Question

Calculate the average rate of change of f(x) = 3x^2 + 2x + 1 for 2 ≤ x ≤ 4

Answer 1

$\bf slope = {{ m}}= \cfrac{rise}{run} \implies \cfrac{{{ f(x_2)}}-{{ f(x_1)}}}{{{ x_2}}-{{ x_1}}}\impliedby \begin{array}{llll} average\ rate\\ of\ change \end{array}\\\\ -------------------------------\\\\ f(x)= 3x^2+2x+1 \qquad \begin{cases} x_1=2\\ x_2=4 \end{cases}\implies \cfrac{f(4)-f(2)}{4-2} \\\\\\ \cfrac{[3(4)^2+2(4)+1]~~-~~[3(2)^2+2(2)+1]}{2} \\\\\\ \cfrac{[48+8+1]~~-~~[12+4+1]}{2}\implies \cfrac{57-17}{2} \\\\\\ \cfrac{60}{2}\implies \cfrac{30}{1}\implies 30$