How do you use $f (x) = sin (x^{2} - 2)$ $f(x) = sin(x^2-2)$ to evaluate $\frac{f (3.0002) - f (3)}{0.0002}$ $(f(3.0002)-f(3))/0.0002$ ?

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Answer 1 · 2018-05-12T01:21:06

Recall the derivative of a function is given by

$f'(x) = lim_(h-> 0) (f(x + h) - f(x))/h$

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

We can find the derivative using the chain rule

$f'(x)= 2xcos(x^2 - 2)$
$f'(3) = 2(3)cos(3^2 - 2) = 6cos(7) = 4.523$

If we plug the given expression into our calculator we get

$(f(3.0002) - f(3))/0.0002 = 4.521$

So our approximation is pretty good.

Hopefully this helps!

How do you use f(x) = sin(x^2-2)f(x)=sin(x2−2)f(x) = sin(x^2-2) to evaluate (f(3.0002)-f(3))/0.0002f(3.0002)−f(3)0.0002(f(3.0002)-f(3))/0.0002?