Math Problem Statement

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On which of the intervals does the function decrease the most?

Solution

The table in the image shows the average rate of change for a function ff over four intervals. The values of the rate of change indicate how much the function is increasing or decreasing in each interval. A negative rate of change means the function is decreasing in that interval.

Here are the average rates of change for each interval:

  • For 1x31 \leq x \leq 3, the rate of change is 1.6-1.6.
  • For 3x53 \leq x \leq 5, the rate of change is 2.8-2.8.
  • For 5x65 \leq x \leq 6, the rate of change is 0.9-0.9.
  • For 6x106 \leq x \leq 10, the rate of change is 1.2-1.2.

Question:

"On which of the intervals does the function decrease the most?"

Solution:

To determine where the function decreases the most, we need to find the interval with the most negative rate of change (the largest negative number).

  • 2.8-2.8 is the most negative value, which corresponds to the interval 3x53 \leq x \leq 5.

Thus, the function decreases the most in the interval 3x53 \leq x \leq 5.

Would you like more details on how to interpret the average rate of change? Here are five related questions to expand your understanding:

  1. What does the rate of change represent in terms of a function's graph?
  2. How do you calculate the average rate of change between two points?
  3. What does it mean if the average rate of change is positive?
  4. How can you determine if a function is increasing or decreasing over an interval?
  5. How would the answer change if one of the intervals had a positive rate of change?

Tip: A more negative rate of change indicates a steeper decrease in the function's value over the given interval.

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Math Problem Analysis

Mathematical Concepts

Average Rate of Change
Functions
Intervals

Formulas

Average Rate of Change = (f(b) - f(a)) / (b - a)

Theorems

Rate of Change Theorem

Suitable Grade Level

Grades 9-11