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Recitation 1

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</p> </p>
</introduction> </introduction>
<!-- Exercises start here. --> <!-- Exercises start here. -->
<exercise workspace="2in"> <exercise>
<statement> <statement>
<p> <p>
The Power Rule for derivatives states: The Power Rule for derivatives states:
<md number="yes"> <md>
<mrow> \frac{d}{dx}\left( x^n \right) = </mrow> <mrow> \frac{d}{dx}\left( x^n \right) = </mrow>
</md> </md>
</p> </p>
</statement> </statement>
</exercise> </exercise>
<exercise>
<statement>
<p>
The Power Rule for antiderivatives states:
<md>
<mrow> \int\left( x^n \right)\ dx = </mrow>
</md>
</p>
</statement>
</exercise>
<exercise>
<statement>
<p>
Practice with the following derivatives:
<md>
<mrow> \frac{d}{dx}\left( 2x^2 + x^{-3} + 4\sqrt{x} \right) \amp = </mrow>
<mrow> \frac{d}{dx}\left( \frac{1}{x} + 3e^{2x} + x^e + \pi^e \right) \amp = </mrow>
</md>
</p>
</statement>
</exercise>
<exercise>
<statement>
<p>
Practice with the following antiderivatives:
<md>
<mrow> \int 2x^2 + x^{-3} + 4\sqrt{x} \ dx \amp = </mrow>
<mrow> \int \frac{1}{x} + 3e^{2x} + x^e + \pi^e \ dx \amp = </mrow>
</md>
</p>
</statement>
</exercise>
<theorem xml:id="thm-FTC">
<statement>
<p>
If <m>F(x)</m> is an antiderivative of <m>f(x)</m><mdash />i.e., if <m>F'(x) = f(x)</m><mdash />then the Fundamental Theorem of Calculus (FTC) says:
<md>
<mrow> \int_a^b f(x)\ dx = F(x)\bigg|_a^b = F(b) - F(a). </mrow>
</md>
If <m>f(x)</m> is a probability density function (don't worry about what that means for now), then integrals such as <m>\int_a^b f(x)\ dx</m> are used to calculate certain probabilities, and so are function values of the form <m>F(x)</m>.
</p>
</statement>
</theorem>
<exercise>
<statement>
<p>
Practice with the following integrals:
<md>
<mrow> \int_0^1 \frac{3}{2} x^{3/2} \ dx \amp = </mrow>
<mrow> \int_0^x 3e^{-3t} \ dt \amp = </mrow>
</md>
</p>
</statement>
</exercise>
</worksheet> </worksheet>