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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 97
Chapter 1, Problem 97

Identify the amplitude and period of the following functions.
g(θ)=3cos(θ3)g\(\left\)(\(\theta\]\right\))=3\(\cos\[\left\)(\(\frac{\theta}{3}\]\right\))

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The given function is g(θ) = 3cos(θ/3). This is a cosine function of the form a * cos(bθ), where a is the amplitude and b affects the period.
Identify the amplitude: In the function g(θ) = 3cos(θ/3), the coefficient of the cosine function is 3. Therefore, the amplitude is the absolute value of this coefficient, which is |3| = 3.
Determine the period: The period of a basic cosine function cos(θ) is 2π. For a function of the form cos(bθ), the period is adjusted by the factor b, and is given by the formula 2π/|b|.
In the function g(θ) = 3cos(θ/3), the value of b is 1/3 (since θ/3 can be rewritten as (1/3)θ). Therefore, the period is 2π divided by 1/3, which simplifies to 2π * 3.
Thus, the amplitude of the function is 3, and the period is 6π.

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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, specifically the height of the wave from its midline to its peak. In the function g(θ) = 3cos(θ/3), the amplitude is represented by the coefficient of the cosine function, which is 3. This means the function oscillates between 3 and -3.
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Period

The period of a periodic function is the length of one complete cycle of the wave. For the cosine function, the standard period is 2π. In the function g(θ) = 3cos(θ/3), the period is affected by the coefficient of θ inside the cosine. The period can be calculated using the formula 2π divided by the coefficient of θ, which in this case is 1/3, resulting in a period of 6π.
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Phase Shift

Phase shift refers to the horizontal shift of a periodic function along the x-axis. While the given function g(θ) = 3cos(θ/3) does not include a phase shift term, understanding this concept is essential for analyzing more complex functions. A phase shift occurs when the argument of the cosine function is adjusted by a constant, affecting where the wave starts on the x-axis.
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Related Practice
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

Area of a circular sector Prove that the area of a sector of a circle of radius r associated with a central angle θ\(\theta\) (measured in radians) is A=12r2θA=\(\frac\)12r^2\(\theta\).

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Textbook Question

Identify the amplitude and period of the following functions.

f(π)=2sin2θf\(\left\)(\(\pi\]\right\))=2\(\sin\)2\(\theta\)

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

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