MASTER’S COMPREHENSIVE EXAM—ADVANCED CALCULUS AUGUST, 1994

1. Let A and B be nonempty sets of real numbers such that a < b for all a ∈ A and all b ∈ B. Show that sup(A) ≤ inf(B).

2. Find all values of x for which the series ∑_{n=0}^{∞}(x cos \frac{nπ}{3} )^{n} converges.

3. Let a be a positive integer and define f : R → R by

f(x) = \{ \begin{matrix}x^{3} sin \frac{1}{x^{a}} & if x ≠ 0 \\ 0 & if x = 0. \end{matrix}

(a) For what values of a is f differentiable at 0 and what is f′(0)?

(b) For what values of a is f′ continuous at 0?

4. Let f : R^{2} → R be continuously differentiable on R^{2} and suppose that f satisfies (D_{1}f)(1, −2) = 2, (D_{2}f)(1, −2) = 3. Define g : R^{3} → R by the formula

g(x, y, z) = f(xyz, −2 cos(xy + xz + yz + 1)).

Find (D_{i}g)(−1, 1, −1) for i = 1, 2, 3.

5. Find all values of h for which the series ∑_{n=1}^{∞} \frac{x}{n(4 + x^{2}n)} converges uniformly on [−h, h]. Does it

converge uniformly on (−∞, ∞)?

6. Given f : R^{2} → R satisfying f(−2, 1) = 1, what additional conditions on f will allow you to solve

the equation

f(xy + xz + yz, e^{xyz}) = 1

for z as a function of x and y in a neighborhood of the point (2, −1, 0)?

7. Let D be the rectangular region in the plane

D = {(u, v) ∈ R^{2} | 0 ≤ u ≤ 3 and 1 ≤ v ≤ 3}

and let T : R^{2} → R^{2} be the transformation

T (u, v) = (u, v(1 + 2u)).

(a) Sketch T (D) and find the equations of the curves that form its boundary.

(b) Evaluate ∫_{T (D)} \frac{dx dy}{1 + 2x}.

8. Evaluate ∫_{C}(y + e^{x^{2}})dx + (ln(1 + y^{2}) − x^{2})dy, where C is the boundary of the unit square 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 oriented counterclockwise. (Hint: Green’s Theorem)