Question

if ( f(x)=6 x^{2}+3 x ), what is the average rate of change over (1,3) ?
( \bigcirc ) a. 30
( \bigcirc ) b. 36
( \bigcirc ) c. 18
( \bigcirc ) d. 27

Explanation:

Step1: Recall the formula for average rate of change

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

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

• For \(x = 1\): \(f(1)=6\times(1)^{2}+3\times(1)=6 + 3=9\).
• For \(x = 3\): \(f(3)=6\times(3)^{2}+3\times(3)=6\times9+9=54 + 9=63\).

Step3: Apply the average - rate - of - change formula

Substitute \(f(1) = 9\), \(f(3)=63\), \(a = 1\), and \(b = 3\) into \(\frac{f(b)-f(a)}{b - a}\). We get \(\frac{f(3)-f(1)}{3 - 1}=\frac{63-9}{2}=\frac{54}{2}=27\).

Answer:

d. 27