Understanding Increasing and Decreasing Intervals
When working with functions, it’s essential to understand the behavior of the function over a given interval. In this worksheet, we’ll explore increasing and decreasing intervals, which will help you analyze and graph functions more effectively.
What are Increasing and Decreasing Intervals?
Increasing and decreasing intervals are used to describe the behavior of a function over a specific interval. A function is said to be:
- Increasing on an interval if the function values increase as the input values increase.
- Decreasing on an interval if the function values decrease as the input values increase.
How to Determine Increasing and Decreasing Intervals
To determine the increasing and decreasing intervals of a function, you can use the following steps:
- Find the critical points of the function by setting the derivative equal to zero and solving for x.
- Create a sign chart to determine the sign of the derivative on each interval.
- Use the sign chart to identify the increasing and decreasing intervals.
📝 Note: Make sure to find all critical points, including those that may be outside the given interval.
Example 1: Finding Increasing and Decreasing Intervals
Suppose we have the function f(x) = x^3 - 6x^2 + 9x + 2. To find the increasing and decreasing intervals, we’ll follow the steps above.
Step 1: Find the critical points
f’(x) = 3x^2 - 12x + 9 = 0
Solving for x, we get:
x = 1, x = 3
Step 2: Create a sign chart
| Interval | Sign of f’(x) |
|---|---|
| (-∞, 1) | - |
| (1, 3) | + |
| (3, ∞) | - |
Step 3: Identify increasing and decreasing intervals
From the sign chart, we can see that:
- f(x) is increasing on the interval (1, 3)
- f(x) is decreasing on the intervals (-∞, 1) and (3, ∞)
Example 2: Finding Increasing and Decreasing Intervals with a Rational Function
Suppose we have the rational function f(x) = (x + 1) / (x - 2). To find the increasing and decreasing intervals, we’ll follow the steps above.
Step 1: Find the critical points
f’(x) = (x - 2)(1) - (x + 1)(1) / (x - 2)^2 = 0
Simplifying, we get:
x = -1, x = 2
Step 2: Create a sign chart
| Interval | Sign of f’(x) |
|---|---|
| (-∞, -1) | + |
| (-1, 2) | - |
| (2, ∞) | + |
Step 3: Identify increasing and decreasing intervals
From the sign chart, we can see that:
- f(x) is increasing on the intervals (-∞, -1) and (2, ∞)
- f(x) is decreasing on the interval (-1, 2)
Conclusion
Understanding increasing and decreasing intervals is crucial for analyzing and graphing functions. By following the steps outlined in this worksheet, you can determine the increasing and decreasing intervals of any function. Remember to find all critical points, create a sign chart, and use the sign chart to identify the increasing and decreasing intervals.
What is the difference between an increasing and decreasing interval?
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An increasing interval is where the function values increase as the input values increase, while a decreasing interval is where the function values decrease as the input values increase.
How do I determine the increasing and decreasing intervals of a function?
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To determine the increasing and decreasing intervals, find the critical points, create a sign chart, and use the sign chart to identify the increasing and decreasing intervals.
What is a critical point, and how do I find it?
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A critical point is a point where the derivative of the function is equal to zero or undefined. To find critical points, set the derivative equal to zero and solve for x.
Related Terms:
- Increasing decreasing positive negative intervals Worksheet
- Increasing and decreasing Functions PDF
