The linear approximation $L ( x )$ to $f ( x ) = \ln x$ near $x _ { 0 } = e$ is

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

Question

The linear approximation $L ( x )$ to $f ( x ) = \ln x$ near $x _ { 0 } = e$ is

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

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 $f(x) = \ln x$ at $x_0 = e$, $f(e) = \ln e = 1$ and $f'(x) = \frac{1}{x}$, so $f'(e) = \frac{1}{e}$. Thus, $L(x) = 1 + \frac{1}{e}(x - e)$.

The Linear Approximation L(x)L ( x )L(x) To f(x)=ln⁡xf ( x ) = \ln xf(x)=lnx

The Linear Approximation $L ( x )$ To $f ( x ) = \ln x$