Question

Complete the average rate of change in the indicated function over the interval provided H(n) = 3/n + 1 on [2, 4]

Answer 1

`\begin{gathered} \text{Given} \\ H(n)=(3)/(n+1)\text{ on }\lbrack2,4\rbrack \end{gathered}`

Recall the average rate of change

`\text{Average Rate of Change}=(\Delta y)/(\Delta x)=(f(b)-f(a))/(b-a)`

Given the function H(n)

b = 4, a = 2.

Substitute the following given and we have the equation

`\begin{gathered} (\Delta y)/(\Delta x)=(H(4)-H(2))/(4-2) \\ (\Delta y)/(\Delta x)=(((3)/(4+1))-((3)/(2+1)))/(2) \\ (\Delta y)/(\Delta x)=(((3)/(5))-((3)/(3)))/(2) \\ (\Delta y)/(\Delta x)=((3)/(5)-1)/(2) \\ (\Delta y)/(\Delta x)=((3)/(5)-(5)/(5))/(2) \\ (\Delta y)/(\Delta x)=((-2)/(5))/(2) \\ (\Delta y)/(\Delta x)=-(2)/(5)\cdot(1)/(2) \\ (\Delta y)/(\Delta x)=-\frac{\cancel{2}}{5}\cdot\frac{1}{\cancel{2}} \\ (\Delta y)/(\Delta x)=-(1)/(5) \end{gathered}`

Therefore, the average rate of change of H(n) in the interval [2.4] is -1/5.