Question

1. if ( f(x)=6 x^{2}+3 x ), what is the average rate of change over (1,3)?

a. 27
b. 30
c. 36
d. 18

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \(y = f(x)\) over the interval \([a,b]\) is given by \(\frac{f(b)-f(a)}{b - a}\). Here, \(a = 1\), \(b=3\), and \(f(x)=6x^{2}+3x\).

Step2: Calculate \(f(3)\) and \(f(1)\)

• For \(x = 3\):

\(f(3)=6\times3^{2}+3\times3=6\times9 + 9=54 + 9=63\)

• For \(x = 1\):

\(f(1)=6\times1^{2}+3\times1=6+3 = 9\)

Step3: Substitute into the average - rate - of - change formula

\(\frac{f(3)-f(1)}{3 - 1}=\frac{63 - 9}{2}=\frac{54}{2}=27\)

Answer:

a. 27