Question

Find the average rate of change on the following equation on the given interval. y=x^2+x+2 on [1,4]

Answer 1

The function is given as,

$y=x^2+x+2$

The interval is given as,

$\lbrack1,4\rbrack$

Consider that the average rate of change of a function f(x), in the interval [a,b] is given by,

$r=\frac{f(b)-f(a)}{b-a}$

According to the given problem,

$\begin{gathered} f(x)=x^2+x+2 \\ a=1 \\ b=4 \end{gathered}$

The value of the function at the end-points is calculated as,

$\begin{gathered} f(b)=f(4)=4^2+4+2=16+6=22 \\ f(a)=f(1)=1^2+1+2=1+3=4 \end{gathered}$

Substitute the values and simplify,

$\begin{gathered} r=\frac{22-4}{4-1} \\ r=\frac{18}{3} \\ r=6 \end{gathered}$

Thus, the average rate of change of the function is 6 units.