Question

Arrange these functions from the greatest to the least value based on the average rate of change in the specified Interval. Rx) = x² + 3x interval: (-2.3] f(x) = 3x -8 interval: (4,5) f(x) = x² - 2x interval: (-3, 4) f(x) = x²-5 interval: (-1, 1] > > >

Answer 1

`\begin{gathered} \text{Arranging from greatest to least value based on value obtained, we have} \\ x^2+3x,3x-8,x^2-2x,x^2-5 \\ 1,2,3,4 \end{gathered}`

To find the average rate of change of a function within a given interval, we use the following formula;

`\frac{f(b)\text{ - f(a)}}{b-a}`

So for the first equation, we have;

`\begin{gathered} f(x)=x^2\text{ + 3x} \\ a\text{ = -2} \\ b\text{ = 3} \\ f(a)=f(-2)=(-2)^2+3(-2)=4-6=-2 \\ f(b)=f(3)=3^2\text{ + 3(3)=9+9 = 18} \\ \\ So; \\ (f(b)-f(a))/(b-a)\text{ = }(18-(-2))/(3-(-2))=(18+2)/(3+2)=(20)/(5)=4 \end{gathered}`

For the second equation, we have;

`\begin{gathered} f(x)\text{ = 3x-8} \\ a=4 \\ b=5 \\ f(a)=f(4)=3(4)-8=12-8=4 \\ f(b)=f(5)=3(5)-8=15-8=7 \\ (f(b)-f(a))/(b-a)=(7-4)/(5-4)=(3)/(1)=3 \end{gathered}`

For the third equation, we have;

`\begin{gathered} f(x)=x^2-2x \\ a=-3 \\ b=4 \\ f(a)=f(-3)=(-3)^2-2(-3)=9+6=15 \\ f(b)=f(4)=4^2-2(4)=16-8=8 \\ \\ (f(b)-f(a))/(b-a)=\text{ }(15-8)/(4-(-3))=(7)/(7)=1 \end{gathered}`

For the last equation, we have;

`\begin{gathered} f(x)\text{ = }x^2-5 \\ a=\text{ -1} \\ b=1 \\ f(a)=f(-1)=(-1)^2-5=1-5=-4 \\ f(b)=f(1)=1^2-5=1-5=-4 \\ \\ (f(b)-f(a))/(b-a)=\text{ }(-4-(-4))/(1-(-1))=(0)/(2)=\text{ 0} \end{gathered}`

So arranging the functions from highest to lowest based on the value obtained, we have;

`x^2+3x,3x-8,x^2-2x,x^2-5`