# How do you find the average rate of change of f(x)=-3x^2+2 between [-2,0]?

Oct 20, 2015

6

#### Explanation:

Summery:
1. Derive equation for rate of change

$$2. By substitution of the values of x  at the two points under
investigation determining the actual rate of change at
those point.

3. Apply the standard method of determining the mean. In this
case it will be


$\frac{{x}_{1} - {x}_{2}}{2}$

$$  Note that rate of change is from left to right on the graph. This is
important!


Solution:

Assumption: your given $\left[- 2 , 0\right]$ is the "inclusive" range for $x$.
Brackets facing outwards represents "exclusive".

Let ${x}_{1} = - 2$
Let ${x}_{2} = 0$

Given that$\text{ } f \left(x\right) = - 3 {x}^{2} + 2$

The rate of change is:$\text{ } {f}^{'} \left(x\right) = - 6 x$

At ${x}_{1}$ the rate of change is: $\left(- 6\right) \times \left(- 2\right) = + 12$
At ${x}_{2}$ the rate of change is: $\left(- 6\right) \times 0 = 0$

So the mean rate of change is $\frac{12 - 0}{2} = 6$