Question

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

Answer 1

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!