2. (a)(b)Let 0 < r < 1 and fi [0, r] → R be continuous. Define, for3Jn+1(a) = | Em Và e [0,r].fn, €Show that fn converges to the zero function uniformly on [0,r].Does part (a) hold for r = 1? Justify your answer.n ≥ 1,

The given data in this question is as follows: We have a continuous function f1:[0,r]→R defined on…

Answered: 2. (a) (b) Let 0 < r < 1 and fi [0, r]…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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3. Define f(x) so that 16) = { exp (교1), |피 1 f(r) = 0, (a) Explain why the function f(x) must be continuous everywhere except perhaps at x = +1. Hint: How is f(x) constructed from known continuous functions? (b) State the definition for f(x) to be continuous at c. Assume that c is not an endpoint. (c) Prove that f(x) is in fact continuous at both –1 and 1. (d) State the limit definition for f(x) to be differentiable at c. Assume that c is not an endpoint. (e) Explain why the function f(x) must be differentiable everywhere except perhaps at r = ±1. Hint: How is f(x) constructed from known differentiable functions? (f) Prove that f(x) is in fact differentiable at both -1 and 1.
The approximated value for e using the Maclaurin series expansion considering a true relative error that is conforming to one significant figure is: O a. 4.052 O b. 3,885 Oc3414 O d. 2414
an Assume f(x) = ) anx" converges on (-R, R). Show that F(x) = > n+1 is defined n+1 n=0 n=0 on (–R, R) and satisfies F' = f on (-R, R).
A function f:R →R, where R'is the set of all real number satisfies the equati on. **- Ax) + f(v)+f(0) 3. for all x,y e R If f(x) is differenti ability at x = 0,then show that f'(x) exist for all x e R. 11
Suppose that f is a continuous, strictly increasing function on [0, 1] and that the •1 range of f is also [0, 1]. If | f(x) dx = 1/3, what is f-'(y) dy? А. 1/3 В. 1/2 С. 2/3 D. 3
The discontinuity can be removede by settinng the f(0,0) at what point?

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2. (a) (b) Let 0 < r < 1 and fi [0, r] → R be continuous. Define, for fn+1(x) = f* ; fn, Vx = [0,r]. Show that fn converges to the zero function uniformly on (0,r]. Does part (a) hold for r = 1? Justify your answer. n ≥ 1,