Comprehensive Examination Advanced Calculus January 1998

A passing mark will be given if 5 problems are completely solved.

1. Let S be a metric space.

(a) Define interior of a set T ⊂ S, int(T).

(b) Prove that if (S_{α})_{α∈A} is a sequence of subsets S, then

∪_{α∈A} int(S_{α}) ⊂ int(∪_{α∈A}S_{α}).

2. Show that if f : R → R is differentiable, with |f′(x)| ≤ 1 for all x ∈ R and f(0) = 0, then |f(x)| ≤ |x| for all x ∈ R.

3. Suppose that ∑_{n=1}^{∞} a_{n} is a convergent series of positive numbers. Prove that lim inf_{n→∞} na_{n} = 0.

4. Determine the radius of convergence and the convergence at the endpoints of the interval of convergence for the power series ∑_{n=1}^{∞} a_{n}x^{n} where

a_{n} = \begin{cases}\frac{1}{n2^{n}} & if n is even, \\ \frac{3^{n}}{n^{2}} & if n is odd. \end{cases}

5. Let f_{n}(x) = nxe^{−nx} for n = 1, 2, 3, ….

(a) Show that for any δ > 0, (f_{n})_{n=1}^{∞} converges uniformly to the function 0 on [δ, ∞).

(b) Show that (f_{n})_{n=1}^{∞} does not converge uniformly on [0, 1].

6. Let

f(x, y) = \begin{cases}\frac{xy^{2}}{x^{2} + y^{2}} & if (x, y) ≠ (0, 0), \\ 0 & else. \end{cases}

(a) For each unit vector u calculate the directional derivative of f at (0, 0) in the direction u.

(b) Explain why f is not differentiable at (0, 0).

7. Evaluate

∫_{C}(y^{2} + log(1 + x^{2})) dx + (e^{sin y} − x^{2}) dy

where C is the boundary of the unit square {(x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1} oriented counterclockwise.