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

College Algebra

10th Edition

ISBN:9781337282291

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Exponential And Logarithmic Functions

Section: Chapter Questions

Problem 39CT: The population P (in millions) of Texas from 2001 through 2014 can be approximated by the model...

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Question

) 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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