Textbook Question
Finding Critical Points
In Exercises 41–50, determine all critical points and all domain endpoints for each function.
g(x) = √(2x − x²)
Ch. 4 - Applications of Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- 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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 4, Problem 4.1.54
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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Identify the natural domain of the function y = 3x + tan(x). The function tan(x) is undefined at odd multiples of π/2, so the natural domain is all real numbers except x = (2n+1)π/2, where n is an integer.
Consider the behavior of the function as x approaches the points where tan(x) is undefined. As x approaches (2n+1)π/2 from the left, tan(x) approaches negative infinity, and from the right, tan(x) approaches positive infinity. This indicates that the function y = 3x + tan(x) has vertical asymptotes at these points.
Analyze the behavior of the function as x approaches positive and negative infinity. As x approaches positive or negative infinity, the linear term 3x dominates, causing the function to increase or decrease without bound.
Since the function has vertical asymptotes and increases or decreases without bound as x approaches infinity or negative infinity, it does not attain a highest or lowest value on its natural domain.
Conclude that the function y = 3x + tan(x) has neither an absolute minimum nor an absolute maximum on its natural domain due to the presence of vertical asymptotes and unbounded behavior at infinity.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Natural Domain
The natural domain of a function is the set of all possible input values (x-values) for which the function is defined. For the function y = 3x + tan x, the natural domain excludes values where tan x is undefined, such as x = π/2 + kπ, where k is an integer. Understanding the domain is crucial for analyzing the behavior of the function across its entire range.
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Absolute Minimum and Maximum
An absolute minimum or maximum refers to the lowest or highest value a function can attain on a given interval or domain. For y = 3x + tan x, determining the existence of absolute extrema involves analyzing the function's behavior and limits, especially considering the periodic nature and asymptotes of the tangent function, which can lead to unbounded behavior.
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Behavior of Trigonometric Functions
Trigonometric functions like tan x have unique properties, such as periodicity and asymptotes, which affect their graphs and values. The tangent function has vertical asymptotes at odd multiples of π/2, causing it to approach infinity. This behavior is key to understanding why y = 3x + tan x may not have absolute extrema, as the function can increase or decrease without bound near these asymptotes.
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Related Practice
Textbook Question
Initial Value Problems
Solve the initial value problems in Exercises 71–90.
ds/dt = 1 + cos t, s(0) = 4
Textbook Question
41. Among all triangles in the first quadrant formed by the x-axis, the y-axis, and tangent lines to the graph of y=3x-x^2, what is the smallest possible area?
Textbook Question
22. A window is in the form of a rectangle surmounted by a semicircle. The rectangle is of clear glass, whereas the semicircle is of tinted glass that transmits only half as much light per unit area as clear glass does. The total perimeter is fixed. Find the proportions of the window that will admit the most light. Neglect the thickness of the frame.
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Textbook Question
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) = x⁴ᐟ⁵, [0, 1]
Textbook Question
Finding Indefinite Integrals
In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.
∫(8y − 2 / y¹ᐟ⁴) dy