Ch. 4 - Applications of Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경

Hass 15th Edition

Ch. 4 - Applications of Derivatives

문제 103

4장, 문제 103

103. A function f(x) has domain (-2, 2). The graph below is a plot of the derivative of f, not a plot of f itself. In other words, this is a graph of y = f'(x). Either use this graph to determine on which intervals the graph of f is concave up and on which intervals the graph of f is concave down, or explain why this information cannot be determined from the graph.

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To determine the concavity of the function f(x), we need to analyze the second derivative, f''(x). The graph provided is of the first derivative, f'(x).

The function f(x) is concave up where its second derivative, f''(x), is positive. This occurs where the graph of f'(x) is increasing.

Similarly, f(x) is concave down where its second derivative, f''(x), is negative. This occurs where the graph of f'(x) is decreasing.

From the graph, observe the intervals where f'(x) is increasing: this happens approximately from x = -2 to x = -1 and from x = 0 to x = 1. In these intervals, f(x) is concave up.

Observe the intervals where f'(x) is decreasing: this happens approximately from x = -1 to x = 0 and from x = 1 to x = 2. In these intervals, f(x) is concave down.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Concavity

Concavity refers to the direction in which a curve bends. A function is concave up on an interval if its second derivative is positive, indicating that the slope of the tangent line is increasing. Conversely, a function is concave down if its second derivative is negative, meaning the slope of the tangent line is decreasing. Understanding concavity helps in analyzing the behavior of the function's graph.

First Derivative Test

The first derivative test involves analyzing the sign of the derivative of a function to determine where the function is increasing or decreasing. If the derivative is positive, the function is increasing; if negative, it is decreasing. This information is crucial for understanding the behavior of the function and can indirectly provide insights into its concavity when combined with the second derivative.

Second Derivative and Its Relation to Concavity

The second derivative of a function provides information about its concavity. If the second derivative is positive, the function is concave up; if negative, it is concave down. In this question, since we are given the graph of the first derivative, we can infer concavity by examining where the first derivative is increasing or decreasing, as these changes indicate the sign of the second derivative.

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