Given the function g(x) = 4(3)^x, section a is from x = 1 to x = 2 and section b is from x = 3 to x = 4.   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. 

Reasoning:   the average rate of change of a function g from x = a to x = b  can be calculated using the expression    it is really an application of the slope formula. it gets the slope of  two points on the graph of function g; the points have the coordinates:   (a, g(a))  and (b, g(solutionpart    a:   avg. rate of change  for section a is 24.  avg. rate of change for section b is 216we're given that g(x) = 4(3)ˣsection a: x = 1 to x = 2section b: x = 3 to x = 4to calculate the average rate of change of function g for section a, apply the average rate of change formula using x = 1 and x = 2. if we let  represent the average rate of change of g for section a, then we calculate it as follows:     the average rate of change of function g for section a is 24.to calculate the average  rate of change of function g for section b, apply the average rate of change  formula using x = 3 and x = 4. if we let  represent the average rate of change of g for section a, then we calculate it as follows:       the average rate of change of function g for section b is 216.part b: the average rate of change of section b is  216. the average rate of change of section a is 24. therefore, the average rate of change of section b is  216/24 = 9 times greater than the average rate of change of section a. this can be attributed to the fact that this is an exponential function, which has an exponential rate of growth. exponential functions grow in value as x increases, hence the increased average rate of change.we can also note that the value of g(2) (at x = 2) increases three-fold when we reach x = 3, and then the value of g(2) would increase nine-fold when we reach x = 4. exponents just imply repeated multiplication, e.g. 3² = 3  · 3 and 3³ = 3  · 3  · 3.