Comprehensive Exam Advanced Calculus August 1998

This exam consists of seven problems to be completed in two hours. A passing score will consist of five problems completed correctly. Please show all work.

1. Let (M, d) be a metric space.

(a) Define what it means for a subset A of M to be compact in M.

(b) Prove that the union of a finite number of compact subsets of M is compact.

2. (a) Define what it means for a function f to be uniformly continuous on an interval

S ⊆ ℝ.

(b) Let f(x) = \frac{1}{x}, let S_{1} = [1, 2], and let S_{2} = (0, 2]. Determine, with proof, whether

or not f is uniformly continuous on S_{1} and S_{2}.

3. Consider the real-valued double sequence { a_{p,q} | p, q ∈ ℕ }. Recall that such a sequence has a limit L, written lim_{p,q→∞} a_{p,q} = L, if for every ϵ > 0, there exists N such that n, m > N imply that |a_{n,m} − L| < ϵ.

(a) If the double limit lim_{p,q→∞} a_{p,q} exists and equals L, prove that

lim_{q→∞} (lim sup_{p→∞} a_{p,q}) = L.

(b) Give an example of a double sequence {a_{p,q}} where the double limit lim_{p,q→∞} a_{p,q} exists but the iterated limit

lim_{q→∞} ( lim_{p→∞} a_{p,q})

does not exist.

4. Suppose that f : ℝ → ℝ has a finite third derivative f′′′ in [a, b], and suppose that

f(a) = f(b) = f′(a) = f′(b) = 0.

Prove that f′′′(c) = 0 for some c ∈ (a, b).

5. Show that the series

∑_{n=0}^{∞} x^{n}(1 − x)^{2}

converges uniformly on [0, 1].

6. Prove that

lim_{n→∞} ∫0^{\frac{1}{n}} cos(nt^{2}) dt = 0.

7. Let f be a real-valued function defined on an open neighborhood of (1, 0) in R^{2} satisfying f(1, 0) = 1. State conditions on f that will ensure the existence of a continuously differentiable function g satisfying g(1, 1) = 0 and the condition

f(xy, xg(x, y)) + f(e^{g(x,y)}, yg(x, y)) = 2

in some neighborhood of (1, 1).