Master’s Comprehensive Exam – Advanced Calculus – January 1997

1. Define the double Riemann integral ∫ ∫_{A} f for a continuous function f defined on a rectangle A in R^{2}.

2. Suppose that u = f(x, y) where f and all of its partial derivatives exist. If x = s^{2} + t and if y = st then find u_{st} in terms of the partial derivatives of f.

3. (a) Define:

i. A set S in R^{n} is open. ii. A set S in R^{n} is closed. iii. A point x ∈ R^{n} is an accumulation point of a set S.

(b) Show that a set is closed if and only if it contains all of its accumulation points.

4. Indicate which of these functions are uniformly continuous on the set (0, ∞). No reasoning is needed, just the answer.

(a) f(x) = x  (b) f(x) = |x|  (c) f(x) = \frac{1}{x}
(d) f(x) = x^{2}  (e) f(x) = e^{x}  (f) f(x) = sin \frac{1}{x}
(g) f(x) = \frac{1}{x^{2}+1}  (h) f(x) = \sqrt{x}  (i) f(x) = sin x

5. (a) Let f_{n} : R → R be a sequence of functions and let A be a subset of R. Let the function f be defined on the set A. Define the concept of the sequence {f_{n}} converging uniformly on the set A to f.

(b) Suppose that, for each n, f_{n} : R → R is continuous and suppose that f_{n} converges uniformly on R to f. Show that, for any a ∈ R, f is continuous at a.

6. (a) For A a subset of a metric space (S, d_{S}), give the definition of the set A being compact.

(b) Let f : S → T be a continuous function from one metric space (S, d_{S}) into another (T, d_{T}). If A is a compact subset of S show that f(A) is a compact subset of T.

7. Suppose that f : [0, ∞) → R is decreasing and f(x) > 0 for all x. Assume there is a constant K such that for each b > 0,

∫_{0}^{b} f(x) dx ≤ K.

Prove that ∑_{n=0}^{∞} f(n) converges.

8. Suppose that f : R → R is continuously differentiable and for some positive number p, f′(x) ≥ p for all x ∈ R. Suppose also that f(0) = 0 (this is not necessary, but may make calculations easier). Show that

(a) f is one-to one.

(b) f is onto.

9. Which of these series converge uniformly on the entire line R? Which converge uniformly on [0, 1]? Why?

(a) ∑_{n=1}^{∞} \frac{1}{x^{2} + n^{2}}

(b) ∑_{n=1}^{∞} \frac{x^{2} + n^{2}}{n^{2}}

(c) ∑_{n=1}^{∞} \frac{x^{2}}{n^{2}}