Question

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

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 \(\frac{f(b)-f(a)}{b - a}\). Here \(a = 1\), \(b=3\), and \(f(x)=2x^{3}-5x^{2}+3x - 7\).

Step2: Calculate \(f(1)\)

Substitute \(x = 1\) into \(f(x)\):

$$
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$$

Step3: Calculate \(f(3)\)

Substitute \(x = 3\) into \(f(x)\):

$$
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$$

Step4: Calculate the average rate of change

Using the formula \(\frac{f(b)-f(a)}{b - a}\), with \(a = 1\), \(b = 3\), \(f(1)=-7\), \(f(3)=11\)

$$ \frac{f(3)-f(1)}{3 - 1}=\frac{11-(-7)}{2}=\frac{11 + 7}{2}=\frac{18}{2}=9 $$

Answer:

c. 9