Question

Given the function f(x) = 4(2)ˣ, section a is from x = 0 to x = 1 and section b is from x = 2 to x = 3. Part a: Find the average rate of change of each section. Include calculations for full credit. Part b: How many times greater is the average rate of change of section b than section a? Explain why one rate of change is greater than the other.

Answer 1

The average rate of change for section a is 4, while for section b it is 16. The rate for section b is 4 times greater than that for section a due to the exponential nature of the function.

Step-by-step explanation:

To find the average rate of change of the function f(x) = 4(2)x, we evaluate it at the endpoints of each section and use the formula for average rate of change, which is ∆f/∆x = [f(x2) - f(x1)] / (x2 - x1).

Section a (x = 0 to x = 1):

f(0) = 4(2)0 = 4
f(1) = 4(2)1 = 8
Average rate of change = (8 - 4) / (1 - 0) = 4

Section b (x = 2 to x = 3):

f(2) = 4(2)2 = 16
f(3) = 4(2)3 = 32
Average rate of change = (32 - 16) / (3 - 2) = 16

Part b: The average rate of change of section b is 16, while for section a is 4. Therefore, the average rate of change of section b is 4 times greater than that of section a. This is because the function is exponential, and its rate of change increases as the value of x increases.