Question

For the polynomial of f(x)=3x^3+4x^2+14x+13 determine the average rate of change between the two given values for x. Round to two decimal places.

x = −6, x = −6.5

Answer 1

`\begin{array}{llll} f(x)~from\\\\ x_1 ~~ to ~~ x_2 \end{array}~\hfill slope = m \implies \cfrac{ \stackrel{rise}{f(x_2) - f(x_1)}}{ \underset{run}{x_2 - x_1}}\impliedby \begin{array}{llll} average~rate\\ of~change \end{array} \\\\[-0.35em] ~\dotfill`

`f(x)= 3x^3+4x^2+14x+13 \qquad \begin{cases} x_1=-6\\ x_2=-6.5 \end{cases}\implies \cfrac{f(-6.5)-f(-6)}{-6.5 - (-6)} \\\\\\ \cfrac{[3(-6.5)^3+4(-6.5)^2+14(-6.5)+13]~~ - ~~[3(-6)^3+4(-6)^2+14(-6)+13]}{-6.5+6} \\\\\\ \cfrac{[-732.875]~~ - ~~[-575]}{-0.5}\implies \cfrac{-157.875}{-0.5}\implies \text{\LARGE 315.75}`