Master’s Comprehensive Exam – Advanced Calculus – January 1995

Part I: Do \underline{all} of these problems.

1. (5 points) Define f : [a, b] → R is a Riemann integrable function.

2. (5 points) Define the function s : [0, 3] → R to be the greatest integer function:

s(x) = \{ \begin{matrix}0, & 0 ≤ x < 1 \\ 1, & 1 ≤ x < 2 \\ 2, & 2 ≤ x < 3 \\ 3, & x = 3 \end{matrix} \}

Define A(x) = ∫_{0}^{x} s(t) dt. At what points in [0,3] is A(x) continuous? At what points is it differentiable?

3. (16 points)

(a) Define a Cauchy sequence.

(b) Define uniform continuity of a function.

(c) Show that if the sequence {p_{n}} is Cauchy and f is a continuous function, then the sequence {f(p_{n})} is also Cauchy,

(d) Give an example of a continuous function and a Cauchy sequence {p_{n}} such that {f(p_{n})} is not Cauchy.

4. (a) (6 points) For what values of x does the series ∑_{n=1}^{∞}(1 − x^{2})^{n} converge?

(b) (6 points) Does this series converge uniformly on (\frac{1}{2}, 1)? Why?

5. (10 points) Suppose that f : [0, 5] → R is continuous and f(0) = 1. Also suppose f is differentiable and |f′(x)| ≤ 3 for x ∈ (0, 5). Show that f(5) < 17.

6. (8 points) Let B = B(0, r) denote the open ball of radius r > 0 centered at the origin 0 in R^{n}. Suppose f = (f_{1}, f_{2}, …, f_{n}) defines a function f : B → R^{n} and that each of the component functions f_{i} has continuous partial derivatives in B. Suppose that for each vector x ∈ B, f(x) = f(−x). Why is it true that the Jacobian determinant J_{f}(0) must be zero at x = 0?

7. (10 points) Consider f(x, y, z) defining a function f : R^{3} → R. Given that the directional derivative of f in the direction u⃗ is given by the dot product ∇f · u⃗, discuss the following.

(a) Why it is that the gradient vector ∇f always points in the direction of maximal directional derivative?

(b) Why is it that level curves of the surface f(x, y, z) always run orthogonal to the gradient?

8. (10 points) Suppose that F (x, y) = f(xe^{y}, xy) where f is a real-valued function of two variables. Express F_{1} = \frac{∂F}{∂x}, F_{2} = \frac{∂F}{∂y} and F_{1,2} = \frac{∂F^{2}}{∂x∂y} in terms of the partials of f.

Part II: Do \underline{only two} of the last three problems.

9. (12 points) Suppose that f_{n} : [0, 1] → R is a sequence of continuous functions and {f_{n}} converges uniformly to f. Show that f is continuous on [0, 1].

10. (12 points) Show that the set of all sequences of 0’s and 1’s is an uncountable set.

11. (12 points)

(a) Define a set S is n-space is compact.

(b) Show from the definition that any compact set is closed. (Hint: A possible approach is to use the fact that if a set is not closed, then there is a point p not in the set which is an accumulation point of the set.)