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

Jan 26, 2018

See explanation.

#### Explanation:

Evaluating $\frac{f \left(3 + h\right) - f \left(3\right)}{h}$:

$\frac{{\left(3 + h\right)}^{2} - {3}^{2}}{h} = \frac{9 + 6 h + {h}^{2} - 9}{h}$

$= 6 + h$

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 $x = 3$.