Answered: Find the linear approximation, L(x), of…

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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) Find the linear approximation, L(x), of at x = 9. Use this to approximate f(11).
Do you think your approximation is too big or too small? Why? Will the approximation of f(10)
be better or worse? Why? (

Expert Solution

Step 1

Let take a function:

$f (x) = \sqrt{x}$

The linear approximation can be calculated as:

$L (x) = f (a) + f' (a) (x - a) . . . . . . . . . . . . . . (1)$

Linear approximation of $f (x) = \sqrt{x}$ at $x=9$ and using the approximation to estimate $f (11)$ :

$f (x) = \sqrt{x} f' (x) = \frac{1}{2 \sqrt{x}}$

Now put x=9 in $f (x), f' (x)$

$f (9) = \sqrt{9} f (9) = 3 f' (9) = \frac{1}{2 \sqrt{9}} f' (9) = \frac{1}{6}$

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