Question

What is the instantaneous rate of change of `f(x)=x^3- x^2+3x` at `x=3`?

Answer 1

24

Explanation:

The instantaneous rate of change at some point is simply the derivative at that point. So to solve this problem, all we do is find the derivative of the function and evaluate it at `x=3`.

Finding the derivative is easy; we use the power rule:
`f'(x) = 3x^2-2x+3`

Now we set `x=3` and find `f'(x)`:
`f'(3) = 3(3)^2-2(3)+3`
`f'(3) = 27-6+3`
`f'(3) = 24`

Therefore our instantaneous rate of change at `x=3` is 24. This means the function is increasing at a rate of 24 at `x=3`.