Cnstruct a taylor series expansion for the function fx

Question

Math 104: Homework 10-

1. (a) Construct a Taylor series expansion for the function f(x) = log(1 + x) at x = 0. You can assume basic properties of logarithm.

(b) Use Taylor's theorem to write down an expression for the remainder R_n(x), and use this to prove that the Taylor series agrees with f in the range x ∈ (-1/2, 1). Note: it can be shown that the Taylor series agrees with f for x ∈ (-1, 1), as discussed in Chapter 26 of Ross. However, here, to illustrate Taylor's theorem, only a subset of this interval is considered.

2. 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].

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

4. (a) For any two numbers u, v ∈ R, prove that uv ≤ (u² + v²)/2. Let f and g be two integrable functions on [a, b]. Prove that if  _a∫^bf² = 1 and _a∫^bg² = 1 then

_a∫^b f_g ≤ 1.

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

| _a∫^bf_g| ≤ (_a∫^bf²)^1/2 (_a∫^bg²)^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.