Science:Math Exam Resources/Courses/MATH307/April 2012/Question 06 (b)

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MATH307 April 2012

  •  Q1 (a)  •  Q1 (b)  •  Q1 (c)  •  Q1 (d)  •  Q2 (a)  •  Q2 (b)  •  Q2 (c)  •  Q3  •  Q4 (a)  •  Q4 (b)  •  Q4 (c)  •  Q4 (d)  •  Q4 (e)  •  Q5 (a)  •  Q5 (b)  •  Q5 (c)  •  Q6 (a)  •  Q6 (b)  •  Q6 (c)  •  Q6 (d)  •  Q7 (a)  •  Q7 (b)  •  Q7 (c)  •  Q8 (a)  •  Q8 (b)  •  Q8 (c)  •  

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Question 06 (b)

Explain the physical significance of the values Failed to parse (Conversion error. Server ("https://wiki.ubc.ca/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle |c_{n}|} and how these relate to Failed to parse (Conversion error. Server ("https://wiki.ubc.ca/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \omega _{n}} . If t is measure in days, and the time period T encompasses several years, which values of cn might you expect to be the largest absolute value? (Think about what time scales you expect temperature to fluctuate over.)

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle c_{n}} is a complex number with real and imaginary part, we write Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle c_{n}} in its complex exponential form Failed to parse (Conversion error. Server ("https://wiki.ubc.ca/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle c_{n}} = Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle \left|c_{n}{}\right|e^{2\pi i\phi _{n}}} Therefore Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle c_{n}} reflects the original temperature function’s frequency spectrum. If we put the above expression back into the Fourier series representation of y(t), we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle y(t)=\sum_{n=-\propto }^{\propto }c_{n}e^{2\pi i\frac{n}{T}t} = \sum_{n=-\propto }^{\propto }\left | c_{n} \right |e^{2\pi i\phi _{n}}e^{2\pi i\frac{n}{T}t} = \sum_{n=-\propto }^{\propto }\left | c_{n} \right |e^{2\pi i(\omega _{n}t+\phi _{n})}} t=0 or t=T would make Failed to parse (Conversion error. Server ("https://wiki.ubc.ca/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle c_{n}} the largest. Because both t=0 and t=T would make Failed to parse (Conversion error. Server ("https://wiki.ubc.ca/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle e^{2\pi i{\frac {n}{T}}t}} = 1. So Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle c_{0}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle c_{T}} have the largest absolute value. The values of Failed to parse (Conversion error. Server ("https://wiki.ubc.ca/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle c_{n}} might you expect to have the largest absolute value is the frequency for the maximum temperature swings over the course of years. This would probably be seasonal fluctuations, so it would be Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wiki.ubc.ca/api/rest_v1/":): {\displaystyle c_{n}} for about 12 months.

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