Math 104: Homework 9 (due April 15)

1. Ross exercise 28.4

2. Suppose that f_{n} is a sequence of real-valued functions defined on an interval [a, b]

that converges uniformly to a function f. Let x_{0} ∈ [a, b], and suppose that

lim_{x→x_{0}} f_{n}(x) = l_{n}

for all n ∈ .

(a) Prove that l_{n} is a Cauchy sequence, and hence that it converges to a limit l.

(b) Prove that lim_{x→x_{0}} f(x) = l.

3. (a) Let f : → be a twice differentiable function, where f(0) = 0 and f′′(x) ≥ 0

for all x > 0. Prove that f(x)/x is increasing for x > 0.

(b) If f : → is twice differentiable, f(0) = 0 and if f(x)/x is increasing for all x > 0, show that f′′(x) ≥ 0 for some x > 0, but not necessarily for all x > 0. [Hint: consider f(x) = x(1 − e^{−x} ).]

4. Ross exercise 29.17

5. Let f(x) = |x|^{3}. Compute f′(x), f′′(x), and show that f^{(3)}(0) does not exist.

6. Ross exercise 30.1

7. Suppose that f is differentiable at a point a. Define

L_{1}(a, h) = \frac{f(a + h) − f(a − h)}{2h} ,
L_{2}(a, h) = \frac{−f(a + 2h) + 8f(a + h) − 8f(a − h) + f(a − 2h)}{12h}  .

(a) Prove that lim_{h→0} L_{i}(a, h) = f′(a) for i = 1, 2.

(b) Consider the case when f(x) = x^{5}. How does |L_{i}(a, h) − f′(a)| behave as h → 0 for i = 1, 2? Is there a difference in the rate of convergence?