Question

if ( f(x)=2 x^{3}-5 x^{2}+3 x - 7 ), what is the average rate of change over (1,3)?
a. 10
b. 9
c. 8
d. 7

if ( f(x)=2 x^{3}-x + 4 ), what is the average rate of change over (0,2)?
a. 8
b. 7
c. 10
d. 6

what is the average rate of change for ( f(x)=x^{3}-2 x + 1 ) over (0,3)?
a. 7
b. 8
c. 9
d. 10

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

Step2: Solve for the first function \(f(x)=2x^{3}-5x^{2}+3x - 7\) over \([1,3]\)

• Calculate \(f(3)\):

$$
LATEXBLOCK0
$$

• Calculate \(f(1)\):

$$
LATEXBLOCK1
$$

• Calculate the average rate of change:

$$
LATEXBLOCK2
$$

Step3: Solve for the second function \(f(x)=2x^{3}-x + 4\) over \([0,2]\)

• Calculate \(f(2)\):

$$
LATEXBLOCK3
$$

• Calculate \(f(0)\):

$$
LATEXBLOCK4
$$

• Calculate the average rate of change:

$$
LATEXBLOCK5
$$

Step4: Solve for the third function \(f(x)=x^{3}-2x + 1\) over \([0,3]\)

• Calculate \(f(3)\):

$$
LATEXBLOCK6
$$

• Calculate \(f(0)\):

$$
LATEXBLOCK7
$$

• Calculate the average rate of change:

$$
LATEXBLOCK8
$$

Answer:

1. b. 9
2. b. 7
3. a. 7