Given the function:
[tex]f(x)=x^{\frac{3}{2}}[/tex]You can rewrite it in this form:
[tex]f(x)=\sqrt[]{x^3}[/tex]Because by definition:
[tex]\sqrt[n]{b^m}=b^{\frac{m}{n}}[/tex]• The formula for calculating the Average Rate of Change over an interval is:
[tex]m=\frac{\Delta y}{\Delta x}=\frac{f(b)-f(a)}{b-a}[/tex]Where these two points are on the function:
[tex](a,f(a)),(b,f(b))[/tex]In this case, given the interval:
[tex]\lbrack4,25\rbrack[/tex]You can identify that:
[tex]a=4[/tex]Then, substituting this value into the function and evaluating, you get:
[tex]f(a)=f(4)=\sqrt[]{(4)^3}=\sqrt[]{64}=8[/tex]You can also identify that:
[tex]b=25[/tex]Then, substituting this value into the function and evaluating, you get:
[tex]f(b)=f(25)=\sqrt[]{(25)^3}=125[/tex]Now you can substitute values into the formula and then evaluate, in order to find the Average Rate of Change over the given interval:
[tex]\frac{\Delta y}{\Delta x}=\frac{125-8}{25-4}=\frac{39}{7}[/tex]• 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:
[tex]f^{\prime}(x)[/tex]By definition:
[tex]\frac{d}{dx}(x^n)=nx^{n-1}[/tex]Therefore, applying this rule, you get:
[tex]\frac{dy}{dx}(x^{\frac{3}{2}})=\frac{3}{2}x^{\frac{3}{2}-1}=\frac{3}{2}x^{\frac{3}{2}-1}=\frac{3}{2}x^{\frac{1}{2}}=\frac{3}{2}\sqrt[]{x}[/tex]Then:
[tex]f^{\prime}(x)=\frac{3}{2}\sqrt[]{x}[/tex]2. Now you have to substitute this value of "x" into the function derivated:
[tex]x=4[/tex]In order to find:
[tex]f^{\prime}(4)[/tex]Then, substituting and evaluating, you get:
[tex]\begin{gathered} f^{\prime}(4)=\frac{3}{2}\sqrt[]{4} \\ \\ f^{\prime}(4)=\frac{3}{2}\sqrt[]{4} \\ \\ f^{\prime}(4)=3 \end{gathered}[/tex]3. Substitute this value of "x" into the function derivated before:
[tex]x=25[/tex]In order to find:
[tex]f^{\prime}(25)[/tex]Then, substituting and evaluating, you get:
[tex]\begin{gathered} f^{\prime}(25)=\frac{3}{2}\sqrt[]{25} \\ \\ f^{\prime}(25)=\frac{15}{2} \end{gathered}[/tex]Hence, the answers are:
[tex]\frac{\Delta y}{\Delta x}=\frac{39}{7}[/tex][tex]\begin{gathered} f^{\prime}(4)=3 \\ \\ f^{\prime}(25)=\frac{15}{2} \end{gathered}[/tex]