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Ch. 1 - Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 1 - FunctionsProblem 96
Chapter 1, Problem 96

Identify the amplitude and period of the following functions.
f(π)=2sin2θf\(\left\)(\(\pi\]\right\))=2\(\sin\)2\(\theta\)

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1
Identify the general form of the sine function, which is \( f(\theta) = a \sin(b\theta) \), where \( a \) is the amplitude and \( \frac{2\pi}{b} \) is the period.
In the given function \( f(\theta) = 2\sin(2\theta) \), compare it with the general form to identify the values of \( a \) and \( b \). Here, \( a = 2 \) and \( b = 2 \).
The amplitude of a sine function is the absolute value of \( a \). Therefore, the amplitude is \( |2| = 2 \).
The period of a sine function is calculated using the formula \( \frac{2\pi}{b} \). Substitute \( b = 2 \) into the formula to find the period.
Calculate the period: \( \frac{2\pi}{2} = \pi \). Thus, the period of the function is \( \pi \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Amplitude

Amplitude refers to the maximum value of a periodic function, particularly in trigonometric functions like sine and cosine. It indicates how far the function reaches above and below its midline. For the function f(θ) = 2sin(2θ), the amplitude is 2, meaning the function oscillates between -2 and 2.
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Period

The period of a function is the length of one complete cycle of the wave. For sine and cosine functions, the period can be determined from the coefficient of the variable inside the function. In f(θ) = 2sin(2θ), the period is calculated as 2π divided by the coefficient of θ, which is 2, resulting in a period of π.
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Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in calculus and describe relationships between angles and sides of triangles. They are periodic functions, meaning they repeat their values in regular intervals. Understanding their properties, including amplitude and period, is essential for analyzing wave-like behaviors in various applications.
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Related Practice
Textbook Question

Even and odd at the origin


a. If ƒ(0) is defined and ƒ is an even function, is it necessarily true that ƒ(0) = 0? Explain.

1
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Textbook Question

{Use of Tech} Polynomial calculations

Find a polynomial ƒ that satisfies the following properties. (Hint: Determine the degree of ƒ; then substitute a polynomial of that degree and solve for its coefficients.)

ƒ(ƒ(x)) = x⁴ - 12x² + 30

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Textbook Question

Identify the amplitude and period of the following functions.

q(x)=3.6cos(πx24)q\(\left\)(x\(\right\))=3.6\(\cos\[\left\)(\(\frac{\pi x}{24}\]\right\))

Textbook Question

{Use of Tech} Polynomial calculations

Find a polynomial ƒ that satisfies the following properties. (Hint: Determine the degree of ƒ; then substitute a polynomial of that degree and solve for its coefficients.)

ƒ(ƒ(x)) = 9x - 8

3
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Textbook Question

Identify the amplitude and period of the following functions.

g(θ)=3cos(θ3)g\(\left\)(\(\theta\]\right\))=3\(\cos\[\left\)(\(\frac{\theta}{3}\]\right\))