MASTER’S COMPREHENSIVE EXAM—ADVANCED CALCULUS AUGUST, 1993

1. Let A be a bounded nonempty set of real numbers. Define −A = {−a | a ∈ A}. Show that

inf(−A) = − sup A.

2. Find all values of x for which ∑_{n=1}^{∞} a_{n}x^{n} converges, where a_{n} = \{ \begin{matrix}sin n & if n is odd \\ 4^{n/2} & if n is even. \end{matrix}

3. Fix a real number r and let f : R → R be defined by

f(x) = \{ \begin{matrix}\frac{|x|^{r}}{x} & if x ≠ 0 \\ 0 & if x = 0. \end{matrix}

(a) For what values of r is f continuous on R?

(b) For what values of r is f differentiable on R and what is f′(0)?

(c) For what values of r is f continuously differentiable on R?

4. Let f : R^{2} → R be of class C^{2} on R^{2} and define g : R^{2} → R by the formula

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

Express D_{1}g and D_{11}g in terms of the D_{i}f and the D_{ij}f, i, j = 1, 2.

5. Consider the series ∑_{n=1}^{∞}(xe^{−x})^{n}.

(a) For which R > 0 does the series converge uniformly on [0, R]?

(b) Does the series converge uniformly on [0, ∞)?

(c) For which R < 0 does the series converge uniformly on [R, 0]?

6. Let f and g be functions on R^{2} such that f(1, 1) = g(1, 1) = 1. Find conditions on f and g that

permit you to solve the equation

f(xy, yz)g(xy, yz) = 1

for y as a differentiable function of x and z near the point x = y = z = 1.

7. Evaluate ∫_{D} \frac{x + y}{xy} dx dy, where D is the region in the first quadrant bounded by the curves xy = 1,

xy = 3, y − x = −1, y − x = 1. (Hint: Make a change of variable to simplify the description of D.)

8. Evaluate ∫_{C}(y^{2} + sin(x^{2}))dx + (cos(y^{2}) − x)dy, where C is the boundary of the unit square 0 ≤ x ≤ 1, 0 ≤ y ≤ 1. (Hint: Green’s Theorem)