</div>
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<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>
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<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>
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</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>
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<div class="para logical" id="recitation-calculus-review-4-1-1">
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<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>
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<div class="para logical" id="recitation-calculus-review-3-2-1-1">
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<div class="para">The Power Rule for antiderivatives states:</div>
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<div class="displaymath process-math" id="recitation-calculus-review-4-1-1-1">
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<div class="displaymath process-math" id="recitation-calculus-review-3-2-1-1-1">
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\begin{gather*}
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\int\left( x^n \right)\ dx =
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\end{gather*}
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</div>
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<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>
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<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>
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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>
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<div class="para logical" id="recitation-calculus-review-5-1-1">
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<div class="para">Practice with the following derivatives:</div>
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<div class="displaymath process-math" id="recitation-calculus-review-5-1-1-1">
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\begin{align*}
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\frac{d}{dx}\left( 2x^2 + x^{-3} + 4\sqrt{x} \right) \amp = \\
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\frac{d}{dx}\left( \frac{1}{x} + 3e^{2x} + x^e + \pi^e \right) \amp =
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\end{align*}
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<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>
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<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>
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</div></div>
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<article class="task exercise-like" id="recitation-calculus-review-3-3-2"><h4 class="heading"><span class="codenumber">(a)</span></h4>
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<div class="para" id="recitation-calculus-review-3-3-2-1">
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<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>
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</div>
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<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>
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<div class="workspace" data-space="2cm"></div>
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<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>
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<div class="para" id="recitation-calculus-review-3-3-3-1">
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<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>
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</div>
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<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>
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<div class="para logical" id="recitation-calculus-review-6-1-1">
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<div class="para">Practice with the following antiderivatives:</div>
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<div class="displaymath process-math" id="recitation-calculus-review-6-1-1-1">
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\begin{align*}
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\int 2x^2 + x^{-3} + 4\sqrt{x} \ dx \amp = \\
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\int \frac{1}{x} + 3e^{2x} + x^e + \pi^e \ dx \amp =
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\end{align*}
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<div class="workspace" data-space="2cm"></div>
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<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>
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<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>
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</div></div>
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<article class="task exercise-like" id="recitation-calculus-review-3-4-2"><h4 class="heading"><span class="codenumber">(a)</span></h4>
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<div class="para" id="recitation-calculus-review-3-4-2-1">
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<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>
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</div>
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<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>
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<div class="workspace" data-space="2cm"></div>
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<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>
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<div class="para" id="recitation-calculus-review-3-4-3-1">
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<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>
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</div>
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<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">
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<div class="workspace" data-space="2cm"></div>
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<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">
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<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>
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</h3>
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<div class="para logical" id="thm-FTC-2-1">
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@@ -222,22 +226,24 @@ eBookConfig.enable_chatcodes = false;
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\int_a^b f(x)\ dx = F(x)\bigg|_a^b = F(b) - F(a).
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\end{gather*}
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</div>
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<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>
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</div>
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<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>
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</div>
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<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>
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<div class="para logical" id="recitation-calculus-review-8-1-1">
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<div class="para">Practice with the following integrals:</div>
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<div class="displaymath process-math" id="recitation-calculus-review-8-1-1-1">
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\begin{align*}
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\int_0^1 \frac{3}{2} x^{3/2} \ dx \amp = \\
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\int_0^x 3e^{-3t} \ dt \amp =
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\end{align*}
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<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>
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</div>
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<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>
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<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>
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<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>
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</div></div>
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<article class="task exercise-like" id="recitation-calculus-review-4-3-2"><h4 class="heading"><span class="codenumber">(a)</span></h4>
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<div class="para" id="recitation-calculus-review-4-3-2-1">
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<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>
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</div>
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<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>
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<div class="workspace" data-space="2cm"></div>
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<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>
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<div class="para" id="recitation-calculus-review-4-3-3-1">
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<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>
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</div>
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<div class="workspace" data-space="2cm"></div>
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<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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