To find the linear approximation of the function [tex]\( f(x) = \sqrt{x} \)[/tex] at [tex]\( c = 9 \)[/tex], follow these steps:
1. Evaluate the function at [tex]\( c = 9 \)[/tex]:
[tex]\[
f(9) = \sqrt{9} = 3.0
\][/tex]
2. Find the derivative of the function [tex]\( f(x) = \sqrt{x} \)[/tex]:
To do this, we use the power rule for differentiation.
[tex]\[
f(x) = x^{1/2}
\][/tex]
The derivative is:
[tex]\[
f'(x) = \frac{1}{2} x^{-\frac{1}{2}} = \frac{1}{2\sqrt{x}}
\][/tex]
3. Evaluate the derivative at [tex]\( c = 9 \)[/tex]:
[tex]\[
f'(9) = \frac{1}{2\sqrt{9}} = \frac{1}{2 \times 3} = \frac{1}{6} \approx 0.1667
\][/tex]
4. Form the linear approximation formula:
The linear approximation of a function at a point [tex]\( c \)[/tex] is given by:
[tex]\[
L(x) = f(c) + f'(c)(x - c)
\][/tex]
Substituting the values we found for [tex]\( f(9) \)[/tex] and [tex]\( f'(9) \)[/tex]:
[tex]\[
L(x) = 3.0 + 0.1667(x - 9)
\][/tex]
Therefore, the linear approximation of [tex]\( f(x) = \sqrt{x} \)[/tex] at [tex]\( c = 9 \)[/tex] is:
[tex]\[
L(x) = 3.0 + 0.1667(x - 9)
\][/tex]
