Question

Arrange these functions from the greatest to the least value based on the average rate of change in the specified interval.

Answer 1

By definition, the average change of rate is given by:
 $AVR = \frac{f(x2)-f(x1)}{x2-x1}$
 We will calculate AVR for each of the functions.
 We have then:

 f(x) = x^2 + 3x interval: [-2, 3]:
 $f(-2) = x^2 + 3x = (-2)^2 + 3(-2) = 4 - 6 = -2 f(3) = x^2 + 3x = (3)^2 + 3(3) = 9 + 9 = 18$
 $AVR = \frac{-2-18}{-2-3}$
 $AVR = \frac{-20}{-5}$
 $AVR = 4$

 f(x) = 3x - 8 interval: [4, 5]:
 $f(4) = 3(4) - 8 = 12 - 8 = 4 f(5) = 3(5) - 8 = 15 - 8 = 7$
 $AVR = \frac{7-4}{5-4}$
 $AVR = \frac{3}{1}$
 $AVR = 3$

 f(x) = x^2 - 2x interval: [-3, 4]
 $f(-3) = (-3)^2 - 2(-3) = 9 + 6 = 15 f(4) = (4)^2 - 2(4) = 16 - 8 = 8$
 $AVR = \frac{8-15}{4+3}$
 $AVR = \frac{-7}{7}$
 $AVR = -1$

 f(x) = x^2 - 5 interval: [-1, 1]
 $f(-1) = (-1)^2 - 5 = 1 - 5 = -4 f(1) = (1)^2 - 5 = 1 - 5 = -4$
 $AVR = \frac{-4+4}{1+1}$
 $AVR = \frac{0}{2}$
 $AVR = 0$


 Answer:
 these functions from the greatest to the least value based on the average rate of change are:
 f(x) = x^2 + 3x
 f(x) = 3x - 8
 f(x) = x^2 - 5
 f(x) = x^2 - 2x