Reference no: EM131081164
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 Rn(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 fg ≤ 1.
(b) Prove the Schwarz inequality that for any two integrable functions f and g on an interval [a, b],
| a∫bfg| ≤ (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.