2 Calculus II

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 and analyze the graph. What order size maximizes the receipts of the company? Give the reasons why the financial management of this company is clearly in incompetent hands.¹. [1, Maximum and Minimum Values]
2. 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 $\pi{r}\sqrt{r^2+h^2}$.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 find the largest distance $r$ from $\pi/3$ so that the error in the three-term Taylor series is no more than 0.000 05 when $\vert x-\frac13\pi\vert<r$. Find $r$ to two significant digits accurate without using trial and error. [1, Taylor and Maclaurin Series]
5. Find $\lim_{x\downarrow0} \ln(\cot x)/e^{\csc^2 x}$. [1, L'Hôpital's Rule]
6. Find $\lim_{x\to\pi/2} (\sec^3 x - \tan^3 x)$. [1, L'Hôpital's Rule]

¹ Include a plot

² Include a plot and picture