Domanda del libro di testo
Let ƒ(x) = 3x - x³ . Show that the equation ƒ(𝓍) = -4 has a solution in the interval [2,3] and use Newton’s method to find it.
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Ch. 4 - Applications of Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 4, Problema 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.
Guida verificata passo dopo passo1
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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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.
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05:59
Determining Concavity Given a Function
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.
Video consigliato:
07:09The First Derivative Test: Finding Local Extrema
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.
Video consigliato:
06:02The Second Derivative Test: Finding Local Extrema
Pratica correlata
Domanda del libro di testo
Sketch the graph of a twice-differentiable function y=f(x) that passes through the points (-2,2), (-1,1), (0,0),(1,1), and (2,2) and whose first two derivatives have the following sign patterns.
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Domanda del libro di testo
In Exercises 121–124, find the inflection points (if any) on the graph of the function and the coordinates of the points on the graph where the function has a local maximum or local minimum value. Then graph the function in a region large enough to show all these points simultaneously. Add to your picture the graphs of the function’s first and second derivatives. How are the values at which these graphs intersect the x-axis related to the graph of the function? In what other ways are the graphs of the derivatives related to the graph of the function?
123. y=(4/5)x^5+16x^2-25
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Domanda del libro di testo
Graph f(x) = 2x^4 -4x^2 + 1 and its first two derivatives together. Comment on the behavior of f in relation to the signs and values of f' and f".
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Domanda del libro di testo
106. Motion Along a Line The graphs in Exercises 105 and 106 show the position s=f(t) of an object moving up and down on a coordinate line. At approximately what times is the (b) velocity equal to zero?
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Domanda del libro di testo
Each of Exercises 89–92 shows the graphs of the first and second derivatives of a function y=f(x). Copy the picture and add to it a sketch of the approximate graph of f, given that the graph passes through the point P.
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