Question

interval | 0 ≤ x ≤ 1 | 1 ≤ x ≤ 4 | 4 ≤ x ≤ 8 | 8 ≤ x ≤ 10
average rate of change | 10 | -5 | 2 | 6
the table gives the average rates of change of a function f over different intervals. in which of the intervals does the function increase the most?
a) 0 ≤ x ≤ 1
b) 1 ≤ x ≤ 4
c) 4 ≤ x ≤ 8
d) 8 ≤ x ≤ 10

Explanation:

Step1: Recall the meaning of average rate of change

The average rate of change of a function over an interval represents how much the function changes on average per unit of \( x \) in that interval. A positive rate means the function is increasing, and a negative rate means it's decreasing. To find where the function increases the most, we compare the positive average rates of change (since negative rates mean decrease, we can ignore the negative one here).

Step2: List the positive average rates of change

• For \( 0\leq x\leq1 \): average rate of change is \( 10 \)
• For \( 4\leq x\leq8 \): average rate of change is \( 2 \)
• For \( 8\leq x\leq10 \): average rate of change is \( 6 \)

Step3: Compare the positive rates

We compare \( 10 \), \( 2 \), and \( 6 \). Among these, \( 10 \) is the largest. So the interval with the largest average rate of change (where the function increases the most) is \( 0\leq x\leq1 \).

Answer:

A. \( 0 \leq x \leq 1 \)