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Problem 12. Determine the amplitude and period of the function y - the function over its single period using five key points. s(x). Then, graph 3 cos

Problem 12. Determine the amplitude and period of the function y - the function over its single period using five key points. s(x). Then, graph 3 cos

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Problem 12

### Problems and Exercises in Mathematics #### Problem 12: Amplitude and Period of a Cosine Function **Task:** Determine the amplitude and period of the function \( y = 3 \cos \left( \frac{1}{4} x \right) \). Then, graph the function over its single period using five key points. --- **Solution Guide:** 1. **Amplitude:** - The amplitude of a cosine function \( y = A \cos(Bx) \) is the absolute value of \( A \). - In this case, \( A = 3 \), so the amplitude is \( 3 \). 2. **Period:** - The period of a cosine function \( y = A \cos(Bx) \) is calculated as \( \frac{2\pi}{|B|} \). - Here, \( B = \frac{1}{4} \), so the period is \( \frac{2\pi}{\frac{1}{4}} = 8\pi \). 3. **Graphing the Function:** - To graph \( y = 3 \cos \left( \frac{1}{4} x \right) \) over one period, identify five key points: the maximum, the zero points, the minimum, and another zero point. - Key points are generally at \( x = 0, \frac{Period}{4}, \frac{Period}{2}, \frac{3 \times Period}{4}, \) and \( Period \). - For \( y = 3 \cos \left( \frac{1}{4} x \right) \), these points would be \( x = 0, 2\pi, 4\pi, 6\pi, \) and \( 8\pi \). 4. **Summary of Key Points:** - \( x = 0 \): \( y = 3 \cos(0) = 3 \) - \( x = 2\pi \): \( y = 3 \cos(\frac{1}{4} \times 2\pi) = 3 \cos(\frac{\pi}{2}) = 0 \) - \( x = 4\pi \): \( y = 3 \cos(\frac{1}{4} \times 4\pi) = 3 \
Transcribed Image Text:### Problems and Exercises in Mathematics #### Problem 12: Amplitude and Period of a Cosine Function **Task:** Determine the amplitude and period of the function \( y = 3 \cos \left( \frac{1}{4} x \right) \). Then, graph the function over its single period using five key points. --- **Solution Guide:** 1. **Amplitude:** - The amplitude of a cosine function \( y = A \cos(Bx) \) is the absolute value of \( A \). - In this case, \( A = 3 \), so the amplitude is \( 3 \). 2. **Period:** - The period of a cosine function \( y = A \cos(Bx) \) is calculated as \( \frac{2\pi}{|B|} \). - Here, \( B = \frac{1}{4} \), so the period is \( \frac{2\pi}{\frac{1}{4}} = 8\pi \). 3. **Graphing the Function:** - To graph \( y = 3 \cos \left( \frac{1}{4} x \right) \) over one period, identify five key points: the maximum, the zero points, the minimum, and another zero point. - Key points are generally at \( x = 0, \frac{Period}{4}, \frac{Period}{2}, \frac{3 \times Period}{4}, \) and \( Period \). - For \( y = 3 \cos \left( \frac{1}{4} x \right) \), these points would be \( x = 0, 2\pi, 4\pi, 6\pi, \) and \( 8\pi \). 4. **Summary of Key Points:** - \( x = 0 \): \( y = 3 \cos(0) = 3 \) - \( x = 2\pi \): \( y = 3 \cos(\frac{1}{4} \times 2\pi) = 3 \cos(\frac{\pi}{2}) = 0 \) - \( x = 4\pi \): \( y = 3 \cos(\frac{1}{4} \times 4\pi) = 3 \
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