Masters Comprehensive Exam - Advanced Calculus - May 1991

1. (4 points each) Provide definitions for the following.

(a) E ⊂ R^{n} is compact.

(b) lim_{x→∞} f(x) = L

(c) f_{n}(x) converges to f(x), uniformly for x ∈ E.

(d) The metric space M is complete.

2. (10 points) Let F (t) = ∫_{0}^{t} \frac{sin(xt)}{x} dx. Find F′(t)

3. (12 points) Let f(x, y) = xy(1 − x − y). Find all critical points of f, and classify them as local maxima, local minima, or saddle points.

4. (10 points) Find all values of x for which the series ∑_{n=0}^{∞} \frac{nx^{n}}{n^{2}+1} converges.

5. (10 points) Let ω = ydx + zdy − dz. Find ∫_{C} ω, where C is the curve of intersection of the cylinder x^{2} + y^{2} = 1 with the plane z = x.

6. (10 points) Consider the equations

x_{1}^{2}x_{2} + x_{2}^{2}x_{3} + x_{3}^{2}x_{1} = 3
x_{1}^{2} + x_{2}^{2} + x_{3}^{2} = 2x_{2} + 2x_{3} − 1

For which of the variables x_{i} can these equations be solved for the other two variables x_{j} and x_{k} in terms of x_{i} in a neighborhood of the solution (1,1,1)? It is not necessary to solve the equations.

7. (10 points) Let T : R^{3} → R^{3} be the transformation defined by T (x, y, z) = (x^{2}y, y + z, z^{2} − x^{2}).

(a) Find the derivative of T.

(b) Near which points (x, y, z) does T have a local inverse?

8. (6 points each) Which of the following series converge conditionally, which converge absolutely, and which diverge? Explain your reasoning.

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

(b) ∑_{n=1}^{∞}(−1)^{n} \frac{(n!)^{3}}{(3n)!}

9. (10 points) Let g(x, y, z) = f(y sin(x), yz). Find g_{12} in terms of the partial derivatives of f.