   Chapter 4.4, Problem 84E Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516

Solutions

Chapter
Section Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516
Textbook Problem

Linear and Quadratic Approximations In Exercises83-86, use a graphing utility to graph the function. Then graph the linear and quadratic approximations P 1 ( x ) = f ( a ) + f ' ( a ) ( x − a ) and P 2 ( x ) = f ( a ) + f ' ( a ) ( x − a ) + 1 2 f n ( a ) ( x − a ) 2 in the same viewing window. Compare the values of f, P 1 , and P 2 , and their first derivatives at x = a . How do the approximations change as you move farther away from x = a ? Function Value of a f ( x ) = 2 ( sin x + cos x ) a = 0

To determine

To graph: The function f(x)=2(sinx+cosx) and its two approximations and then compare values of the 3 functions and the derivative of f at the point provided.

Explanation

Given: The function f(x)=2(sinx+cosx) and the value a=0.

Graph:

Differentiate the function twice to obtain:

f'(x)=2(cosxsinx)f''(x)=2(sinxcosx)

Now use these to get both the linear and quadratic approximations at a.

P1(x)=2(sin(0)+cos(0))+2(cos(0)sin(0))(x0)=2(0+1)+2(10)(x0)=2+2x

And,

P2(x)=2(sin

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