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

Question

Elijah Abram · Accepted Answer

I found #14# The average rate of change can be evaluated as: #color(red)((f(x_2)-f(x_1))/(x_2-x_1))# Where: #x_1=0# #x_2=4# #f(x_1)=f(0)=0-0=0# #f(x_2)=f(4)=4^3-2(4)=56# So the average rate of change will be: #(56-0)/(4-0)=56/4=14#

Elijah Abram · Answer

undefined To find the average rate of change of the function $ f(x) = x^3 - 2x $ on the interval $[0, 4]$, you first need to find the values of the function at the endpoints of the interval and then calculate the change in the function value divided by the change in the input variable over that interval.

1. Evaluate $ f(x) $ at $ x = 0 $ and $ x = 4 $ to find the function values at the endpoints.
   $ f(0) = (0)^3 - 2(0) = 0 $
   $ f(4) = (4)^3 - 2(4) = 64 - 8 = 56 $

2. Calculate the change in the function value: $ f(4) - f(0) = 56 - 0 = 56 $

3. Calculate the change in the input variable: $ 4 - 0 = 4 $

4. Finally, divide the change in the function value by the change in the input variable to get the average rate of change:
   $ \text{Average rate of change} = \frac{f(4) - f(0)}{4 - 0} = \frac{56}{4} = 14 $

Luke Alvarez · Answer

undefined To find the average rate of change for the function $ f(x) = x^3 - 2x $ on the interval $[0, 4]$, you first find the values of the function at the endpoints of the interval, then calculate the difference in the function values, and finally divide by the difference in the input values.

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

$ f(4) = (4)^3 - 2(4) = 64 - 8 = 56 $

Difference in function values: $ 56 - 0 = 56 $

Difference in input values: $ 4 - 0 = 4 $

Average rate of change: $ \frac{56}{4} = 14 $