Question

How to find instantaneous rate of change for `x^3 +2x^2 + x` at x from 1 to 2?

Answer 1

The instantaneous rate of change is the same as taking the derivative. This is defined at a point.

We can easily find that function via power rule:
`f(x) = x^3 + 2x^2 + x implies f'(x) = 3x^2 + 4x + 1`

I hope it is clear here that the concept of "instantaneous rate of change" from one point to another doesn't make any sense. However, you can think about the average rate of change very easily. We just do that in the way we'd think: the change in y divided by the change in x. This gives the following:
`(f(2) - f(1))/(2 - 1) = 14/1 = 14`