Question

if ( f(x)=6 x^{2}+3 x ), what is the average rate of change over ( 1,3 )?
a. 30
b. 18
c. 27
d. 36
which interval has a positive average rate of change for ( f(x)=-x^{2}+2 x+5 )?
a. ( 1,2 )
b. ( 0,3 )
c. ( 2,3 )
d. ( 0,1 )
if ( f(x)=-x^{2}+4 x-3 ), what is the average rate of change over ( 0,2 )?
a. 0
b. 1
c. 2
d. -2

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 \(\frac{f(b)-f(a)}{b - a}\).

Step2: Solve for \(f(x)=6x^{2}+3x\) over \([1,3]\)

• First, find \(f(3)\):

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

• Then, find \(f(1)\):

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

• Now, calculate the average rate of change:

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

Step3: Solve for \(f(x)=-x^{2}+2x + 5\)

• For interval \([1,2]\):

\(f(2)=-(2)^{2}+2\times2+5=-4 + 4+5=5\)
\(f(1)=-(1)^{2}+2\times1+5=-1 + 2+5=6\)
\(\frac{f(2)-f(1)}{2 - 1}=\frac{5 - 6}{1}=-1\)

• For interval \([0,3]\):

\(f(3)=-(3)^{2}+2\times3+5=-9+6 + 5=2\)
\(f(0)=-(0)^{2}+2\times0+5=5\)
\(\frac{f(3)-f(0)}{3 - 0}=\frac{2-5}{3}=-1\)

• For interval \([2,3]\):

\(f(3)=2\) (calculated above), \(f(2)=5\) (calculated above)
\(\frac{f(3)-f(2)}{3 - 2}=\frac{2-5}{1}=-3\)

• For interval \([0,1]\):

\(f(1)=6\) (calculated above), \(f(0)=5\) (calculated above)
\(\frac{f(1)-f(0)}{1 - 0}=\frac{6-5}{1}=1\)

Step4: Solve for \(f(x)=-x^{2}+4x-3\) over \([0,2]\)

• Find \(f(2)\):

\(f(2)=-(2)^{2}+4\times2-3=-4 + 8-3=1\)

• Find \(f(0)\):

\(f(0)=-(0)^{2}+4\times0-3=-3\)

• Calculate the average rate of change:

\(\frac{f(2)-f(0)}{2 - 0}=\frac{1-(-3)}{2}=\frac{4}{2}=2\)

Answer:

1. c. \(27\)
2. d. \([0,1]\)
3. c. \(2\)