Let a < b and let f be a function defined on [a, b]. Suppose that f is continuous on [a, b] and fis differentiable on (a, b). (a) Let z € (a, b) and let h> 0 be such that r+h≤ b. Prove that there is 0 € (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). Fixz> 0 and h> 0. Find in terms of r and h, as in the previous formula. (c) Fix r> 0. Find lim 0(z, h). Hint: You may need to use L'Hopital's rule here.
Let a < b and let f be a function defined on [a, b]. Suppose that f is continuous on [a, b] and fis differentiable on (a, b). (a) Let z € (a, b) and let h> 0 be such that r+h≤ b. Prove that there is 0 € (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). Fixz> 0 and h> 0. Find in terms of r and h, as in the previous formula. (c) Fix r> 0. Find lim 0(z, h). Hint: You may need to use L'Hopital's rule here.
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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Please answer Part C 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 x > 0 and h> 0. Find in terms of x and h, as in the previous
formula.
(c) Fix x > 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%2Ffb84e696-1661-4099-a3fc-bc1ca9e36019%2Ffdn66k_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 x > 0 and h> 0. Find in terms of x and h, as in the previous
formula.
(c) Fix x > 0. Find
lim 0(x, h).
h→0
Hint: You may need to use L'Hopital's rule here.
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