Question

What is the average rate of change of the function f (x) = 20 (1/4)^x from x=1 to x = 2

Answer 1

$\bf slope = {{ m}}= \cfrac{rise}{run} \implies \cfrac{{{ f(x_2)}}-{{ f(x_1)}}}{{{ x_2}}-{{ x_1}}}\impliedby \begin{array}{llll} average\ rate\\ of\ change \end{array}\\\\ -------------------------------$

$\bf f(x)= 20\left( \frac{1}{4} \right)^x \qquad \begin{cases} x_1=1\\ x_2=2 \end{cases}\implies \cfrac{f(2)-f(1)}{2-1} \\\\\\ \cfrac{20\left( \frac{1}{4} \right)^2~~-~~20\left( \frac{1}{4} \right)^1}{2-1}\implies \cfrac{20\cdot \frac{1^2}{4^2}~~-~~20\cdot \frac{1}{4}}{1}\implies \cfrac{\frac{20}{16}~~-~~5}{1} \\\\\\\cfrac{\frac{5}{4}~~-~~5}{1}\implies \cfrac{\frac{5-20}{4}}{1}\implies \cfrac{-\frac{15}{4}}{1}\implies -\cfrac{15}{4}\implies -3\frac{3}{4}$