Answered: f(x) 2 ;a = 1

Algebra and Trigonometry (MindTap Course List)

4th Edition

ISBN:9781305071742

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter2: Functions

Section2.4: Average Rate Of Change Of A Function

Problem 4.2E: bThe average rate of change of the linear function f(x)=3x+5 between any two points is ________.

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Question

Linear approximation and concavity Carry out the following steps for the given functions ƒ and points a.
a. Find the linear approximation L to the function ƒ at the point a.
b. Graph ƒ and L on the same set of axes.
c. Based on the graphs in part (b), state whether linear approximations
to ƒ near x = a are underestimates or overestimates.
d. Compute ƒ"(a) to confirm your conclusions in part (c).

Expert Solution

Step 1

$G i v e n f u n c t i o n i s f (x) = \frac{2}{x}; a = 1 w e h a v e t o f i n d l i n e a r a p p r o x i m a t i o n . a s w e k n o w t h e f r o m u l a f o r l i n e a r a p p r o x i m a t i o n i s L (x) = f (a) + f' (a) (x - a)$

Step 2

$a) f (x) = \frac{2}{x}; a = 1 L (x) = f (1) + f' (1) (x - 1) f (1) = 2 f' (x) = \frac{- 2}{x^{2}} f' (1) = - 2 T h e r e f o r e, L (x) = 2 - 2 (x - 1) \Rightarrow L (x) = 2 - 2 x + 2 \Rightarrow L (x) = 4 - 2 x \Rightarrow L (x) = 2 (2 - x) L i n e a r a p p r o x i m a t i o n L t o t h e f u n c t i o n f a t t h e p o i n t a = 1 i s L (x) = 2 (2 - x)$

Step 3

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