How do you find the average rate of change for the function #f(x) = x^3 - 4x^2# on the indicated intervals [0,10]?

Question

Christopher Allers · Accepted Answer

The average rate of change of function #f# on interval #[a,b]# is #(f(b)-f(a))/(b-a)#

The average rate of change for the function #f(x) = x^3 - 4x^2# on the interval #[0,10]# is:

#(f(10)-f(0))/(10-0) = ([1000-400]-[0])/(10-0) = 60#

Keep in mind that a line's slope corresponds to the average rate of change.

Christopher Agresta · Answer

undefined To find the average rate of change of the function $ f(x) = x^3 - 4x^2 $ on the interval $[0,10]$, 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. In this case, $ a = 0 $ and $ b = 10 $. Substituting these values into the formula:

$$ \text{Average Rate of Change} = \frac{f(10) - f(0)}{10 - 0} $$

Calculate $ f(10) $ and $ f(0) $ by substituting $ x = 10 $ and $ x = 0 $ into the function $ f(x) = x^3 - 4x^2 $:

$$ f(10) = (10)^3 - 4(10)^2 $$
$$ f(10) = 1000 - 400 = 600 $$

$$ f(0) = (0)^3 - 4(0)^2 $$
$$ f(0) = 0 - 0 = 0 $$

Now substitute these values back into the formula:

$$ \text{Average Rate of Change} = \frac{600 - 0}{10 - 0} = \frac{600}{10} = 60 $$

So, the average rate of change of the function $ f(x) = x^3 - 4x^2 $ on the interval $[0,10]$ is 60.