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

Ch. 1 - Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 1 - Functions

문제 97

1장, 문제 97

Identify the amplitude and period of the following functions.
$g$\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.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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.

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.

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

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Amplitude

Period

Phase Shift

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