How do you find the average rate of change for the function #f(x)= x^3 +2x^2 + x# on the indicated intervals [-1,2]?

Question

Isaiah Allen · Accepted Answer

The average rate of change is #6# (#f# units per #x# unit)

The ratio of one quantity's change to another's change is known as the rate of change. Average rate of change of a function #f(x)# (with respect to #x#) is the change in #f(x)# divided by the change in #x#, so Average rate of change of a function #f(x)# on an interval #[a,b]# is: #(f(b)-f(a))/(b-a)#. In this question: the function is #f(x) = x^3+2x^2+x# and the interval is #[a,b] = [-1,2]# So the average rate of change of #f# on the interval is: #(f(2)-f(-1))/(2-(-1)) = (18-0)/(2+1) = 18/3 = 6#. If we had units for #x# and #f(x)# we would use them as #f# units per #x# unit

Evelyn Agard · Answer

undefined To find the average rate of change of the function $ f(x) = x^3 + 2x^2 + x $ on the interval $[-1, 2]$, you can use the formula:

$$ \text{Average rate of change} = \frac{f(b) - f(a)}{b - a} $$

where $ a $ and $ b $ are the endpoints of the interval.

Substitute $ a = -1 $ and $ b = 2 $ into the formula:

$$ \text{Average rate of change} = \frac{f(2) - f(-1)}{2 - (-1)} $$

Now, evaluate $ f(2) $ and $ f(-1) $ by plugging in the values of $ x $ into the function:

$$ f(2) = (2)^3 + 2(2)^2 + 2 = 8 + 8 + 2 = 18 $$
$$ f(-1) = (-1)^3 + 2(-1)^2 - 1 = -1 + 2 - 1 = 0 $$

Substitute these values back into the formula:

$$ \text{Average rate of change} = \frac{18 - 0}{2 - (-1)} = \frac{18}{3} = 6 $$

So, the average rate of change of $ f(x) = x^3 + 2x^2 + x $ on the interval $[-1, 2]$ is $ 6 $.