82 lines
2.6 KiB
XML
82 lines
2.6 KiB
XML
<?xml version="1.0" encoding="UTF-8"?>
|
|
|
|
<!-- When creating a new activity, make a copy of this file with appropriate name -->
|
|
<worksheet xml:id="recitation-calculus-review">
|
|
<title>Calculus Review</title>
|
|
|
|
<!-- Optional introduction -->
|
|
<introduction>
|
|
<p>
|
|
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.
|
|
</p>
|
|
</introduction>
|
|
<!-- Exercises start here. -->
|
|
<exercise>
|
|
<statement>
|
|
<p>
|
|
The Power Rule for derivatives states:
|
|
<md>
|
|
<mrow> \frac{d}{dx}\left( x^n \right) = </mrow>
|
|
</md>
|
|
</p>
|
|
</statement>
|
|
</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> |