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➗calculus ii review

Monotonically Decreasing

Written by the Fiveable Content Team • Last updated September 2025
Written by the Fiveable Content Team • Last updated September 2025

Definition

A function is said to be monotonically decreasing if its value never increases as the input variable increases. In other words, the function's value either stays the same or decreases as the input variable becomes larger.

5 Must Know Facts For Your Next Test

  1. A monotonically decreasing function has a non-positive derivative, meaning the derivative is either negative or zero at every point in the function's domain.
  2. Monotonically decreasing functions are often used to model quantities that decrease over time, such as radioactive decay, population decline, or the depreciation of an asset.
  3. In the context of alternating series, a monotonically decreasing sequence of positive terms is a necessary condition for the series to converge.
  4. The terms in a convergent alternating series must approach zero, and a monotonically decreasing sequence ensures that the absolute value of each term is less than the previous term.
  5. Monotonically decreasing functions have important applications in optimization problems, where they can be used to find the minimum value of a function.

Review Questions

  • Explain how the concept of a monotonically decreasing function relates to the convergence of an alternating series.
    • For an alternating series to converge, the absolute value of the terms must approach zero. A monotonically decreasing sequence of positive terms ensures that each term is less than the previous term, which is a necessary condition for the series to converge. The decreasing nature of the terms guarantees that the series will eventually become small enough to satisfy the convergence criteria.
  • Describe the relationship between the derivative of a monotonically decreasing function and its behavior.
    • A function is monotonically decreasing if and only if its derivative is non-positive, meaning it is either negative or zero at every point in the function's domain. This implies that the function's value never increases as the input variable increases, as the derivative represents the rate of change of the function. The non-positive derivative ensures that the function is either strictly decreasing or constant, which is the defining characteristic of a monotonically decreasing function.
  • Discuss the practical applications of monotonically decreasing functions in the context of optimization problems.
    • Monotonically decreasing functions are often used in optimization problems to find the minimum value of a function. Since the function is guaranteed to either decrease or remain constant as the input variable increases, the minimum value must occur at either the lower bound of the function's domain or at a point where the derivative is zero (a critical point). This property simplifies the optimization process, as the search for the minimum can be confined to a specific range or set of critical points, rather than requiring a more complex analysis of the function's behavior.