How to do this Linear Approximation?

Question

this question has been giving me a little trouble:

Use a linear approximation to estimate the number $8.07^{2/3}$

I tried using $f(a)+f'(a)(x-a)$ but the answer I get ($4.02$) is apparently wrong. Any help would be appreciated!

Answer 1

Take $f(x)=x^{2/3}$ and $a=8$. Then $$f(x)=x^{2/3} \Rightarrow f'(x)=\frac{2}{3}x^{-1/3} \Rightarrow f'(a)=\frac{2}{3}\cdot 8^{-1/3}=\frac{1}{3}$$

So $f(a)+f'(a)(x-a)=f(8)+\frac{1}{3}(.07)=4+\frac{1}{3}(.07)\approx4.023$

Answer 2

If you use $f(x)=x^{2/3}$, you have $f(x)\approx f(a)+f'(a)(x-a)$. $f'(x)=\frac 23 x^{-1/3}$. If you plug in $a=8$, $f'(8)=\frac 13$, so $f(8.07)=4+0.07/3=4.02333$. The real answer is $4.023299$