Question

What is the average rate of change of the function over the interval x = 0 to x = 4?

f(x)= 2x−1/3x+5

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}\\\\[-0.35em] \rule{34em}{0.25pt}\\\\ f(x)= \cfrac{2x-1}{3x+5}\qquad \begin{cases} x_1=0\\ x_2=4 \end{cases}\implies \cfrac{f(4)-f(0)}{4-0}`

`\bf \cfrac{~~\left( (2(4)-1)/(3(4)+5) \right)-\left( (2(0)-1)/(3(0)+5) \right)~~}{4}\implies \cfrac{~~\left( (7)/(17) \right)-\left( (-1)/(5) \right)~~}{4}\implies\cfrac{~~\left( (7)/(17) \right)+\left( (1)/(5) \right)~~}{4} \\\\\\ \cfrac{~~(35+17)/(85)~~}{4}\implies \cfrac{~~(52)/(85)~~}{(4)/(1)}\implies \cfrac{52}{85}\cdot \cfrac{1}{4}\implies \cfrac{13}{85}`