The average rate of change of a given function f(x) in the interval (a,b) is given by the following formula:
[tex]AROC=\frac{f(b)-f(a)}{b-a}[/tex]This is valid only if f(x) is defined at x=a and x=b.
In our case the interval is (-2,5) and the function is:
[tex]f(x)=x^2+5x-12[/tex]So we need to find f(-2) and f(5):
[tex]\begin{gathered} f(-2)=(-2)^2+5\cdot(-2)-12=-18 \\ f(5)=5^2+5\cdot5-12=38 \end{gathered}[/tex]Then the average rate of change of this function in the interval (-2,5) is given by:
[tex]AROC=\frac{f(5)-f(-2)}{5-(-2)}=\frac{38-(-18)}{5+2}=\frac{38+18}{7}=\frac{56}{7}=8[/tex]AnswerThen the answer is that the Average Rate of Change of f(x) in (-2,5) is 8.
