Explanation Given: The function A ( x ) = ∫ 0 x f ( t ) d t The following figure describing f ( x ) Key concepts applied: FTC II or The second Fundamental Theorem of Calculus deals with derivatives of definite integrals with respect to variable upper limits of integration. It states that if F ( x ) = ∫ a x f ( t ) d t and f is continuous, then F ' ( x ) = f ( x ) . Concavity of a graph A graph is said to be concave up at a point if the tangent line to the graph at that point lies below the graph in the vicinity of the point Regardless of the sign of the slope (positive, negative or zero), the slopes of the tangent are increasing (from left to right) when it is concave up. Therefore, when a function is concave up, the graph of its derivative is increasing. A graph is said to be concave down at a point if the tangent line to the graph at that point lies above the graph in the vicinity of the point. Regardless of the sign of the slope (positive, negative or zero), the slopes of the tangent are decreasing as we move from left to right when the graph is concave down. Therefore, when a function is concave down, the graph of its derivative is decreasing. A point where the concavity changes (from up to down or down to up) is called a point of inflection (POI); note that the tangent line to a graph at a point of inflection must cross the graph at that point. A function f has a local maximum or relative maximum at a point x o if the values f ( x ) of f for x ‘near’ x o are all less than f ( x o ) . Thus, the graph of f near x o has a peak at x o . A function f has a local minimum or relative minimum at a point x o if the values f ( x ) of f for x ‘near’ x o are all greater than f ( x o ) . Thus, the graph of f near x o has a trough at x o . The basic properties of the maxima and minima are summarized in the following table Behavior Graph Derivative at point slightly to the left of the extremum point x o Derivative at extremum point x o Derivative at point slightly to the right of the extremum point x o Second derivative at extremum point x o Local maximum f ' ( x o − ) > 0 Positive Increasing f ' ( x o ) = 0 Zero f ' ( x o + ) < 0 Negative Decreasing f ' ' ( x o ) < 0 Local Minimum f ' ( x o − ) < 0 Negative Decreasing f ' ( x o ) = 0 Zero f ' ( x o + ) > 0 Positive Increasing f ' ' ( x o ) > 0 From the above table we understand that the minimum values of a function occur where its derivate goes from negative to positive, similarly the maximum values of a function occur where its derivate goes from positive to negative. Correspondingly, the minimum values of a function occur where its second derivate is negative, and the maximum values of a function occur where its second derivate is positive. Some functions have an inflexion point which is a point at which a change in the direction of curvature occurs. At inflexion points, the second derivative is equal to zero. The function of the form A ( x ) = ∫ 0 x f ( t ) d t is called the Area function. Some properties of the area function are as follows, The Area Function A ( x ) = ∫ 0 x f ( t ) d t is Positive when the graph of the function is above the x-axis. This means that for the region where f ( x ) > 0 , the area function is increasing. The Area Function A ( x ) = ∫ 0 x f ( t ) d t is negative when the graph of the function is below the x-axis. This means that for the region where f ( x ) < 0 , the area function is decreasing. From the above two properties it follows that when the function f ( x ) changes from positive to negative, the area function stop increasing and starts decreasing. Therefore, at this point the area function A ( x ) reaches a local maximum. Similarly, when the function f ( x ) changes from negative to positive, the area function stop decreasing and starts increasing...

Loose-Leaf Version for Calculus: Early Transcendentals Single Variable

4th Edition

ISBN: 9781319312886

Author: Rogawski, Jon, Adams, Colin, FRANZOSA, Robert

Publisher: W. H. Freeman

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Question

Chapter 5.5, Problem 45E

To determine

a) The intervals on which the given function is increasing and decreasing

b) Where the given function has a local min or max

c) The inflection points of the given function

d) Where the given function is concave up or concave down

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Chapter 5.5, Problem 44E

Chapter 5.5, Problem 46E

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Chapter 5 Solutions

Loose-Leaf Version for Calculus: Early Transcendentals Single Variable

Ch. 5.1 - Prob. 1PQ Ch. 5.1 - Prob. 2PQ Ch. 5.1 - Prob. 3PQ Ch. 5.1 - Prob. 4PQ Ch. 5.1 - Prob. 5PQ Ch. 5.1 - Prob. 1E Ch. 5.1 - Prob. 2E Ch. 5.1 - Prob. 3E Ch. 5.1 - Prob. 4E Ch. 5.1 - Prob. 5E

Knowledge Booster

a) The intervals on which the given function is increasing and decreasing b) Where the given function has a local min or max c) The inflection points of the given function d) Where the given function is concave up or concave down

a) The intervals on which the given function is increasing and decreasing b) Where the given function has a local min or max c) The inflection points of the given function d) Where the given function is concave up or concave down

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Chapter 5 Solutions

a) The intervals on which the given function is increasing and decreasing

b) Where the given function has a local min or max

c) The inflection points of the given function

d) Where the given function is concave up or concave down