Use the linear approximation of the function $f(x)=\sqrt{9-x}$ at $a=0$ to approximate the number $\sqrt{9.08}$ .
A) $7.4445$
B) $2.2556$
C) $3.0133$
D) $7.4556$
E) $0.1556$

Question

Quizplus · Accepted Answer

The linear approximation of $f(x)=\sqrt{9-x}$ at $a=0$ is $f(x) \approx f(0) + f'(0)x$. Here, $f(0) = \sqrt{9} = 3$ and $f'(x) = -\frac{1}{2\sqrt{9-x}}$, so $f'(0) = -\frac{1}{6}$. Thus, the approximation is $3 - \frac{1}{6}x$. For $x = -0.08$ (since $9-x = 9.08$), the approximation is $3 - \frac{1}{6}(-0.08) = 3 + 0.0133 = 3.0133$.

Use the Linear Approximation of the Function f(x)=9−xf(x)=\sqrt{9-x}f(x)=9−x​ At a=0a=0a=0 To Approximate the Number

Use the Linear Approximation of the Function $f(x)=\sqrt{9-x}$ At $a=0$ To Approximate the Number