Question

if ( f(x)=x^{2}+3x + 2 ), what is the average rate of change over ( -1,1 )?
a. 3
b. 5
c. 0.4
d. 2

if ( f(x)=x^{3}-3x^{2}+2x ), what is the average rate of change over ( 1,4 )?
a. 5
b. -3
c. -2
d. 8

what is the average rate of change of ( f(x)=7x ) over ( 0,5 )?
a. 2
b. 5
c. 3
d. 7

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 \(f(x)=x^{2}+3x + 2\) over \([-1,1]\)

• First, find \(f(1)\):

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

• Then, find \(f(-1)\):

\(f(-1)=(-1)^{2}+3\times(-1)+2=1-3 + 2=0\)

• Now, calculate the average rate of change:

\(\frac{f(1)-f(-1)}{1-(-1)}=\frac{6 - 0}{2}=3\)

Step3: Solve for \(f(x)=x^{3}-3x^{2}+2x\) over \([1,4]\)

• First, find \(f(4)\):

\(f(4)=4^{3}-3\times4^{2}+2\times4=64-48 + 8=24\)

• Then, find \(f(1)\):

\(f(1)=1^{3}-3\times1^{2}+2\times1=1-3 + 2=0\)

• Now, calculate the average rate of change:

\(\frac{f(4)-f(1)}{4 - 1}=\frac{24-0}{3}=8\)

Step4: Solve for \(f(x)=7x\) over \([0,5]\)

• First, find \(f(5)\):

\(f(5)=7\times5 = 35\)

• Then, find \(f(0)\):

\(f(0)=7\times0=0\)

• Now, calculate the average rate of change:

\(\frac{f(5)-f(0)}{5-0}=\frac{35 - 0}{5}=7\)

Answer:

1. A. 3
2. D. 8
3. D. 7