Master’s Comprehensive Exam Advanced Calculus August 1997

1. (a) Suppose that A ⊂ ℝ and f : A → ℝ. Define:

i. “f is continuous on A”

ii. “f is uniformly continuous on A”

(b) Give an example of a continuous function f : (0, 1) → ℝ which is not uniformly continuous. Explain.

(c) Fill in the blank to make a true statement (no proof required): “Suppose that A ⊂ ℝ and f : A → ℝ is continuous. If A is a set then f is uniformly continuous on A.”

2. Suppose that f : ℝ → ℝ is continuously differentiable (that is, the derivative exists and is

continuous). For n = 1, 2, ..., define g_{n}(x) = \frac{f(x + 1/n) − f(x)}{1/n}. Show that g_{n}(x) → f′(x)

uniformly for 0 ≤ x ≤ 1 as n → ∞. (Note that the pointwise convergence is just the definition of the derivative, so you should make certain you prove uniform convergence. You may use basic facts about continuity and compact sets.)

3. (a) Suppose that g(s, t) is a differentiable function of two variables and let f(x) = g(x^{2}, x). Give a formula for the derivative f′(x) in terms of the partial derivatives of g.

(b) Define f(x) = ∫_{0}^{x} sin(x^{2}t^{2}) dt. Find the derivative f′(x). (The answer will involve some

definite integrals you cannot and need not evaluate, just as in the definition of f.)

4. Suppose that f : ℝ^{2} → ℝ.

(a) Define what it means to say that f is differentiable at the origin.

(b) Give an example where f is not differentiable at the origin, even though both partial derivatives f_{x}(0, 0) and f_{y}(0, 0) exist.

5. Suppose that the series ∑_{n=1}^{∞} |a_{n}| converges and show that ∑_{n=1}^{∞} a_{n} converges. (In other words,

show that absolute convergence implies convergence.)

6. Prove true or false: Any subset of ℝ is either open or closed.

7. Let f(x) = sin(e^{(x^{2})} + cos(1/(1 + x^{2}))). Find ∫_{0}^{1} f′(x) dx.