1.
\(\displaystyle \frac{d}{dx}\int_4^x e^{3t}\, dt\)
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Worksheet Some Exercises for this section
I included some practice problems that cover some main concepts in this section. You don’t need to turn it in, but I highly encourage you to work on this with your classmates. I may take problems here to be your in-class practice problems, homework problems, and/or exam problems. Reach out to Richard for help if you get stuck or have any questions.
I will only include the final answer to some of the problems for you to check your result. If you want to check your work, talk to Richard and he is happy to discuss the process with you.
Exercise Group.
Calculate the following derivative.
2.
\(\displaystyle \frac{d}{dx}\int_x^{-\frac{\pi}{4}} \sec^2\left(\theta\right)\, d\theta\)
3.
\(\displaystyle \frac{d}{dx}\int_1^\frac{1}{x} \cos^3(t)\, dt\)
4.
\(\displaystyle \frac{d}{du}\int_{-u}^{3u} \sqrt{x^2 + 1} \, dx\)
5.
Prove the formula
\begin{equation*}
\frac{d}{dx}\int_{u(x)}^{v(x)} f(t)\, dt = f\left(v(x)\right)\cdot v'(x) - f\left(u(x)\right)\cdot u'(x)
\end{equation*}
Note: By saying "prove", Richard means to justify this formula informally. The goal here is to make sure you know how the formula is derived and why each component is necessary. Richard will never look for a technical mathematical proof in this class, but you are certainly more than welcome to try coming up with a proof-y proof and Richard is always happy to chat about it.
6.
Let \(A\) be the accumulation function defined as
\begin{equation*}
A(x) = \int_0^x t\sin(t)\, dt
\end{equation*}
Below is the graph of \(f(t) = t\sin(t)\) on \([0,3\pi]\text{.}\)
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Determine the intervals on which the accumulation function, \(A\) is increasing (and decreasing)
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Determine the value of \(x\) where the accumulation function, \(A\text{,}\) has a peak (and valley).
