andy/Fall-2026-Math-1041
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity

Better recitation formatting

Browse Source
This commit is contained in:
andy
2026-01-09 13:38:46 -05:00
parent c878378375
commit 4a82a1ed7a
1 changed files with 108 additions and 67 deletions
Download Patch File Download Diff File Expand all files Collapse all files
source/recitations/calculus-review.ptx
+106 -65
View File
@@ -11,75 +11,116 @@
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">
<title>Fundamental Theorem of Calculus</title>
<statement>
<page>
<!-- Exercises start here. -->
<exercise>
<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:
<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>.
The Power Rule for derivatives states:
<md>
<mrow> \frac{d}{dx}\left( x^n \right) = </mrow>
</md>
</p>
</statement>
</theorem>
</statement>
</exercise>
<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>
<exercise>
<statement>
<p>
The Power Rule for antiderivatives states:
<md>
<mrow> \int\left( x^n \right)\ dx = </mrow>
</md>
</p>
</statement>
</exercise>
<exercise>
<introduction>
<p>
Practice with the following derivatives:
</p>
</introduction>
<task workspace="2cm">
<p>
<m>\displaystyle \frac{d}{dx}\left( 2x^2 + x^{-3} + 4\sqrt{x} \right) = </m>
</p>
</task>
<task workspace="2cm">
<p>
<m>\displaystyle \frac{d}{dx}\left( \frac{1}{x} + 3e^{2x} + x^e + \pi^e \right) = </m>
</p>
</task>
</exercise>
<exercise>
<introduction>
<p>
Practice with the following antiderivatives:
</p>
</introduction>
<task workspace="2cm">
<p>
<m>\displaystyle \int 2x^2 + x^{-3} + 4\sqrt{x} \ dx = </m>
</p>
</task>
<task workspace="2cm">
<p>
<m>\displaystyle \int \frac{1}{x} + 3e^{2x} + x^e + \pi^e \ dx = </m>
</p>
</task>
</exercise>
</page>
<page>
<theorem xml:id="thm-FTC">
<title>Fundamental Theorem of Calculus</title>
<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:
<md>
<mrow> \int_a^b f(x)\ dx = F(x)\bigg|_a^b = F(b) - F(a). </mrow>
</md>
</p>
</statement>
</theorem>
<p>
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, as are values of the form <m>F(x)</m>.
</p>
<exercise>
<introduction>
<p>
Practice with the following integrals:
</p>
</introduction>
<task workspace="2cm">
<p>
<m> \displaystyle \int_0^1 \frac{3}{2} x^{3/2} \ dx = </m>
</p>
</task>
<task workspace="2cm">
<p>
<m> \displaystyle \int_0^x 3e^{-3t} \ dt = </m>
</p>
</task>
</exercise>
</page>
</worksheet>