Given:
Consider the given function is:
[tex]f(x)=x^2-8x+13[/tex]
To find:
The average rate of change of the function over the interval [tex]-1\leq x\leq 6[/tex].
Solution:
The average rate of change of the function f(x) over the interval [a,b] is:
[tex]m=\dfrac{f(x_2)-f(x_1)}{x_2-x_1}[/tex]
We have,
[tex]f(x)=x^2-8x+13[/tex]
At [tex]x=-1[/tex],
[tex]f(-1)=(-1)^2-8(-1)+13[/tex]
[tex]f(-1)=1+8+13[/tex]
[tex]f(-1)=22[/tex]
At [tex]x=6[/tex],
[tex]f(6)=(6)^2-8(6)+13[/tex]
[tex]f(6)=36-48+13[/tex]
[tex]f(6)=1[/tex]
Now, the average rate of change of the function f(x) over the interval [tex]-1\leq x\leq 6[/tex] is:
[tex]m=\dfrac{f(6)-f(-1)}{6-(-1)}[/tex]
[tex]m=\dfrac{1-22}{7}[/tex]
[tex]m=\dfrac{-21}{7}[/tex]
[tex]m=-3[/tex]
Therefore, the average rate of change of the function f(x) over the interval [tex]-1\leq x\leq 6[/tex] is -3.
