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Sg(x) sin() a#0 f(x) = x = 0 where is differentiable at x 0 with g(0) = g'(0) = 0. Show that f is also differentiable at x = 0 and find f'(0). (You may either prove it in your own way or use the road map below.) (a) Use the definition of f to write out the difference quotient of f at x = f(0+h)-f(0) ? 0, i.e. what is (b) Try to find a lower bound and an upper bound for the absolute value of the difference quotient in part (a). (Remember | sin(c)| < 1 for all cE R) f(0+h)-f(0) (c) Use pinching theorem to show lim-070+=70 exists. What is the limit? What does this tell you about f'(0)?

Sg(x) sin() a#0 f(x) = x = 0 where is differentiable at x 0 with g(0) = g'(0) = 0. Show that f is also differentiable at x = 0 and find f'(0). (You may either prove it in your own way or use the road map below.) (a) Use the definition of f to write out the difference quotient of f at x = f(0+h)-f(0) ? 0, i.e. what is (b) Try to find a lower bound and an upper bound for the absolute value of the difference quotient in part (a). (Remember | sin(c)| < 1 for all cE R) f(0+h)-f(0) (c) Use pinching theorem to show lim-070+=70 exists. What is the limit? What does this tell you about f'(0)?

Advanced Engineering Mathematics
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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Problem 3. Let g(x) sin() x # 0 f (x) = x = 0 where g is differentiable at x = 0 with g(0) = g'(0) = 0. Show that f is also differentiable at r = 0 and find f'(0). (You may either prove it in your own way or use the road map below.) %D (a) Use the definition of f to write out the difference quotient of f at x = 0, i.e. what is f (0+h)–ƒ(0) 7 (b) Try to find a lower bound and an upper bound for the absolute value of the difference quotient in part (a). (Remember | sin(c)| <1 for all c e R) (c) Use pinching theorem to show limp-0 |70+2)=70 exists. What is the limit? What does this tell you about f'(0)? f(0+h)
Transcribed Image Text:Problem 3. Let g(x) sin() x # 0 f (x) = x = 0 where g is differentiable at x = 0 with g(0) = g'(0) = 0. Show that f is also differentiable at r = 0 and find f'(0). (You may either prove it in your own way or use the road map below.) %D (a) Use the definition of f to write out the difference quotient of f at x = 0, i.e. what is f (0+h)–ƒ(0) 7 (b) Try to find a lower bound and an upper bound for the absolute value of the difference quotient in part (a). (Remember | sin(c)| <1 for all c e R) (c) Use pinching theorem to show limp-0 |70+2)=70 exists. What is the limit? What does this tell you about f'(0)? f(0+h)
Expert Solution
Step 1

The given function is fx=gxsin1xx≠00x=0

where g is differentiable function at x=0 such that g0=g'0=0.

(a)

Now consider:

f0+h-f0h=g0+hsin10+h-0h=ghsin1hh

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