(7) 5. a) find the linear approximation to $f(x)=sqrt3{x}$ near $x = 27$.\nb) use part (a) to approximate…

Explanation:

Step1: Recall linear - approximation formula

The linear approximation of a function $y = f(x)$ near $x = a$ is given by $L(x)=f(a)+f^{\prime}(a)(x - a)$. First, find $f(a)$ and $f^{\prime}(x)$. Given $f(x)=x^{\frac{1}{3}}$, then $f^{\prime}(x)=\frac{1}{3}x^{-\frac{2}{3}}=\frac{1}{3x^{\frac{2}{3}}}$. When $a = 27$, $f(27)=27^{\frac{1}{3}} = 3$ and $f^{\prime}(27)=\frac{1}{3\times27^{\frac{2}{3}}}=\frac{1}{3\times9}=\frac{1}{27}$.

Step2: Write the linear - approximation formula

Substitute $f(27) = 3$ and $f^{\prime}(27)=\frac{1}{27}$ into the linear - approximation formula $L(x)=f(a)+f^{\prime}(a)(x - a)$. We get $L(x)=3+\frac{1}{27}(x - 27)=3+\frac{1}{27}x - 1=\frac{1}{27}x + 2$.

Step3: Use the linear - approximation to estimate $\sqrt[3]{26.46}$

Let $x = 26.46$. Then, using the linear approximation $L(x)=\frac{1}{27}x + 2$, we substitute $x = 26.46$ into it. $L(26.46)=\frac{1}{27}\times26.46+2$. First, $\frac{26.46}{27}=0.98$. Then $L(26.46)=0.98 + 2=2.98$.

Answer:

a) $L(x)=\frac{1}{27}x + 2$ b) $2.98$