Masters Comprehensive Exam Advanced Calculus January 1993

Each question is worth 10 points.

1. Give an example of a subset D ⊂ R and a continuous function f with domain D such that f is not uniformly continuous on D. Prove your assertion that f is continuous but not uniformly continuous.

2. Let n be a non-negative integer, and P (x) a polynomial in x. Consider the function

f(x) = \{ \begin{matrix}\frac{P (x)e^{(−1/x^{2})}}{x^{n}} & if x ≠ 0 \\ 0 & otherwise \end{matrix}

(a) Show that f ∈ C^{1} and f′(0) = 0.

(b) Show that f ∈ C^{∞}.

(c) Show that f is not analytic unless P (x) = 0.

3. Consider the function u = f(x, y), where x = s^{2}t and y = st. Here, s, t are independent variables, and f ∈ C^{2}. Find u_{st} in terms of the partial derivatives of f.

4. True or false:

(a) If f_{n}(x) is a continuous function of x ∈ R, and lim_{n→∞} f_{n}(x) = f(x) pointwise, then f(x) is continuous.

(b) With the same hypotheses as in part (a), lim_{n→∞} ∫ f_{n} = ∫ f.

For each part: if the answer is “true”, give a proof. If the answer is “false”, give a counterexample, with justification.

5. Let C ⊂ R^{n}.

(a) Define what it means for C to be compact.

(b) Prove that if C is compact, then it is closed and bounded.

6. Let f be a function defined on a set D ⊂ R^{n}.

(a) Define ∫_{D}f.

(b) Give as many conditions as you can on D and f for this integral

to exist.

7. Prove that if a series converges absolutely, then it converges.

8. Let a_{n} be a sequence of real numbers. Give the definitions of lim inf_{n→∞} a_{n} and lim sup_{n→∞} a_{n}. Prove that if they are equal and finite, then lim_{n→∞} a_{n} exists, and is equal to this common value.

9. Let C be the curve bounding the semicircular region R in R^{2} defined by the inequalities y ≥ 0, x^{2} + y^{2} ≤ 1. Evaluate the integral ∫_{C} y(dx + dy) in two ways:

(a) Directly from the definition of the line integral.

(b) As an integral over R, using Green’s theorem. The curve C is to be traversed in the counterclockwise direction.