Question

question 1 (1 point)
the linear approximation of $f(x)=ln(x)$ at $x = 1$ is:
$l(x)=x + 1$
$l(x)=1$
$l(x)=x$
none of these options are correct.
$l(x)=x - 1$
view hint for question 1

Explanation:

Step1: Recall linear - approximation formula

The linear approximation of a function $y = f(x)$ at $x = a$ is given by $L(x)=f(a)+f^{\prime}(a)(x - a)$.

Step2: Find $f(1)$

Given $f(x)=\ln(x)$, then $f(1)=\ln(1) = 0$.

Step3: Find the derivative of $f(x)$

The derivative of $f(x)=\ln(x)$ is $f^{\prime}(x)=\frac{1}{x}$. So, $f^{\prime}(1)=\frac{1}{1}=1$.

Step4: Calculate the linear - approximation

Substitute $a = 1$, $f(1)=0$ and $f^{\prime}(1)=1$ into the linear - approximation formula $L(x)=f(a)+f^{\prime}(a)(x - a)$. We get $L(x)=0 + 1\times(x - 1)=x - 1$.

Answer:

E. $L(x)=x - 1$