Question

Use the difference quotient to estimate the instantaneous rate of change in f(x)=`x^2` at x=3?

Answer 1

See explanation.

Explanation:

Evaluating `(f(3+h)-f(3))/h`:

`((3+h)^2-3^2)/h = (9+6h+h^2-9)/h`

`=6+h`

You can then estimate the instantaneous rate of change by selecting "small" values of `h`.

For example, if `h=.01`, the estimate would be `6.01`.

The key idea is that once we've simplified the difference quotient we can let `h` tend to `0`. If `h to 0`, the difference quotient tends to 6, which is the exact instantaneous rate of change at `x=3`.