Intervals of Increase/Decrease

Definition

Intervals of increase and decrease refer to the regions within the domain of a function where the function's value is either increasing or decreasing. These intervals are crucial in understanding the behavior and characteristics of a function, as they provide insights into the function's overall shape and trends.

5 Must Know Facts For Your Next Test

1. Intervals of increase and decrease are determined by examining the sign of the function's derivative on a given interval.
2. If the derivative is positive on an interval, the function is increasing on that interval.
3. If the derivative is negative on an interval, the function is decreasing on that interval.
4. The points where the function's derivative changes sign are called critical points, and they mark the boundaries between intervals of increase and decrease.
5. Analyzing the intervals of increase and decrease is essential for sketching the graph of a function and understanding its overall behavior.

Review Questions

• Explain how the sign of the derivative determines the intervals of increase and decrease for a function.
  • The sign of the derivative of a function determines whether the function is increasing or decreasing on a given interval. If the derivative is positive, the function is increasing on that interval. If the derivative is negative, the function is decreasing on that interval. The points where the derivative changes sign are called critical points, and they mark the boundaries between the intervals of increase and decrease.
• Describe the relationship between critical points and the intervals of increase and decrease.
  • Critical points of a function are the points where the function changes from increasing to decreasing, or vice versa. These critical points divide the function's domain into intervals of increase and decrease. By identifying the critical points, you can determine the exact intervals where the function is increasing and decreasing, which is crucial for understanding the function's behavior and sketching its graph.
• How can the analysis of intervals of increase and decrease be used to make inferences about the overall shape and behavior of a function?
  • The analysis of intervals of increase and decrease provides valuable insights into the overall shape and behavior of a function. By identifying the regions where the function is increasing and decreasing, you can determine the function's local maxima and minima, as well as any points of inflection. This information can be used to sketch the function's graph, understand its critical points, and make predictions about the function's behavior within its domain. The intervals of increase and decrease are fundamental to the study of a function's properties and characteristics.

Related terms

Increasing Function: A function is said to be increasing on an interval if its value increases as the input variable increases within that interval.

Decreasing Function: A function is said to be decreasing on an interval if its value decreases as the input variable increases within that interval.

Critical Point: A critical point of a function is a point where the function changes from increasing to decreasing, or vice versa, within its domain.