assignment 13 math 2280 dylan zwick spring 2013math 2280-assignment 13 dylan zwick spring 2013...
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![Page 1: Assignment 13 Math 2280 Dylan Zwick Spring 2013Math 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](https://reader034.vdocument.in/reader034/viewer/2022042600/5f4b5dd68d29451afc4a3dc3/html5/thumbnails/1.jpg)
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
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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.
_/5 /L
f(t) = sin3t.
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2
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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.
r , -
3
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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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5< / I 11 ( 11I c(11ytLcyL
i2)f
2
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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.
—<toO<t<ir
7-+ i()+
01
17
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cox(n)d
jt1
)
— (Co5fl) C05(QJ)
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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<
Zli JT1
o
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( Li c(i))
/
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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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5I
$1 I RC ) (- (f()3
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(1)+
+5)1 (i;) ++
f(t)—{—2 —3<t<O
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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
V € So 1) Ji ‘ I 15arQ 0,
tto
vl
1V7 C’1 c5 (n
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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
111
1—+—+.
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4
+ s (Tht)
flT
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CJ) CC(r
a4
17
each
5()- 6J()
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9.2.20 - Derive the Fourier series given below, and graph the period 2irfunction to which the series converges.
Je he
‘l U C
)oj(
_______
— I / c 3 y— J LI IL1T —-
2 2cosnt 3t —6irt+2ir— 12
n=1
(0 <t < 2r)
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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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it:
(cuie ii,’e5
O<t<Tr.
C jtcIc
f(t)=1,
(c
z
C Y4 pi//iii ii1)
=1
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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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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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More room for Problem 9.3.13, if you need it.
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