Question

Help Please! Given the function f(x) = 5x, 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. 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 question is defective, or at least is trying to lead you down the primrose path.

The function is linear, so the rate of change is the same no matter what interval
(section) of it you're looking at.

The "rate of change" is just the slope of the function in the section.  That's

         (change in f(x) ) / (change in 'x') between the ends of the section.

In Section A:
Length of the section = (1 - 0) = 1
f(1) = 5
f(0) = 0
change in the value of the function = (5 - 0) = 5
Rate of change =
           (change in the value of the function) / (size of the section) = 5/1  =  5

In Section B:
Length of the section = (3 - 2) = 1
f(3) = 15
f(2) = 10
change in the value of the function = (15 - 10) = 5
Rate of change =
           (change in the value of the function) / (size of the section) = 5/1  =  5

Part A:
The average rate of change of each section is 5.

Part B:
The average rate of change of Section B is equal to
the average rate of change of Section A.

Explanation:
The average rates of change in every section are equal
because the function is linear, its graph is a straight line,
and the rate of change is just the slope of the graph.