Calculus II, Spring 2014

Midterm 1

April 01, 2014

Show all details.

1. (12 pts) Find the solution of 3xy′ − y = 1 + ln x on x > 0 with y(1) = −2.

2. (16 pts) Show that ∑_{k=1}^{∞} k^{−2} converges and evaluate lim_{n→∞} \frac{log ( ∑_{k=n}^{∞} k^{−2})}{log n}. Give details.

Hint: If the limit is p, this means that the sum in the limit is approximately n^{p}. Find p and prove it.

3. (10+10 pts)

(a) Show that the series 1− \frac{π^{2}}{4·2!} + \frac{π^{4}}{16·4!} −⋯+(−1)^{n} \frac{π^{2n}}{2^{2n}·(2n)!} +⋯ converges absolutely. Explain.

(b) Find the sum of the series in (a). \underline{Prove your answer}.

Hint: it is related to the Taylor series of an elementary function.

4. (16 pts) Give an approximation of ∫0^{\frac{1}{2}} sin(x^{2})dx to within 10^{−8}. Give the formula of the approximation, need not give the numerical value. Explain why the error is less than 10^{−8}.

5. (10+10 pts) True or False? Prove it if true, give a counter example if false.

(a) If f(x) = a_{0} + a_{1}x + a_{2}x^{2} + ⋯ on |x| < 1, then a_{n} = f^{(n)}(0)/n!.

(b) If g(x) = f(0) + ∑_{n=1}^{∞} \frac{f^{(n)}(0)}{n!}x^{n} on |x| < 1, then f(x) = g(x) on |x| < 1.

6. (16 pts) Let the region R be given as ’inside of r = 1’ and ‘outside of r = 1 − cos θ’. Find the area of R, the perimeter of R and surface area obtained from evolving R around the x-axis. Be careful about graphing R as it affects all that follows There are certain symmetry in the graph of R that you should notice.