Question

1. let f(x) = x³ - x + 1. find the average range of change from 1 to 3.

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \( f(x) \) from \( x = a \) to \( x = b \) is given by \( \frac{f(b)-f(a)}{b - a} \). Here, \( a = 1 \), \( b = 3 \), and \( f(x)=x^{3}-x + 1 \).

Step2: Calculate \( f(3) \)

Substitute \( x = 3 \) into \( f(x) \):
\( f(3)=3^{3}-3 + 1=27-3 + 1=25 \)

Step3: Calculate \( f(1) \)

Substitute \( x = 1 \) into \( f(x) \):
\( f(1)=1^{3}-1 + 1=1-1 + 1=1 \)

Step4: Calculate the average rate of change

Using the formula \( \frac{f(3)-f(1)}{3 - 1} \), substitute the values of \( f(3) \) and \( f(1) \):
\( \frac{25 - 1}{3 - 1}=\frac{24}{2}=12 \)

Answer:

12