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AP Calculus AB: 8.2.3 Regions Where a Function Increases or Decreases

Mathematics12 CardsCreated 3 months ago

This set of flashcards explains how critical points divide a function’s graph into intervals where it either increases or decreases. By analyzing the sign of the first derivative, you can determine these intervals and understand how critical points signal potential changes in the function’s behavior.

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Regions Where a Function Increases or Decreases

  • Critical points divide the curve of a function into regions where the function either increases or decreases.

  • On an interval, the sign of the first derivative of a function indicates whether that function is increasing or decreasing

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Key Terms

Term
Definition

Regions Where a Function Increases or Decreases

  • Critical points divide the curve of a function into regions where the function either increases or decreases.

  • On an interval...

note

  • Consider the regions of this graph that are defined by the critical points.

  • Between any two adjacent critical points the cur...

Find the interval(s) where the function y=(x^2−9)^2/3 is decreasing.

x < −3 or 0 < x < 3

Based on this graph of f ‘(x), where is the function f(x) increasing?

−1 < x < 1

Find the interval(s) where the function g(x)=x^3−3x^2+2 is decreasing.

0 < x < 2

Suppose you know that a continuous function f (x) is increasing at x = 2, and decreasing at x = 4, and that x = 3 is a critical point. Which of the following could describe f ′(3)?

f ′(3) does not exist.

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AP Calculus AB: 8.2.3 Regions Where a Function Increases or Decreases
TermDefinition

Regions Where a Function Increases or Decreases

  • Critical points divide the curve of a function into regions where the function either increases or decreases.

  • On an interval, the sign of the first derivative of a function indicates whether that function is increasing or decreasing

note

  • Consider the regions of this graph that are defined by the critical points.

  • Between any two adjacent critical points the curve behaves in only one way. It is either increasing or it is decreasing.

  • Notice that there is no way for the behavior of the curve to change without it either leveling off or making a sharp turn. When either situation occurs, there is a critical point, indicating a possible change from increasing to decreasing, or decreasing to increasing.

  • However, a critical point does not always force a curve to change its behavior. For example, x 5 is a critical point, but the function is increasing on both the right and the left.

  • Recall that a critical point of a function is a point where the derivative of the function is either zero or undefined. Thus it is possible to have a point where a function is continuous but not differentiable.

  • Here is an example of a function whose critical point you have already determined.

  • To study the behavior of the curve, make a sign chart. To keep it simple, just label the critical point.

  • Next, choose any x-value that is less than the critical value. Here an easy value is x = –1. Since the derivative at x = –1 is negative, all the values to the left of the critical point will make the derivative negative. The function is decreasing on that region. Mark a down-arrow on the sign chart.

  • You must also choose an x-value to the right of the critical point. Here an easy value is x = 0. Since that value makes the derivative positive, the function is increasing on that region. Mark an up-arrow on the sign chart.

  • Since the function is decreasing to the left of the critical point and increasing to the right, you can conclude that the curve reaches a minimum point at the critical point.

Find the interval(s) where the function y=(x^2−9)^2/3 is decreasing.

x < −3 or 0 < x < 3

Based on this graph of f ‘(x), where is the function f(x) increasing?

−1 < x < 1

Find the interval(s) where the function g(x)=x^3−3x^2+2 is decreasing.

0 < x < 2

Suppose you know that a continuous function f (x) is increasing at x = 2, and decreasing at x = 4, and that x = 3 is a critical point. Which of the following could describe f ′(3)?

f ′(3) does not exist.

Find the interval(s) where the function f(x)=1/x^2 is increasing.

x < 0

Suppose you are told that H ′(x) is positive on the intervals −3 < x < −1 and x > 5 (and nowhere else). Which of the following could be a graph of H (x)?

Since H ′(x) is positive on the intervals −3 < x < − 1 and x > 5, H (x) will be increasing on those intervals. Graph C describes such a function

Suppose f(x)is a continuous differentiable function and you are given that f′′(x) is always positive. Which of the following statements must be true?

f′(x) is always increasing

On what interval is this function increasing?

x < 0 or x > 0

On what interval is the function f(x)=xlnx+2 decreasing?

0

Find the interval(s) where f (x) = x ^3 − 3x is increasing.

x < −1 or x > 1