Question

Given the function f(x) = 2(3)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. (4 points) 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. (6 points)

Answer 1

$\bf slope = {{ m}}= \cfrac{rise}{run} \implies \cfrac{{{ f(x_2)}}-{{ f(x_1)}}}{{{ x_2}}-{{ x_1}}}\impliedby \begin{array}{llll} average\ rate\\ of\ change \end{array}\\\\ -------------------------------\\\\ f(x)=2(3)^x \qquad \begin{cases} x_1=0\\ x_2=1 \end{cases}\implies \cfrac{f(1)-f(0)}{1-0}\implies \cfrac{2(3)^1-2(3)^0}{1}\\\\ -------------------------------\\\\ f(x)=2(3)^x \qquad \begin{cases} x_1=2\\ x_2=3 \end{cases}\implies \cfrac{f(3)-f(2)}{3-2}\implies \cfrac{2(3)^3-2(3)^2}{1}$

part B)

how many times? 9 times larger

how so?  well, is an exponential function, anything raised at the 0 exponent, is 1, no matter what the number is

and anything raised at the 1 exponent is itself, no matter what number it is

after 0 and 1, the base number is multiplying itself that many times, and that's a larger value.