if ( f(x)=6 x^{2}+3 x ), what is the average rate of change over (1,3) ?\n( \bigcirc ) a. 30\n( \bigcirc )…

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

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

Answer

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