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AP Calculus AB: 8.3.1 Concavity and Inflection Points

Mathematics9 CardsCreated 3 months ago

This flashcard set explores how the second derivative reveals the concavity of a function—whether it curves upward or downward. It explains how to identify inflection points, where the concavity changes, and emphasizes that these occur only where the second derivative is zero or undefined.

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Concavity and Inflection Points

  • The concavity of a graph can be determined by using the second derivative.

  • If the second derivative of a function is positive on a given interval, then the graph of the function is concave up on that interval. If the second derivative of a function is negative on a given interval, then the graph of the function is concave down on that interval.

  • Points where the graph changes concavity are called inflection points.

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Key Terms

Term
Definition

Concavity and Inflection Points

  • The concavity of a graph can be determined by using the second derivative.

  • If the second derivative of a function is positiv...

note

  • Given a function, you can determine where it is increasing and where it is decreasing. The next property to examine is curvature, or concav...

Suppose you are told that in the interval a < x < b, the slope of the function h (x) is decreasing as x increases. Is h (x) concave up or concave down in this interval?

Concave down

Given the graph of g(x), find the intervals where g(x) is concave up.

x < x1

Is the function h (x) = 2 cos x + sin^2 x concave up or down at the point x=π?

Concave up

Suppose you are given that f ″(x) < 0 on the intervals x < −1 and x > 1 (and nowhere else). Which of the following could be a graph of f (x)?

Graph C describes a function which is concave down on the appropriate intervals. The following is a good rule of thumb: If the graph is shaped like...

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AP Calculus AB: 8.3.1 Concavity and Inflection Points
TermDefinition

Concavity and Inflection Points

  • The concavity of a graph can be determined by using the second derivative.

  • If the second derivative of a function is positive on a given interval, then the graph of the function is concave up on that interval. If the second derivative of a function is negative on a given interval, then the graph of the function is concave down on that interval.

  • Points where the graph changes concavity are called inflection points.

note

  • Given a function, you can determine where it is increasing and where it is decreasing. The next property to examine is curvature, or concavity.

  • Notice that the graph on the left is decreasing and then increasing, but it is curved upward. The graph is said to be concave up. It resembles the outline of a coffee cup that is upright.

  • On the right, the graph is curved downward. It is said to be concave down. This time it resembles the outline of an overturned coffee cup.

  • To determine the concavity of a function, you will need to study the behavior of its derivative.

  • Notice that the slopes of the tangent lines start out negative, then become zero, and finally become positive. They are increasing. Therefore the derivative is increasing.

  • Another way of saying that the derivative is increasing is to say that the second derivative is positive.

  • You can conclude that if the second derivative is positive, the function is concave up. Similarly, if the second derivative is negative, the function is concave down.

  • This graph is concave down on the left and concave up on the right. The point where the concavity changes is called an inflection point.

  • An inflection point can only occur where the second
    derivative is zero or undefined.

Suppose you are told that in the interval a < x < b, the slope of the function h (x) is decreasing as x increases. Is h (x) concave up or concave down in this interval?

Concave down

Given the graph of g(x), find the intervals where g(x) is concave up.

x < x1

Is the function h (x) = 2 cos x + sin^2 x concave up or down at the point x=π?

Concave up

Suppose you are given that f ″(x) < 0 on the intervals x < −1 and x > 1 (and nowhere else). Which of the following could be a graph of f (x)?

Graph C describes a function which is concave down on the appropriate intervals. The following is a good rule of thumb: If the graph is shaped like a bowl on an interval, then the function is concave up there. If the graph is shaped like an upside-down bowl, then the function is concave down there.

To be more precise, notice that on the intervals x < −1 and x > 1, the slope of the tangent line to the graph decreases as x increases. That means that the function is concave down on the intervals x < −1 and x > 1.

Suppose you are given the function s (t) = t ^3 − 5t − 1. Is s (t) concave up or concave down at t = 2?

Concave up

If the graph of the second derivative of f(x)is shown, on which of the following intervals is f(x) concave up?

(q,s)

If the graph of the second derivative is shown, on which of the following intervals is f(x) concave down?

(s,t)