2. Let a < b and let f be a function defined on [a, b]. Suppose that f is continuous on [a,b] and f is differentiable on (a, b). (a) Let ze [a, b) and let h> 0 be such that a+h≤b. Prove that there is € (0,1 such that: 1 f(a+h)-f(x) h = f'(x + 0h) Hint: r, h are fixed here. Define a new function and apply the MVT to this new function. (b) Consider f(y) = ln(y). Fix z> 0 and h> 0. Find in terms of z and h, as in the previous formula.
2. Let a < b and let f be a function defined on [a, b]. Suppose that f is continuous on [a,b] and f is differentiable on (a, b). (a) Let ze [a, b) and let h> 0 be such that a+h≤b. Prove that there is € (0,1 such that: 1 f(a+h)-f(x) h = f'(x + 0h) Hint: r, h are fixed here. Define a new function and apply the MVT to this new function. (b) Consider f(y) = ln(y). Fix z> 0 and h> 0. Find in terms of z and h, as in the previous formula.
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter3: Functions
Section3.3: More On Functions; Piecewise-defined Functions
Problem 98E: Determine if the statemment is true or false. If the statement is false, then correct it and make it...
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Please answer part b of this question, thank you.
![2. Let a < b and let f be a function defined on [a, b]. Suppose that f is continuous on [a, b] and f is
differentiable on (a, b).
(a) Let x = [a, b) and let h> 0 be such that r+h≤ b. Prove that there is € (0, 1) such that:
f(x+h)-f(x)
h
= f'(x + 0h)
Hint: x, h are fixed here. Define a new function and apply the MVT to this new function.
(b) Consider f(y) = ln(y). Fix xz> 0 and h> 0. Find in terms of r and h, as in the previous
formula.
(c) Fix > 0. Find
lim 0(x, h).
h→0
Hint: You may need to use L'Hopital's rule here.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff4893c46-a94c-4c81-a9fc-277ce07300a6%2Fd38413f2-772e-4d9f-8355-801ef9ed47db%2Fb5r6bml_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2. Let a < b and let f be a function defined on [a, b]. Suppose that f is continuous on [a, b] and f is
differentiable on (a, b).
(a) Let x = [a, b) and let h> 0 be such that r+h≤ b. Prove that there is € (0, 1) such that:
f(x+h)-f(x)
h
= f'(x + 0h)
Hint: x, h are fixed here. Define a new function and apply the MVT to this new function.
(b) Consider f(y) = ln(y). Fix xz> 0 and h> 0. Find in terms of r and h, as in the previous
formula.
(c) Fix > 0. Find
lim 0(x, h).
h→0
Hint: You may need to use L'Hopital's rule here.
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