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Root Finding
2. Use Newton's method to estimate the one real solution of x^3 +3x + 1 = 0. Start with x_0 = 0 and then find x_2.
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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 4.1.11
Finding Extrema from Graphs
In Exercises 11–14, match the table with a graph.
Guida verificata passo dopo passo1
Identify the critical points from the table where the derivative f'(x) is zero. These points are x = a and x = b.
Since f'(a) = 0 and f'(b) = 0, these points could be local maxima, minima, or saddle points. We need to analyze the graph to determine the nature of these points.
For x = c, f'(c) = 5, which indicates that the function is increasing at x = c. This means the slope of the tangent line at x = c is positive.
Examine each graph to see where the slope of the tangent is zero at x = a and x = b, and positive at x = c. This will help us match the table to the correct graph.
In graph (d), the function has horizontal tangents (f'(x) = 0) at x = a and x = b, and the slope is positive at x = c, matching the conditions given in the table.
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Critical Points
Critical points occur where the derivative of a function is zero or undefined. These points are essential for finding local extrema, as they indicate where the function's slope changes, potentially leading to local maxima or minima. In the given table, points 'a' and 'b' are critical points since their derivatives are zero.
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04:50
Critical Points
First Derivative Test
The First Derivative Test is a method used to determine whether a critical point is a local maximum, local minimum, or neither. By analyzing the sign of the derivative before and after the critical point, one can conclude the nature of the extremum. For instance, if the derivative changes from positive to negative at a critical point, it indicates a local maximum.
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07:09The First Derivative Test: Finding Local Extrema
Extrema
Extrema refer to the maximum and minimum values of a function within a given interval. Local extrema are found at critical points, while absolute extrema are the highest or lowest values over the entire domain. In the context of the provided graphs and table, identifying the extrema involves analyzing the critical points and their corresponding function values.
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05:58Finding Extrema Graphically
Pratica correlata
Domanda del libro di testo
Checking the Mean Value Theorem
Which of the functions in Exercises 7–12 satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers.
f(x) = {2x − 3, 0 ≤ x ≤ 2
6x − x² − 7, 2 < x ≤ 3
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Domanda del libro di testo
Finding Extrema from Graphs
In Exercises 7–10, find the absolute extreme values and where they occur.
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Domanda del libro di testo
Each of Exercises 67–88 gives the first derivative of a continuous function y=f(x). Find y'' and then use Steps 2–4 of the graphing procedure described in this section to sketch the general shape of the graph of f.
82. y' = sin t, for 0 ≤ t ≤ 2π
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Domanda del libro di testo
99. In Exercises 99 and 100, the graph of f' is given. Determine x-values corresponding to inflection points for the graph of f.
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Domanda del libro di testo
Checking the Mean Value Theorem
Which of the functions in Exercises 7–12 satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers.
f(x) = x²ᐟ³, [−1, 8]
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