Reference no: EM131081598
Math 104: Homework 10-
1. Construct a Taylor series expansion for the function f(x) = log(1 + x) at x = 0. By considering the remainder Rn(x), prove that the Taylor series agrees with f in the range -1/2 < x < 1.
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∫b f2)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.
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