Question

a function is given. f(t) = 6t^2 + t; t = 3, t = 5 (a) determine the net change between the given values of the variable. (b) determine the average rate of change between the given values of the variable.

Explanation:

Step1: Calculate f(5)

Substitute \(t = 5\) into \(f(t)=6t^{2}+t\).
\[

$$\begin{align*} f(5)&=6\times5^{2}+5\\ &=6\times25 + 5\\ &=150+5\\ &=155 \end{align*}$$

\]

Step2: Calculate f(3)

Substitute \(t = 3\) into \(f(t)=6t^{2}+t\).
\[

$$\begin{align*} f(3)&=6\times3^{2}+3\\ &=6\times9+3\\ &=54 + 3\\ &=57 \end{align*}$$

\]

Step3: Find net - change

The net - change of the function \(y = f(t)\) from \(t=a\) to \(t = b\) is \(f(b)-f(a)\). Here \(a = 3\), \(b = 5\), so the net - change is \(f(5)-f(3)=155 - 57=98\).

Step4: Find average rate of change

The average rate of change of the function \(y = f(t)\) from \(t=a\) to \(t = b\) is \(\frac{f(b)-f(a)}{b - a}\). Here \(a = 3\), \(b = 5\), and \(f(5)-f(3)=98\), \(b - a=5 - 3 = 2\). So the average rate of change is \(\frac{98}{2}=49\).

Answer:

(a) 98
(b) 49