Calculus II, Spring 2013

Midterm Exam 1

Mar 28, 2013, 10:10 AM. Show all details.

1. (16 pts) Is the integral ∫_{0}^{∞} \frac{1}{\sqrt[3]{x + x^{2}}} dx convergent? Find all possible locations of divergence and check out one by one. Do not try to find the anti-derivative.

2. (10+6 pts) True or False? Prove it if true, give a counter example if false. (a) If a_{0} +a_{1}x+a_{2}x^{2} +⋯ converges at x = −2, then it converges absolutely at x = 1.

(b) If a_{0} + a_{1}x + a_{2}x^{2} + ⋯ converges at x = −1, then it converges at x = 1.

3. (16 pts) Evaluate lim_{n→∞} \frac{log ( ∑_{k=1}^{n} k^{\frac{−1}{2}} )}{log n}. Give details.

Hint: If the limit is p, this means that the sum is approximately n^{p}. Find p. To prove it, recall the proof of integral test.

4. (6+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.

5. (10 pts) Suppose that y(x) is given by

\frac{dy}{dx} = y + cos(y − 1), y(0) = 1

and y(x) admits a Taylor series expansion around x = 0. Find a_{0}, a_{1}, a_{2} and a_{3} for the expansion of y(x).

6. (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.

7. (10 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}.

8. (10 pts) Let a parametrized curve be given by r = e^{2θ} in polar coordinate. Find \frac{dy}{dx} and \frac{d^{2}y}{dx^{2}} of the curve at θ = π/2.