# How do you use f(x) = sin(x^2-2) to evaluate (f(3.0002)-f(3))/0.0002?

May 12, 2018

Recall the derivative of a function is given by

$f ' \left(x\right) = {\lim}_{h \to 0} \frac{f \left(x + h\right) - f \left(x\right)}{h}$

If we let $x = 3$ and $h = 0.0002$ (very close to $0$), we get that the given expression is equal to $f ' \left(3\right)$.

We can find the derivative using the chain rule

$f ' \left(x\right) = 2 x \cos \left({x}^{2} - 2\right)$
$f ' \left(3\right) = 2 \left(3\right) \cos \left({3}^{2} - 2\right) = 6 \cos \left(7\right) = 4.523$

If we plug the given expression into our calculator we get

$\frac{f \left(3.0002\right) - f \left(3\right)}{0.0002} = 4.521$

So our approximation is pretty good.

Hopefully this helps!