The linear approximation $L ( x )$ to $f ( x ) = \sqrt [ 3 ] { x }$ near $x _ { 0 } = 8$ is

A) $- \frac { 1 } { 2 } + \frac { 1 } { 12 } ( x - 8 )$
B) $- 2 - \frac { 1 } { 12 } ( x - 8 )$
C) $2 + \frac { 1 } { 12 } ( x - 8 )$
D) $2 - \frac { 1 } { 12 } ( x - 8 )$
E) $- 2 + \frac { 1 } { 12 } ( x - 8 )$

Question

The linear approximation $L ( x )$ to $f ( x ) = \sqrt [ 3 ] { x }$ near $x _ { 0 } = 8$ is

A) $- \frac { 1 } { 2 } + \frac { 1 } { 12 } ( x - 8 )$
B) $- 2 - \frac { 1 } { 12 } ( x - 8 )$
C) $2 + \frac { 1 } { 12 } ( x - 8 )$
D) $2 - \frac { 1 } { 12 } ( x - 8 )$
E) $- 2 + \frac { 1 } { 12 } ( x - 8 )$

Quizplus · Accepted Answer

The linear approximation of a function $f(x)$ at a point $x_0$ is given by $L(x) = f(x_0) + f'(x_0)(x - x_0)$. For $f(x) = \sqrt[3]{x}$, $f'(x) = \frac{1}{3}x^{-2/3}$. At $x_0 = 8$, $f(8) = \sqrt[3]{8} = 2$ and $f'(8) = \frac{1}{3} \cdot 8^{-2/3} = \frac{1}{3} \cdot \frac{1}{4} = \frac{1}{12}$. Thus, $L(x) = 2 + \frac{1}{12}(x - 8)$.

The Linear Approximation L(x)L ( x )L(x) To f(x)=x3f ( x ) = \sqrt [ 3 ] { x }f(x)=3x​

The Linear Approximation $L ( x )$ To $f ( x ) = \sqrt [ 3 ] { x }$