Question

Find the average rate of change for the given function on the given interval.

f(x) = 2^x  − 1; [0, 3]

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^x-1 \qquad \begin{cases} x_1=0\\ x_2=3 \end{cases}\implies \cfrac{f(3)-f(0)}{3-0} \\\\\\ \cfrac{[2^3-1]~~-~~[2^0-1]}{3}\implies \cfrac{[8-1]~~-~~[1-1]}{3} \\\\\\ \cfrac{7-0}{3}\implies \cfrac{7}{3}`

Answer 2

The average rate of change is (f(3)-f(0))/3=(7-0)/3=7/3.