Question

Use an appropriate local linear approximation to estimate 3√9

Answer 1

The linear approximation to $f(x)=\sqrt[3]{x}$  at  $x_{0}=9$ .

A linear approximation is given by  $L(x) \approx f\left(x_{0}\right)+f^{\prime}\left(x_{0}\right)\left(x-x_{0}\right) .$

We are given that  $x_{0}=9$ .

Firstly, find the value of the function at the given point:  $y_{0}=f\left(x_{0}\right)=3^{\frac{2}{3}}$ .

Secondly, find the derivative of the function, evaluated at the point:  $f^{\prime}(9)$ .

Find the derivative:  $f^{\prime}(x)=\frac{1}{3 x^{\frac{2}{3}}}$  

Next, evaluate the derivative at the given point to find slope.

 

$f^{\prime}(9)=\frac{3^{\frac{2}{3}}}{27}$  

Plugging the values found, we get that  $L(x) \approx 3^{\frac{2}{3}}+\frac{3^{\frac{2}{3}}}{27}(x-(9))$ .

Or, more simply: $L(x) \approx \frac{3^{\frac{2}{3}}}{27} x+\frac{2 \cdot 3^{\frac{2}{3}}}{3}$ .

Answer:  $L(x) \approx \frac{3^{\frac{2}{3}}}{27} x+\frac{2 \cdot 3^{\frac{2}{3}}}{3} \approx 0.077040141594515 x+1.38672254870127$ .