Textbook Question
Graph the functions in Exercises 37–56.
y = |x − 2|
Ch. 1 - Functions
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 1, Problem 1.3.36
Using the Addition Formulas
Use the addition formulas to derive the identities in Exercises 31–36.
sin (A − B) = sin A cos B − cos A sin B
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Start by recalling the addition formula for sine, which is: .
To derive the formula for , use the identity for sine of a negative angle: .
Apply the identity to the addition formula: .
Substitute into the addition formula: .
Use the identities and to simplify: .
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Addition Formulas
Addition formulas are trigonometric identities that express the sine and cosine of the sum or difference of two angles in terms of the sines and cosines of the individual angles. For example, the sine of the difference of two angles, sin(A - B), can be expressed as sin A cos B - cos A sin B. These formulas are essential for simplifying expressions and solving trigonometric equations.
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Additional Rules for Indefinite Integrals
Trigonometric Identities
Trigonometric identities are equations that hold true for all values of the variables involved, provided they are within the domain of the functions. They are fundamental in calculus and trigonometry for transforming and simplifying expressions. The addition formulas are a subset of these identities, allowing for the manipulation of sine and cosine functions in various mathematical contexts.
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Derivation Techniques
Derivation techniques involve the methods used to prove or derive mathematical identities from established formulas. In the context of trigonometric identities, this often includes algebraic manipulation, substitution, and the application of known identities. Understanding these techniques is crucial for effectively deriving new identities, such as the sine and cosine addition formulas.
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Related Practice
Textbook Question
In Exercises 5–8, determine whether the graph of the function is symmetric about the 𝔂-axis, the origin, or neither.
𝔂 = e⁻ˣ²
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Textbook Question
Even and Odd Functions
In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.
f(x) = x² + x
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Textbook Question
In Exercises 19–32, find the (a) domain and (b) range.
_____
𝔂 = -1 + ∛ 2 - x
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Textbook Question
In Exercises 9–16, determine whether the function is even, odd, or neither.
𝔂 = x - sin x
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Textbook Question
Use graphing software to graph the functions specified in Exercises 31–36.
Select a viewing window that reveals the key features of the function.
Graph two periods of the function f (x) = 3 cot (x/2) + 1.