MASTER’S COMPREHENSIVE EXAM—ADVANCED CALCULUS AUGUST, 1989

1. Let A, B be bounded nonempty sets of real numbers. Define C = {a + b | a ∈ A, b ∈ B}. Show that sup C = sup A + sup B.

2. Find all values of x for which ∑_{n=1}^{∞} a_{n}x^{n} converges, where a_{n} = \{ \begin{matrix}(1/2)^{(n+1)/2} & if n is odd \\ (1/3)^{n/2} & if n is even. \end{matrix}

3. Let f : R → R be defined by

f(x) = \{ \begin{matrix}x + 2x^{2} sin(\frac{1}{x}) & if x ≠ 0 \\ 0 & if x = 0. \end{matrix}

(a) Find f′(0).

(b) Is f continuously differentiable on R?

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

g(x, y, z) = f(xy + xz + yz, e^{xyz}).

find (D_{i}g)(2, −1, 0) for i = 1, 2, 3.

5. Determine all values of h for which

∑_{n=1}^{∞} \frac{x}{n(4 + x^{2}n)}

converges uniformly on [−h, h]. Does this series converge uniformly on (−∞, ∞)?

6. Let f be a function of two variables for which f(1, 0) = 1. What additional conditions on f will

allow the equation

f(xy, xz) + f(e^{z}, yz) = 2

to be solved for z in a neighborhood of (1, 1, 0)?

7. Evaluate ∫_{D} exp(\frac{x − y}{x + y} ) dx dy, where D is the triangle bounded by the lines x = 0, y = 0, x + y = 1.

(Hint: Change of variable u = x + y, v = x − y.)

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