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Root Finding
5. Use Newton's method to find the positive fourth root of 2 by solving the equation x^4 -2 = 0. Start with x_0 = 1 and 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 4.4.71
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.
71. y' = x(x² - 12)
Guida verificata passo dopo passo1
First, find the second derivative y'' by differentiating y' = x(x² - 12). Use the product rule for differentiation, which states that if you have a function u(x)v(x), its derivative is u'(x)v(x) + u(x)v'(x). Here, let u(x) = x and v(x) = x² - 12.
Differentiate u(x) = x to get u'(x) = 1. Differentiate v(x) = x² - 12 to get v'(x) = 2x.
Apply the product rule: y'' = u'(x)v(x) + u(x)v'(x) = 1(x² - 12) + x(2x). Simplify this expression to find y''.
Now, analyze the critical points and concavity using y'' to understand the general shape of the graph of f(x). Critical points occur where y'' = 0 or is undefined. Determine these points and test intervals around them to see if y'' is positive (indicating concave up) or negative (indicating concave down).
Use the information from the second derivative test to sketch the general shape of the graph of f(x). Consider the behavior at critical points and the concavity to draw a rough sketch of the graph, showing where it is increasing, decreasing, concave up, and concave down.
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First Derivative
The first derivative of a function, denoted as y' or f'(x), represents the rate of change of the function with respect to x. It provides information about the slope of the tangent line to the graph of the function at any given point. Analyzing the first derivative helps identify critical points, where the function may have local maxima, minima, or points of inflection.
Video consigliato:
07:09
The First Derivative Test: Finding Local Extrema
Second Derivative
The second derivative, denoted as y'' or f''(x), is the derivative of the first derivative. It indicates the rate of change of the slope of the function, providing insights into the concavity of the graph. If y'' is positive, the graph is concave up, suggesting a local minimum; if y'' is negative, the graph is concave down, indicating a local maximum.
Video consigliato:
06:02The Second Derivative Test: Finding Local Extrema
Graphing Procedure
The graphing procedure involves several steps to sketch the function based on its derivatives. Steps typically include finding critical points from the first derivative, determining concavity and inflection points from the second derivative, and analyzing the behavior of the function at these points. This systematic approach helps create an accurate representation of the function's overall shape.
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Percorso guidato
06:15Graphing The Derivative
Pratica correlata
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Finding Extrema from Graphs
In Exercises 1–6, determine from the graph whether the function has any absolute extreme values on [a, b]. Then explain how your answer is consistent with Theorem 1.
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Finding Extreme Values
In Exercises 1–10, find the extreme values (absolute and local) of the function over its natural domain, and where they occur.
y = 𝓍³ + 𝓍² ― 8𝓍 + 5
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Domanda del libro di testo
119. Find the values of constants a, b, and c such that the graph of y = ax^3 + bx^2 + cx has a
local maximum at x = 3, local minimum at x =- 1, and inflection point at (1, 11).
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Domanda del libro di testo
Sketch the graphs of the rational functions in Exercises 53–60.
𝓍⁴ ― 1
y = ------------------
𝓍²
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Theory and Examples
In Exercises 53 and 54, show that the function has neither an absolute minimum nor an absolute maximum on its natural domain.
y = 3x + tan x
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