MASTER’S COMPREHENSIVE EXAM—ADVANCED CALCULUS JUNE, 1997 All problems have equal value

1. Let a be a positive integer and define f(x) = \{ \begin{matrix}x^{a} sin \frac{1}{x} & 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?

2. (a) Let f : R^{2} → R. Define the phrase “f is differentiable at the point (a, b) ∈ R^{2}.”

(b) Suppose f : R^{2} → R and g : R^{2} → R are differentiable at (a, b). Define F : R^{2} → R by the formula

F (x, y) = f(x, y) + g(x, y).

Prove directly from the definition that F is differentiable at (a, b).

3. Let f : R^{2} → R be a differentiable function satisfying D_{1}f(0, 0) = D_{2}f(0, 0) = 10. Define g : R^{3} → R

by the formula

g(x, y, z) = f((y − x)z, (z − x)y).

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

4. Find all real values of x for which the series ∑_{n=0}^{∞} a_{n}x^{n} converges, where

a_{n} = \{ \begin{matrix}\frac{1}{n} & if n is odd \\ (\frac{1}{2})^{n} & if n is even. \end{matrix}

5. Consider the sequence of functions f_{n}(x) = (x/n)e^{−x/n}.

(a) Show that this sequence converges pointwise for all x ∈ R.

(b) Show that this sequence converges uniformly on [0, C] for all C > 0.

(c) Does this sequence converge uniformly on [0, +∞)?

6. Let f : R → R and g : R → R. Suppose (x_{0}, y_{0}, u_{0}) ∈ R^{3} is a solution of the system

f(x) + f(y) = u
g(x) + g(y) = u^{2}.

State conditions on f and g that guarantee that this system has solutions x = p(u) and y = q(u) defined in a neighborhood of u_{0} and satisfying p(u_{0}) = x_{0} and q(u_{0}) = y_{0}.

7. Let f : [a, b] → R.

(a) Define the phrase “f is Riemann integrable on [a, b].”

(b) Prove directly from the definition that the function f : [0, 1] → R defined by

f(x) = \{ \begin{matrix}0 & if x is rational \\ 1 & if x is irrational \end{matrix}

is not Riemann integrable on [0, 1].