Master’s Comprehensive Exam – Advanced Calculus – May 1996

Do 9 of these 10 problems

1. (a) Define Riemann integrability on [a, b] for a function f : [a, b] → R.

(b) Consider f : [0, 1] → R defined by

f(x) = \{ \begin{matrix}1, & for x = 0 \\ 0, & for x ≠ 0 \end{matrix} \}

\underline{Using only the definition}, show that f is Riemann integrable on [0, 1].

2. Suppose that f : R → R is differentiable, f′(x) ≥ 3, and f(0) = 0. Show that f(10) ≥ 30.

3. Consider the function f : R^{2} → R defined by

f(x, y) = 2x^{4} + y^{4} + x^{2} − 2y^{2}.

Find the critical or stationary points and discuss the nature of each one.

4. (a) Define what it means for a sequence p_{n} of points to be a Cauchy sequence.

(b) Show that any convergent sequence is a Cauchy sequence.

5. (a) Define compactness for a subset S of the real line R.

(b) Show that any compact set is bounded.

6. For the relationship x^{2} + y^{2} − x^{3} = 0 \underline{explicitly} find a solution of the form y = f(x) valid near the point (5, −10).

7. Does the following series converge pointwise on the reals R? Does it converge uniformly on R?

∑_{n=1}^{∞} \frac{1}{n^{2} + x^{2}}

8. Suppose that a_{n} ≥ 0. Show that if ∑_{n=1}^{∞} a_{n} converges then so does ∑_{n=1}^{∞} a_{n}^{2}.

9. (a) Define bounded variation on [a, b] for a function f : [a, b] → R.

(b) Suppose that f : [a, b] → R is continuous and f′ is bounded. Show that f is of bounded

variation.

10. (a) Define what it means for a point x to be a boundary point of a set S ⊂ R^{n}.

(b) Show that if x is a boundary point of S ⊂ R^{n} then there is a sequence {s_{n}} of points from S such that lim_{x→∞} s_{n} = x.

(c) Is it true or false that if x is a boundary point of S then it is also possible to find a sequence {s_{n}} with s_{n} ∉ S such that lim_{x→∞} s_{n} = x?