Question

Given the function f(x)=-x^2-5x+12f(x)=−x 2 −5x+12, determine the average rate of change of the function over the interval -5\le x \le 4−5≤x≤4.

Answer 1

Given:

The function is

$f(x)=-x^2-5x+12$

To find:

The average rate of change of the function over the interval −5 ≤ x ≤ 4.

Solution:

The average rate of change of a function f(x) over the interval [a,b] is

$m=\dfrac{f(b)-f(a)}{b-a}$

Putting x=-5 in the given function, we get

$f(-5)=-(-5)^2-5(-5)+12$

$f(-5)=-25+25+12$

$f(-5)=12$

Putting x=4 in the given function, we get

$f(4)=-(4)^2-5(4)+12$

$f(4)=-16-20+12$

$f(4)=-24$

Now, the average rate of change of the function over the interval −5 ≤ x ≤ 4 is

$m=\dfrac{f(4)-f(-5)}{4-(-5)}$

$m=\dfrac{-24-12}{4+5}$

$m=\dfrac{-36}{9}$

$m=-4$

Therefore, the average rate of change of the function over the interval −5 ≤ x ≤ 4 is -4.