Question

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

Answer 1

we know that

The Average Rate of Change is the slope of a line or a curve on a given range. It is defined as the ratio of the difference in the function f(x) as it changes from 'a' to 'b' to the difference between 'a' and 'b'.

So

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

case a) $f(x) = x^{2} + 3x$

in the interval $[-2, 3]$

$a=-2\\ b=3\\ f(a)=(-2)^{2} +3*(-2)\\ f(a)=-2\\ f(b)=(3)^{2} +3*(3)\\ f(b)=18$

Find the Average Rate of Change

$A=\frac{(18-(-2))}{(3+2)}$

$A=\frac{20}{5}$

$A=4$

case b) $f(x) = 3x-8$

in the interval $[4, 5]$

$a=4\\ b=5\\ f(a)=3*4-8\\ f(a)=4\\ f(b)=3*5-8\\ f(b)=7$

Find the Average Rate of Change

$A=\frac{(7-(4))}{(5-4)}$

$A=\frac{3}{1}$

$A=3$

case c) $f(x) = x^{2} -2x$

in the interval $[-3, 4]$

$a=-3\\ b=4\\ f(a)=(-3)^{2} -2*(-3)\\ f(a)=15\\ f(b)=(4)^{2} -2*(4)\\ f(b)=8$

Find the Average Rate of Change

$A=\frac{(8-(15))}{(4+3)}$

$A=\frac{-7}{7}$

$A=-1$

case d) $f(x) = x^{2} -5$

in the interval $[-1, 1]$

$a=-1\\ b=1\\ f(a)=(-1)^{2} -5\\ f(a)=-4\\ f(b)=(1)^{2} -5\\ f(b)=-4$

Find the Average Rate of Change

$A=\frac{(-4-(-4))}{(1+1)}$

$A=\frac{0}{2}$

$A=0$

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

so

the answer is

1) $f(x) = x^{2} + 3x$ -------> $A=4$

2) $f(x) = 3x-8$ -----> $A=3$

3) $f(x) = x^{2} -5$ -----> $A=0$

4) $f(x) = x^{2} -2x$ -----> $A=-1$

Answer 2

Answer:

b,d,c,a

Step-by-step explanation:

a) $f(x) = x^2 + 3x$

interval: [-2, 3]

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

=$\frac{f(3)-f(-2)}{3-(-2)} =\frac{18+2}{5} -4$

b) $f(x) = 3x - 8$

interval: [4, 5]

Average value =$\frac{15-7}{1} =8$

c)

$f(x) = x^2 - 2x$

interval: [-3, 4]

Average value =$\frac{8-15}{7} =-1$

d) $f(x) = x^2 - 5$

interval: [-1, 1]

Average value =$\frac{-4+4}{2} =0$

Hence arranging we get

b, d, c, a