Math 104: Homework 10 (due April 22)

1. Construct a Taylor series expansion for the function f(x) = log(1 + x) at x = 0. By considering the remainder R_{n}(x), prove that the Taylor series agrees with f in the range −^{1}/2 < x < 1.

2. Ross exercise 32.7

3. Suppose f is a continuous function on [a, b], and f(x) ≥ 0 for all x ∈ [a, b]. Prove that if ∫_{a}^{b}f = 0, then f(x) = 0 for all x ∈ [a, b].

4. Construct an example of a function where f(x)^{2} is integrable on [0, 1] but f(x) is not.

5. Ross exercise 33.7

6. Ross exercise 33.8

7. (a) For any two numbers u, v ∈ , prove that uv ≤ (u^{2} + v^{2})/2. Let f and g be two integrable functions on [a, b]. Prove that if ∫_{a}^{b}f^{2} = 1 and ∫_{a}^{b}g^{2} = 1 then

∫_{a}^{b}fg ≤ 1.

(b) Prove the Schwarz inequality, that for any two integrable functions f and g on an interval [a, b],

|∫_{a}^{b}fg| ≤ ( ∫_{a}^{b}f^{2}) ^{1/2} ( ∫_{a}^{b}g^{2}) ^{1/2} .

(c) Let X be the set of all continuous functions on the interval [a, b]. For any f, g ∈ X, define

d(f, g) = ( ∫_{a}^{b}|f − g|^{2}) ^{1/2} .

Prove that d is a metric.