2 Calculus II

In this class,

• Questions must be answered in order asked.
• Solutions must be neat.
• Solutions must be exact and fully simplified.
• You must use the given symbols.
• You must show all reasoning.
• Copying is never allowed, even when working together.

1. For a conical tent of given volume, find the ratio of the height $h$ to the radius of the base $r$ that requires the least amount of material. Note: verify first that the surface area of a cone is $\pi{}r\sqrt{r^2+h^2}$.¹. [1, Maximum and Minimum Values]
2. Inside a conical tent of height $h$ and radius of the base $r$, a living space is to be partioned in the shape of a circular cylinder with a flat top of radius $R$ and height $H$. (a) Find the living space with the largest possible volume. (b) Find the living space with the largest curved surface.². [1, Maximum and Minimum Values]
3. Find the MacLaurin series for $1/(1+x^5)$. Hint: you may not want to crunch this out. Explain why not. Use a suitable trick instead. [1, Taylor and Maclaurin Series]
4. Write out the Taylor series for $\cos x$ around $\pi/3$ using the exact values of $\cos(\pi/3)$ and $\sin(\pi/3)$. Now assume that you approximate the Taylor series by its first three terms. What is the exact expression for the error in that approximation? How can you approximate this error at values of $x$ close to $\pi/3$? Use the approximate error to find the distance $r$ from $\pi/3$ so that the error is no more than 0.000 05 for all $x$ for which $\vert x-\frac13\pi\vert<r$. [1, Taylor and Maclaurin Series]
5. Find $\lim_{x\to0^+} \frac{\ln\cot x}{e^{csc^2 x}}$. [1, L'Hôpital's Rule]
6. Find $\lim_{x\to1} \left(\frac{1}{\ln(x)} - \frac{x}{x-1}\right)$. [1, L'Hôpital's Rule]

¹ Include a plot and picture

² Include graphs of the functions being maximized and a picture of the tent and cylinder