Math 2280 - Assignment 13
Dylan Zwick
Spring 2013
Section 9.1 - 1, 8, 11, 13, 21
Section 9.2 - 1, 9, 15, 17, 20
Section 9.3 - 1, 5, 8, 13, 20
1
Section 9.1 - Periodic Functions and TrigonometricSeries
9.1.1 - Sketch the graph of the function f defined for all t by the givenformula, and determine whether it is periodic. If so, find its smallestperiod.
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f(t) = sin3t.
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2
9.1.8 - Sketch the graph of the function f defined for all t by the givenformula, and determine whether it is periodic. If so, find its smallestperiod.
f(t) sinhirt.
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3
9.1.11 - The value of a period 2ir function f(t) in one full period is givenbelow. Sketch several periods of its graph and find its Fourier series.
f(t) =1, —71 <t < 7i.
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9.1.13 - The value of a period 2ir function f(t) in one full period is givenbelow. Sketch several periods of its graph and find its Fourier series.
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7-+ i()+
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17
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9.1.21 - The value of a period 2ir function f(t) in one full period is givenbelow. Sketch several periods of its graph and find its Fourier series.
f(t)t2, —<t<
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6
Section 9.2 - General Fourier Series and Convergence
9.2.1 - The values of a periodic function f(t) in one full period are givenbelow; at each discontinuity the value of f(t) is that given by theaverage value condition. Sketch the graph of f and find its Fourierseries.
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(1)+
+5)1 (i;) ++
f(t)—{—2 —3<t<O
— 2 O<t<3
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7
9.2.9 - The values of a periodic function f(t) in one full period are givenbelow; at each discontinuity the value of f(t) is that given by theaverage value condition. Sketch the graph of f and find its Fourierseries.
f(t) = —1<t<’
3
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9.2.15 -
(a) - Suppose that f is a function of period 27r with f(t) = t2 foro <t < 2n. Show that
f(t)= 42
+4cosnt
—
sinnt
and sketch the graph of f, indicating the value at each discontinuity.
(b) - Deduce the series summations
i 2
in= 6
n=’
and
I in+1 21)
12n=1
from the Fourier series in part (a).
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9.2.17 -
(a) - Supose that f is a funciton of period 2 with f(t) = t for 0 <t <
2. Show that
f(t) = 1— 2 sin nirt
and sketch the graph of f, indicating the value at each discontinuity.
(b) - Substitute an appropriate value of t to deduce Leibniz’s series
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9.2.20 - Derive the Fourier series given below, and graph the period 2irfunction to which the series converges.
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)oj(
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— I / c 3 y— J LI IL1T —-
2 2cosnt 3t —6irt+2ir— 12
n=1
(0 <t < 2r)
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13
Section 9.3 - Fourier Sine and Cosine Series
9.3.1 - For the given function f(t) defined on the given interval find theFourier cosine and sine Series of f and sketch the graphs of the twoextensions of f to which these two series converge.
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10
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9.3.5 - For the given function f(t) defined on the given interval find theFourier cosine and sine series of f and sketch the graphs of the twoextensions of f to which these two series converge.
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0<t<11<t<22<t<3
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9.3.8 - For the given function f(t) defined on the given interval find theFourier cosine and sine series of f and sketch the graphs of the twoextensions of f to which these two series converge.
f(t) = t — t2,
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