Given the function:
[tex]f\mleft(x\mright)=3x^3+2[/tex]You can use the following formula to find the Average Rate of Change:
[tex]Average\text{ }Rate\text{ }of\text{ }Change=\frac{f(b)-f(a)}{b-a}[/tex]Where the following are two points on the function:
[tex]\begin{gathered} (a,f(a)) \\ \\ (b,f(b)) \end{gathered}[/tex]1. You know that you must determine the Average Rate of Change between:
[tex]x=3\text{ and }x=5[/tex]Then, you can set up that:
[tex]\begin{gathered} a=3 \\ b=5 \end{gathered}[/tex]2. In order to find the corresponding value for:
[tex]\begin{gathered} f(a)=f(3) \\ f(b)=f(5) \end{gathered}[/tex]You can follow these steps:
- Substitute the value of "a" into the function and evaluate:
[tex]f(3)=3(3)^3+2=3(27)+2=81+2=83[/tex]- Substitute the value of "b" into the function and then evaluate:
[tex]f(5)=3(5)^3+2=3(125)+2=377[/tex]3. Knowing all the values, you can substitute into the formula for calculating the Average Rate of Change and evaluate:
[tex]Average\text{ }Rate\text{ }of\text{ }Change=\frac{377-83}{5-3}=\frac{294}{2}=147[/tex]Hence, the answer is:
You can determine the average rate of change by finding the corresponding output values (y-values) for:
[tex]x=3\text{ and }x=5[/tex]
After finding those values, you can substitute them into the formula for calculating the Average Rate of Change, and then evaluate. It is:
[tex]Average\text{ }Rate\text{ }of\text{ }Change=147[/tex]