Question

Find the average rate of change of h(x) = 3x ^ 3 - 3x ^ 2 - 2 from x = - 2 to x = 1

Answer 1

54

Step -By-Step
Exp
H(x) = 3x ^3-3x when 2-2

Answer 2

The average rate of change is given by;

`(\Delta h)/(\Delta x)=(h_2-h_1)/(x_2-x_1)_{}`

Given:

`h(x)=3x^3-3x^2-2`
`\begin{gathered} h_2=h(1) \\ h_1=h(-2) \\ x_2=1 \\ x_1=-2 \end{gathered}`
`\begin{gathered} h(1)\text{ means susbstituting when x =1 into h(x);} \\ h(x)=3x^3-3x^2-2 \\ h(1)=3(1^3)-3(1^2)-2 \\ h(1)=3(1)-3(1)-2 \\ h(1)=3-3-2 \\ h(1)=-2 \end{gathered}`

`\begin{gathered} h(-2)\text{ means susbstituting when x =-2 into h(x);} \\ h(x)=3x^3-3x^2-2 \\ h(-2)=3(-2^3)-3(-2^2)-2 \\ h(-2)=3(-8)-3(4)-2 \\ h(-2)=-24-12-2 \\ h(-2)=-38 \end{gathered}`

`\begin{gathered} (\Delta h)/(\Delta x)=(h_2-h_1)/(x_2-x_1)_{} \\ (\Delta h)/(\Delta x)=(h_2-h_1)/(x_2-x_1)_{}=(h(1)-h(-2))/(x_2-x_1) \\ (\Delta h)/(\Delta x)=\frac{-2-(-38)_{}}{1-(-2)} \\ (\Delta h)/(\Delta x)=(-2+38)/(1+2) \\ (\Delta h)/(\Delta x)=(36)/(3) \\ (\Delta h)/(\Delta x)=12 \end{gathered}`

Therefore, the average rate of change of h(x) from x = -2 to x = 1 is 12.