(a) For f(r) = x² sin(x), find a partition P of [-5, ) such that U(f, P) – L(f, P) < Note sin() = 1/2. (b) Use only limit definition of a derivative and the fact that sin(x) lim 1, 240 to directly find f'(7) for f(x) = x² sin(x). (c) Use only the e – 8 definition of continuous functions to directly prove f(x) = x² sin(x) is continuous at a = 0. Do not assume/use continuity of æ² or sin(x).
(a) For f(r) = x² sin(x), find a partition P of [-5, ) such that U(f, P) – L(f, P) < Note sin() = 1/2. (b) Use only limit definition of a derivative and the fact that sin(x) lim 1, 240 to directly find f'(7) for f(x) = x² sin(x). (c) Use only the e – 8 definition of continuous functions to directly prove f(x) = x² sin(x) is continuous at a = 0. Do not assume/use continuity of æ² or sin(x).
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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Solve for b but using the info in a)
![(a) For f(r) = x² sin(x), find a partition P of [-5, ] such that U(f, P) – L(f, P) < : Note
sin() = 1/2.
200
(b) Use only limit definition of a derivative and the fact that
sin(r)
lim
= 1,
to directly find f'(7) for f(x) = x² sin(x).
(c) Use only the e – 8 definition of continuous functions to directly prove f(x) = x² sin(x) is
continuous at r = 0. Do not assume/use continuity of x² or sin(x).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F23a2a698-a79b-4afa-a566-39831aa3db56%2Fb300bfca-17c5-4b9d-8af8-301259862a53%2Fb69v8bh_processed.png&w=3840&q=75)
Transcribed Image Text:(a) For f(r) = x² sin(x), find a partition P of [-5, ] such that U(f, P) – L(f, P) < : Note
sin() = 1/2.
200
(b) Use only limit definition of a derivative and the fact that
sin(r)
lim
= 1,
to directly find f'(7) for f(x) = x² sin(x).
(c) Use only the e – 8 definition of continuous functions to directly prove f(x) = x² sin(x) is
continuous at r = 0. Do not assume/use continuity of x² or sin(x).
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