Question

Consider the following function. F(x) = x^3/2 Find the average rate of change over the interval [4,25]

Answer 1

Given the function:

`f(x)=x^{(3)/(2)}`

You can rewrite it in this form:

`f(x)=\sqrt[]{x^3}`

Because by definition:

`\sqrt[n]{b^m}=b^{(m)/(n)}`

• The formula for calculating the Average Rate of Change over an interval is:

`m=(\Delta y)/(\Delta x)=(f(b)-f(a))/(b-a)`

Where these two points are on the function:

`(a,f(a)),(b,f(b))`

In this case, given the interval:

`\lbrack4,25\rbrack`

You can identify that:

`a=4`

Then, substituting this value into the function and evaluating, you get:

`f(a)=f(4)=\sqrt[]{(4)^3}=\sqrt[]{64}=8`

You can also identify that:

`b=25`

Then, substituting this value into the function and evaluating, you get:

`f(b)=f(25)=\sqrt[]{(25)^3}=125`

Now you can substitute values into the formula and then evaluate, in order to find the Average Rate of Change over the given interval:

`(\Delta y)/(\Delta x)=(125-8)/(25-4)=(39)/(7)`

• In order to find the Instantaneous Rate of Change at the endpoints of the interval, you need to:

1. Derivate the function. Then, you need to find:

`f^(\prime)(x)`

By definition:

`(d)/(dx)(x^n)=nx^(n-1)`

Therefore, applying this rule, you get:

`(dy)/(dx)(x^{(3)/(2)})=(3)/(2)x^{(3)/(2)-1}=(3)/(2)x^{(3)/(2)-1}=(3)/(2)x^{(1)/(2)}=(3)/(2)\sqrt[]{x}`

Then:

`f^(\prime)(x)=(3)/(2)\sqrt[]{x}`

2. Now you have to substitute this value of "x" into the function derivated:

`x=4`

In order to find:

`f^(\prime)(4)`

Then, substituting and evaluating, you get:

`\begin{gathered} f^(\prime)(4)=(3)/(2)\sqrt[]{4} \\ \\ f^(\prime)(4)=(3)/(2)\sqrt[]{4} \\ \\ f^(\prime)(4)=3 \end{gathered}`

3. Substitute this value of "x" into the function derivated before:

`x=25`

In order to find:

`f^(\prime)(25)`

Then, substituting and evaluating, you get:

`\begin{gathered} f^(\prime)(25)=(3)/(2)\sqrt[]{25} \\ \\ f^(\prime)(25)=(15)/(2) \end{gathered}`

Hence, the answers are:

`(\Delta y)/(\Delta x)=(39)/(7)`
`\begin{gathered} f^(\prime)(4)=3 \\ \\ f^(\prime)(25)=(15)/(2) \end{gathered}`