find the linear approximation of $f(x)=ln x$ at $x = 1$ and use it to estimate $ln(1.47)$. $l(x)=square$ $ln…

Explanation:

Step1: Recall linear - approximation formula

The linear - approximation of a function $y = f(x)$ at $x = a$ is given by $L(x)=f(a)+f^{\prime}(a)(x - a)$. First, find $f(1)$ and $f^{\prime}(x)$ for $f(x)=\ln x$. For $f(x)=\ln x$, when $x = 1$, $f(1)=\ln(1)=0$. The derivative of $f(x)=\ln x$ is $f^{\prime}(x)=\frac{1}{x}$. When $x = 1$, $f^{\prime}(1)=\frac{1}{1}=1$.

Step2: Find the linear - approximation formula

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 $L(x)=0 + 1\times(x - 1)=x - 1$.

Step3: Estimate $\ln(1.47)$

To estimate $\ln(1.47)$, we use the linear - approximation. Let $x = 1.47$. Since $L(x)=x - 1$, then $L(1.47)=1.47-1=0.47$.

Answer:

$L(x)=x - 1$ $\ln(1.47)\approx0.47$