Question

Find the average value of f(x)=2/x over the interval [1, 3].

Answer 1

$\bf slope = m = \cfrac{rise}{run} \implies \cfrac{ f(x_2) - f(x_1)}{ x_2 - x_1}\impliedby \begin{array}{llll} average~rate\\ of~change \end{array}\\\\ -------------------------------\\\\ f(x)= \cfrac{2}{x} \qquad \begin{cases} x_1=1\\ x_2=3 \end{cases}\implies \cfrac{f(3)-f(1)}{3-1}\implies \cfrac{\quad \frac{2}{3}-\frac{2}{1}\quad }{2} \\\\\\ \cfrac{\quad \frac{2-6}{3}\quad }{2}\implies \cfrac{\quad \frac{-4}{3}\quad }{\frac{2}{1}}\implies \cfrac{-4}{3}\cdot \cfrac{1}{2}\implies -\cfrac{2}{3}$