Let ƒ : [a, b] → R be Riemman integrable, ƒ = R([a, b]). Define F : [a, b] → R by F(x) = f* ƒ(t)dt, Vx ¤ [a,b]. a 1. Give an example of a function f such that F is not differentiable at some point xo € (a, b). 2. Give an example of a function f such that F is differentiable in (a, b), but F'(x) ‡ f(x) for some x = (a, b). oes this contradict the Fundamental Theorem of Calculus? Explain.
Let ƒ : [a, b] → R be Riemman integrable, ƒ = R([a, b]). Define F : [a, b] → R by F(x) = f* ƒ(t)dt, Vx ¤ [a,b]. a 1. Give an example of a function f such that F is not differentiable at some point xo € (a, b). 2. Give an example of a function f such that F is differentiable in (a, b), but F'(x) ‡ f(x) for some x = (a, b). oes this contradict the Fundamental Theorem of Calculus? Explain.
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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![Let f: [a, b] → R be Riemman integrable, ƒ = R([a, b]). Define F : [a, b] → R by
F(x) = f* f(t)\ dt, \x € [a,b].
a
1. Give an example of a function f such that F is not differentiable at some point
xo € (a, b).
2. Give an example of a function f such that F is differentiable in (a, b), but F'(x) ‡
f(x) for some x ≤ (a, b).
Does this contradict the Fundamental Theorem of Calculus? Explain.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0a6878de-8200-48ed-837b-864fc15b0c65%2F3f719f6c-1f03-450c-b55e-b0f9eff1f633%2Fey0fll6_processed.png&w=3840&q=75)
Transcribed Image Text:Let f: [a, b] → R be Riemman integrable, ƒ = R([a, b]). Define F : [a, b] → R by
F(x) = f* f(t)\ dt, \x € [a,b].
a
1. Give an example of a function f such that F is not differentiable at some point
xo € (a, b).
2. Give an example of a function f such that F is differentiable in (a, b), but F'(x) ‡
f(x) for some x ≤ (a, b).
Does this contradict the Fundamental Theorem of Calculus? Explain.
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