Announcements 10/6/10 Prayer Exam goes until Saturday a. a.Correction to syllabus: on Saturdays, the...
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Transcript of Announcements 10/6/10 Prayer Exam goes until Saturday a. a.Correction to syllabus: on Saturdays, the...
Announcements 10/6/10 Prayer Exam goes until Saturday
a. Correction to syllabus: on Saturdays, the Testing Center gives out last exam at 3 pm, closes at 4 pm.
Homework survey—survey closes tonight. Please respond this afternoon/evening if you haven’t already.
Taylor’s Series review:a. cos(x) = 1 – x2/2! + x4/4! – x6/6! + …b. sin(x) = x – x3/3! + x5/5! – x7/7! + …c. ex = 1 + x + x2/2! + x3/3! + x4/4! + …
Reminder What is ? What is k? How do they relate to the velocity? Relationship between and T Relationship between k and Consistency check: /k = ?
Reading Quiz What’s the complex conjugate of:
a.
b.
c.
d.
1 3
4 5
i
i
1 3
4 5
i
i
1 3
4 5
i
i
1 3
4 5
i
i
1 3
4 5
i
i
Complex Numbers – A Summary What is “i”? What is “-i”? The complex plane Complex conjugate
a. Graphically, complex conjugate = ? Polar vs. rectangular coordinates
a. Angle notation, “A” Euler’s equation…proof that ei = cos +
isina. must be in radiansb. Where is 10ei(/6) located on complex
plane?
What is the square root of 1… 1 or -1?
Complex Numbers, cont. Adding
a. …on complex plane, graphically? Multiplying
a. …on complex plane, graphically?b. How many solutions are there to x2=1? c. What are the solutions to x5=1?
(xxxxx=1) Subtracting and dividing
a. …on complex plane, graphically?
Polar/rectangular conversion Warning about rectangular-to-polar
conversion: tan-1(-1/2) = ?a. Do you mean to find the angle for (2,-1)
or (-2,1)?
Always draw a picture!!
Using complex numbers to add sines/cosines
Fact: when you add two sines or cosines having the same frequency (with possibly different amplitudes and phases), you get a sine wave with the same frequency! (but a still-different amplitude and phase)
a. “Proof” with Mathematica… (class make up numbers)
Worked problem: how do you find mathematically what the amplitude and phase are?
Another worked problem?
Using complex numbers to solve equations
Simple Harmonic Oscillator (ex.: Newton 2nd Law for mass on spring)
Guess a solution like
what it means, really: (and take Re{ … } of each side)
2
2
d x kx
mdt
( ) i tx t Ae
( ) cos( )x t A t