question\nfind the average rate of change of the function\n$f(x)$, given below, from $x = 3$ to $x =…

question\nfind the average rate of change of the function\n$f(x)$, given below, from $x = 3$ to $x = t$.\n$f(x)=4x^{2}+x - 2$\ngive your answer in terms of $t$.\nprovide your answer below:

question\nfind the average rate of change of the function\n$f(x)$, given below, from $x = 3$ to $x = t$.\n$f(x)=4x^{2}+x - 2$\ngive your answer in terms of $t$.\nprovide your answer below:

Answer

Answer:

$4t + 13$

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function (y = f(x)) from (x=a) to (x = b) is (\frac{f(b)-f(a)}{b - a}). Here, (a = 3) and (b=t), and (f(x)=4x^{2}+x - 2).

Step2: Calculate (f(3))

Substitute (x = 3) into (f(x)): [ \begin{align*} f(3)&=4\times(3)^{2}+3- 2\ &=4\times9+3 - 2\ &=36+3-2\ &=37 \end{align*} ]

Step3: Calculate (f(t))

Substitute (x = t) into (f(x)): [ f(t)=4t^{2}+t - 2 ]

Step4: Calculate the average rate of change

[ \begin{align*} \frac{f(t)-f(3)}{t - 3}&=\frac{(4t^{2}+t - 2)-37}{t - 3}\ &=\frac{4t^{2}+t-39}{t - 3}\ &=\frac{4t^{2}-12t+13t - 39}{t - 3}\ &=\frac{4t(t - 3)+13(t - 3)}{t - 3}\ &=\frac{(t - 3)(4t + 13)}{t - 3}\ &=4t+13\quad(t\neq3) \end{align*} ]