Thanks for clarifying the function
[tex] f(x) = 4\cdot 2^x[/tex]
Part A.
The average rate of change is the change in y over the interval divided by the change in x over the interval. The change in x is just the length of the interval.
So section A, the average from 1 to 2 is
[tex] a =\dfrac{ f(2) - f(1) }{2 -1 } = \dfrac{ f(2)-f(1)}{1} = 4 (2^2) - 4(2^1) = 16 - 8 = 8[/tex]
Section B, the average from 3 to 4 is
[tex]b =\dfrac{f(4) - f(3)}{4-3} = \dfrac{f(4)-f(3)}{1} = 4 (2^4) - 4(2^3) = 4(16)-4(8) =64-32=32[/tex]
Part B.
Section B is changing (32/8=) four times faster than section A.
f is an exponential function that not only increases its value but increases the rate at which it increases its value. Each step of one unit in x doubles the rate of increase. Here we stepped two units, from [1,2] to [3,4], so quadrupled the rate.
