2 Calculus II

In this class,

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

1. A company charges orders as follows
   • Orders of 50,000 items or less are charged at $30 per thousand.
   • For orders above 50,000, the charge per thousand is reduced by 37${\textstyle\frac{1}{2}}\hbox{\rlap{\small /}c}$ for each thousand above 50,000. (This reduced charge per thousand is applied to the complete order.)
   Plot the receipts of this company versus order size and analyze the graph for all features. What order size maximizes the receipts of the company? Give the reasons why the financial management of this company is clearly incompetent.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 $\sin(x^5)$. 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 $\vert x-\frac13\pi\vert<r$. Then improve on your error estimate to find a more accurate value of $r$. [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 $\lim_{x\to\pi/2} (\sec^3 x - \tan^3 x)$. [1, L'Hôpital's Rule]
7. Variable $z$ is given in terms of the measurable variables $f$ and $g$ as
   
   $$\frac1z = \frac1f + \frac1g$$
   
   
   The values of $f$ and $g$ and their uncertainties are:
   
   $$f = 4 \pm 0.01 \qquad g = 8 \pm 0.02$$
   
   
   What are the maximum relative and absolute errors in the computed $z$? Are you stunned by the value of the relative error in $z$? Explain why not. [1, Total Differential]

¹ Include a plot

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