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<h1 class="heading"><a href="root-1-2.html"><span class="title">Introduction to Probability and Statistics for the Life Sciences</span> <span class="subtitle">Spring 2026 MATH 1044</span></a></h1>
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<p class="byline">Andy Eisenberg</p>
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<ul class="structural toc-item-list contains-active"><li class="toc-item toc-worksheet active"><div class="toc-title-box"><a href="recitation-calculus-review.html" class="internal"><span class="title">Calculus Review</span></a></div></li></ul>
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<main class="ptx-main"><div id="ptx-content" class="ptx-content"><section class="worksheet" id="recitation-calculus-review" data-margins="0.75in 0.75in 0.75in 0.75in"><h2 class="heading hide-type">
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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 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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\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>
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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="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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||
\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>
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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>
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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>
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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>
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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>
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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">
|
||
<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>
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||
</h3>
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||
<div class="para logical" id="thm-FTC-1-1">
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||
<div class="para">If <span class="process-math">\(F(x)\)</span> is an antiderivative of <span class="process-math">\(f(x)\)</span>—i.e., if <span class="process-math">\(F'(x) = f(x)\)</span>—then the Fundamental Theorem of Calculus (FTC) says:</div>
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||
<div class="displaymath process-math" id="thm-FTC-1-1-6">
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\begin{gather*}
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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>
|
||
</div>
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||
<div class="autopermalink" data-description="Paragraph"><a href="#thm-FTC-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="Theorem 1"><a href="#thm-FTC" title="Copy heading and permalink for Theorem 1" aria-label="Copy heading and permalink for Theorem 1">🔗</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>
|
||
<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>
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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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</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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