How do you find the average rate of change of the function #f(x)=4 ·x^2 + 2 ·x−4# over the interval [3, 3.17]?

Question

Emily Ammerman · Accepted Answer

The average rate of change of a function #f# over an interval #[a,b]# is #(f(b)-f(a))/(b-a)# So, here's where the math comes in: #(f(3.17)-f(3))/(3.17-3)#

William Abdalla · Answer

undefined To find the average rate of change of the function $ f(x) = 4x^2 + 2x - 4 $ over the interval $[3, 3.17]$, you calculate the difference in the function values at the endpoints of the interval and divide by the difference in the x-values.

1. Evaluate the function at the endpoints of the interval:
   - $ f(3) = 4(3)^2 + 2(3) - 4 = 36 + 6 - 4 = 38 $
   - $ f(3.17) = 4(3.17)^2 + 2(3.17) - 4 = 40.0568 + 6.34 - 4 = 42.3968 $

2. Calculate the difference in function values: $ 42.3968 - 38 = 4.3968 $.

3. Calculate the difference in x-values: $ 3.17 - 3 = 0.17 $.

4. Divide the difference in function values by the difference in x-values: $ \frac{4.3968}{0.17} \approx 25.861 $.

So, the average rate of change of the function over the interval $[3, 3.17]$ is approximately $25.861$.