(a) Explanation Let us define a function f : R → R f ( x ) = cos ( x ) Now, let us define an interval [ 0 , x ] .Since, f ( x ) = cos ( x ) is trigonometry function and therefore, is continuous on ( 0 , x ) and differentiable on [ 0 , x ] . It is evident from the function, f ( x ) = cos ( x ) that f ( 0 ) = 1 From Mean Value Theorem f ' (b) To determine lim x → 0 ( cos ( x ) − 1 x )

6th Edition

ISBN: 9781465208880

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topic…

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Question

Chapter 4.2, Problem 45PS

(a)

To determine

The proof that there is a number ‘w’ between 0 and ‘x’ for which −sin(w)=cos(x)−1x

(b)

To determine

limx→0(cos(x)−1x)

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Chapter 4.2, Problem 44PS

Chapter 4.2, Problem 46PS

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A. Let e = 2x, then the identity cos 2x = 1-2 sin2 x becomes cos 0 = 1–2 sin2 (0/2). a. Use this fact and lim0 ine = 1 to prove that lim.0 g- 0. b. Using the lim,-0 definition of a derivative and the result from part (a), show that * cos r = - sin r. c. use the result from part (a), the equation sin x+ cos x = 1 and the product rule to prove that E sin z = cos r.

Chapter 4 Solutions

Calculus

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The proof that there is a number ‘w’ between 0 and ‘x’ for which − sin ( w ) = cos ( x ) − 1 x

Solution Summary: The author explains the proof that there is a number ‘w’ between 0 and ‘x’.