Master’s Comprehensive Exam – Advanced Calculus – August 1996

1. (a) Define “f : [a, b] → R is Riemann integrable”.

(b) Give an example of a function which is not Riemann integrable and show, using your defini-

tion in part (a), that it is not.

2. Provide a counterexample to the converse of each of these theorems.

(a) If a set S in R is compact, then S is bounded.

(b) If a function f : R → R is differentiable, then f is continuous.

(c) If f : [0, 1] → R is continuous then f is bounded.

3. Show that if f′(x) > 0 for all x ∈ (a, b) then f is strictly increasing on (a, b).

4. What conditions on a and b are sufficient to guarantee that the equation

x^{2}y − y^{3} + x^{3} − a^{2}b − a^{3} + b^{3} = 0

has a solution y = f(x) valid near (a, b)?

5. (a) For a function f : [a, b] → R define “f is of bounded variation on [a, b]” and define the total

variation of f on [a, b].

(b) Suppose that f : [a, b] → R is monotonically increasing. Show that f is of bounded variation

on [a, b] and determine it total variation.

6. (a) For a function f : R → R define “f is uniformly continuous on R.”

(b) Assume that f : R → R is bounded and assume that f is Riemann integrable on any finite interval. Define, for each x ∈ R, F (x) = ∫_{0}^{x} f(s) ds. Show that F is uniformly continuous on the set R.

7. Suppose that, for each n ≥ 1, f_{n} : [0, 1] → R is a continuous function and suppose that the sequence {f_{n}} converges uniformly to a function f on [0, 1]. Show that

lim_{n→∞} ∫_{0}^{1} f_{n}(x) dx = ∫_{0}^{1} f(x) dx

8. Suppose that the series ∑_{n=0}^{∞} a_{n}x^{n} has a radius of convergence equal to r, a finite number. What is the radius of convergence of these two series?

(a) ∑_{n=0}^{∞} na_{n}x^{n}
(b) ∑_{n=0}^{∞} \frac{a_{n}}{2^{n}} x^{n}

9. (a) For what values of x does the series

∑_{n=1}^{∞}(1 − x^{2})^{n}

converge?

(b) Determine those intervals of the form [a, b] where the series converges uniformly.