I: Multiple Choice

(1)

1.(3pts) Consider the f_u11etio11

2+2‘: —2<:::<0

f(@1)=

₂ _0§:13-<2

The co1.'1'ect Fourier series 1'eprese11tetio11 of f is:

KP.-5' 33i1%(1 " ("1)“]Si11(E'?)

F

®g + 2,;;;;1%§(1 - (—1)'"') @@S(“—;@) + %(-1)n+iS1n(%)

[Q1 3 + E,§°+1WF(1 — (—1)'"’) (105%) + E(—1)'”*+ s111("""—?)

(D1 3 + 2;-:.11@<1 ~ <~11»":»

% + Z}f1°:1%(1 -- (--1)“) cos(m'ra?) + -,;§;(—1)?'*+1 si11('n:rr$) _

P32, Wig 'p'<1W¢i*r’@w

rs neiilqew eve]/1 new

gjci] £0

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_-1

_{_2}

1‘”i%]:f:‘”“">@:§s

2. (3pts) Consider the fu11etio11

f(:r.1]:?'r2—$2, —7r<:L*<1i1T. The correct Fourier series rep1'esentetiou of f is:

{A}§§_i1§g si11('n,:1:) $64) E3. é.’_._\r/_§:_} ) § C9 [/4 )3 ( {f )! ( D) (Q,/~@

T’? + Ef=1$(--1)""+1 c0s(12.x) _

[C]? + E,j°=1§‘§[—1]'"'+1 cos(nm) + ﬁg si11(n:r:) W PO V’)? ' 70 Lin c>c>_g 4:: i"<..=J<5.’(-vi 5 /3) Q,/1

[D]% + E§°:1§;(-1)"+1 si11(n$) Mi, I 7L

+ 2?=1$(_]_)n+1 c0S(.m.G)

(__i‘_'.';-..) )

[M6 ﬂééai

0:/I 2

L0 I/VJ/J!/1 cf

()0 I

%:%&%@&:%£%ﬁ%uW$2[F@+§]rHT

:17

₂

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‘Z.

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TF

'*" ~?T£7/”'"" “E1

I-'1

L/-1';

-'--,__r___""---..-!' 1:‘v-‘'1

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(2)

F

3, (3pts) Consider the function f : 21-rt -— 152, O < t < 'rr. Assume that if we extend f to the negative t-axis in s periodic lnsnner, the resulting function is even. If we ﬁnd e 2'rr-periodic solution, mp (15), of the ODE

0I2$ ,

E'£§+$:..i(t)

in the forln of a, Fourier series, then the correct Fourier coeﬂicients of mp (2?) ere:

‘Ii’

.r

£A3»A@=1gi, A-T1,:

Be-:0

FBIAO = An = 0, BR = (~1)“(%;1 ~ $1 + is

2

U:%“:A-n: :-B’-'1-:0

4112 ___ 4 B__ 1n211' 4 +4

'[D:' )AU : T: An ‘*' \ ‘F1 P" (— ) _ “R3

1E3»)AU=2—*;5,An:

_{(H 11}

Bﬂ.=0

{>5

§

_~**;'.'3"'

W1

_"ii _{Vi C/=:v5(1/1+)}

X

\l

><FcII

FCIIZ

/If +

/In C05 (I/II}

:42"!

O5’

:7

41 */Iﬂ K12 .(jr:~5(:$/PI“)

tag /I

/I

“7

4 Au

_..-

|-TN»

.--"12

("I/I$.l”LI)/Iy1£..<.>5(v1I‘) 1‘

"

#

J1 505 (WI)

- 4/7;“

,_

_

4 Z

Hg

/'Ir1 T

1/:7

-

5-

.__i_

I-Sn 7' O

I

Pa

ﬁx

'=--...

“"=‘"’I

’

4. (3pts) Consider the function f = 2e_"TI“’I, —1 <1 m < 1. The correct complex Fourier series representation I

for is iven b : ; so _,_

f g 2(c—€’—1)“) 'm1':1.: A in TIII }(

0 . -

--

s

.1-_

772;!"

$_ 7fY£'I"‘1V1l I "II‘<'LI+mh> -; . )

S

6 ci)(

If F0

4?,

AX

7|'[[-Iwiii 6

,(- 1

-1

fjihi-EDO.._.__OU'—if'T€% ) ‘Z 2

L5 J 11- e(1+n 1r ] 7( . I

@920“

r "n

»-sms<1+%

2

1

.9

1/l“‘eD ~"

-__

C ‘Z

— “+1 -—'*.=r$

°°=~ s»r<1iiTa(1+%I “L

.-"'

=-O0 e(1+n21r2) 8 _

oo 1 (“1)n+1 EH T I ..

——@‘3 1|‘(1+*rr§} B"

icign 00 5 J 1 I I

r [)0 — "— n —' m -" -.,- " H X “'. -TI‘

KD} En kiln mrr in _, _,~, S 2 ‘Z 6 ii/I Y

{E} 3“

2 ( +

)6 2

1

I'r'H_> X20

I,

.-. ,_I-I-*

‘ | _ | ,

. J III

: I

I "'I+:‘1»=1r§ ’\”'I I _ "" » Min I i___M_____' +_

_,I..-._.__L

~T><¢-

--""_

._'!''_,.---""’_,',,._,_..-@-

F.-is

'1'".-i*.

TTZIWTB "7

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I

H.»

T

;.: MI

=.'

--

’

*‘

___

’ ll -I-I

e/%i(]); _

Mir ~71 6“CIIniI'

I ~ ~ ~

(3)

I | At r : g, the extension converges to:

I" ‘I

IO;

f \

J

I D Z"

I-53)

ll GD

re): 0 égmj

‘PT U<.."3<l I

1

; ' _ _{__-1 '3""'°""_@

. = _ C

5- J ,1 = _. _ o-- . .. I

"‘; W

-1.

1

I

..* "1" ‘I

‘I

'%.‘@l-e

9--“"'"'Q ,.. O’/‘O

E"l'\

6.(3pts) Let the curve C be the ellipse centered at the origin in the my-plane, with major diameter of 14 along the $—axis and minor diameter of 10 along the y-axis, oriented counter-clockwise. A correct parametrization of C is

iven by:

F (t) I Tcost, 'y(t) = Ssint, t lj U II :r(t) : 14 cost, 'y(t) : 10 sint, t Z O [C)$[t) = 5cost, y(t) : Tsint, 25 Z O [D]$I[t} I 1U cos t, y(t) = 14 sint, t Z O {E).r(f;) = 7cos 15, y[t) = —5sint, t Z U

7 W

I\AJ' _?T I

r *~ ‘

U31 -g

I

T

(4)

7, (3pts)The length of the curve given by 'r(t) :< 7152, t2\ﬁ, Int >, fro1n 1 § t 5 2, is:

I ‘-. . _.I,

gZ+h11

F6“:

4 ILIJFJ

2 ‘FL;

’r' >

r 1 3 __

“-er 923

,

,_

vi

, I _, ,

+ln2 l" 4.2;'g+i—z "*" I J ML?/+'.2g,";%:[7

I f(/LIIMI I m_-;~

C

/I/1:2;

L ;-

S2 II r‘(/1/MI in '1‘

Slﬂflils +,@;I'll1/I-'1 12u2+¢Qp»qull/W

"‘ ”7""I-It-ﬂwfzfl -"" Z”7~I +*/QACII]

"I 2I+iL»(,2}

8.(3pts) Consider the curve given by *r[t) =4; sint — tcost,cost + tsin t,3t2 — 8 Ir:-. At t : 3, the value of the curvature .'r.(t) is:

"I

I F

..

“W373?

F dI'§§\/1);

ICns.I'J

“

' 1

I I/C -: 4 ‘i ')L,.§_'_@_.§-l- C(.1~"‘ -l*£_j'l‘y1'1I'-'1

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. I _ _

' ______..._..-.-.____--—"-|

____..-II mxll

J:wI1:-.i;'%-

1» mi‘

-I JEM!

,

-1 111,, . ' IL I

f >< r _- J l 1 W p _

+giWI_ _- _]_'CG_‘2+ . ‘I T" 1'-' ii ‘“' é,}‘{:<'.1$i’ I‘ IZQIF-’!ﬂI) _ _. J.

- <_;iI/!‘I;»\*I‘I"¢.3(.rSi' acsi" I'L'.iv'l'I' (9 I E-(_7I7!i.l/‘Jig LL 2c"oSIH-S, __

J

:;gf-.=~i'Zé.1£/II‘)

é-Iiaestj

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IIr’v ~11 5 J%@vc.»++%@+Lr@$?+*YiLI -I 4%‘?

{A}, L _ X 1

|rBl|§7 F 4 6[_;'1;,'l' -i'l§>f*f1‘l.)-§‘;5"-l-l- ‘i’ S1./l‘l 'i'({)‘El’. |

' ' . - I I4” Ii Zjll--g,%.1::-El’ -"»I—£Siv11l' " ""I"?(;<.>S'2‘II'l

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ll WII3 +3 '§»*)—*',]"}r§_'? 5'?" I )1] _

.-I.

(5)

9. (Bpts) C011sider the curve given by r(t) =< arctan t, %ln(1 -1- t2) >. For a given t, the tangential component of

l 'ation is: .

r'?.C(,‘:,.i3B11 PM y_,q[,,_) d

$1

|[1+t2

@~-

A-{BY 1

I110.

- ’ ll r"‘C=I~}II

’ I[1+r1]F.-Am

1 Z

.1

L

1| _-r

, 4IJ>s

.+H-,5

‘ij

f“4I5‘i* <’**%‘I 3"?”

>

I-'"I+I'*

qy _%.

I

A

_

p r 1

I§l?3;>

r*‘@tI 1

vzt

_{_L}

_=.-

_.1.

cz+

_I

_--—

z=+

“IEJIC

_,5',",,',v)'P» QINLF‘ J’(],,z) ~i57:I7.

£|+;2)4u++“3)

£:+_,L'*‘-Ev

—§-

U”l

t~+

*

1 -,1, ______. ..l 2+ -' ._ __.l-_

‘P L "'-"1 A} T ,1)? T ) ,

'1',/‘II _ I _____l___ 1

L____rT___ _. __.. _ _ [I‘l"l1)

II r"‘II

(_

A‘ g __ 1::

I.‘-'

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_-—" ' ,4. -1 1"

-ii-F'-_- I . - T

KB "2

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w,.n,0\¢ 5(1.—,a~w.¢1.0\ "'I5§(5'IZI\/$."~'@@>)

*

1 L—2)(q) + <~a)<f)

u

.-Ill ﬁl_|l

_-r

i ' 1*

.j—"

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,

rill’

,

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*,.

,

10. (3pts)Let W(s,t) = F(*u.(s,t),'u(s,t]) and 'u.(1,0) = 5, ’u.S(l,U) =: 4, 'u,¢(1,0) = 3, '0(1,0) I 2, 'uS(l,U) = 'T_,'u;(1,U) = -1, F,,,(5, 2) : -2, F,,,(5, 2) = -3. Then the quantity VI/'S(l,O) is equal to:

I

r ' l F (:%__f ~ 2-Y

“(go

/Us 1

%

T

is

T

5;;

Q5

- I

(6)

Q:

_il— _I= ' 1

11 3pts)Cons1der the function F('i: y, 2) = wyz at the pomt (1 2, 3 The minimum rate of change at the given

point is

(A) \/ii

(B) 7

imam,

D)-7

"T;

Z

2 X2

X 7

WE

W

9

I?-'53: (9 J;

“ ‘~;__-|-annnuian

1‘

I1

(|i“OI]~

pl"...

‘=-#~»I

(I311

""*

~

(:2 ,,,1_;,,,,'|,,,n.»I{_,i1I,»'l/l 1'39‘ C C’ {...» t§1l/1516 T Ii}

‘-12. 3pts)Cons1de1 the surface given by the equation $2 + yg + 2:2 = 25 The tangent plane to this suiface is parallel to the plane :1: + 21.; + 2: : 1 at the point

I’

2':-‘xM"'— ‘—-_h._.l"

_‘\

If F

\A) ~. 3 1 3 1 2)

If ‘i IKE)

{Cl_J

f_|;h'\ cQum

‘U1...

w|—*

we

0.1%--U, wai- 10»-0.1%

a,

5-"U‘! we

VF

"Z?>< 29,22

J

LEE

HI’

I/I gd JD II/I gm Jr If :2‘ Q 2

O

CD1 or

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s.<>rI...<.§

II/its IQ? I/1*‘,/"6""/If-’¢/T

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SC.

vdo

Qm(

CBO

15 rm ea

I

(7)

are

H, Z 3

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21> +

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|lOl|-215 " < " ' 1 l I )

'~.3'§L:..%_

.

H 2* r

s>(:

Q-~ zwl"

\,

W2), 1;

l—~§t

L9 ,1 ._ .

Ir?‘ .3 | -_,_,-1-— 1; J Z. J I la la

' 1

S

_if

j‘1” _,__r"

I

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+ ll

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ac

Q

lln+

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2'2‘

2

=0

I w st» ~,/rs 1~+ll+"-“fl 1~+~@>l

14. (313-ts) Consider the two circles in the my-plane given by equations $2 + y2 I 8 and (ii) $2 + 192 — 9. At any point, when compared to the curvature of circle (i), tl1e curvature of circle (ii) is:

-{A} larger ‘B the same

smaller [D)opposite [1-B) no way to tell

.1 IIIn -i—l1-|I\-IIHI -II".-In

i

4'.

I

i |. 1

1-—__-r-

—_-1--'1'

I

(8)

'Z— ' -Ir

I

15.(3pts) -The value of I0. G($, y)d_-5, Where G'[:c, y) : and C is given by 23; = 323% with 1 § :1? § 8, is:

-{All 8(5% -— 2%

r “'8 3

>

_

@i<5§"2%>

2 r10:§;<3§>

X21 1

Q "

1 Z"

2 ) E, "1 ,

[D]i(5‘- "

_____,M_

1 1E:»§l5%~

I53wmltﬁ

1,

1 1+ +”"r it

"-._»-" I300}

1

= 5?

<5

I

i"*“:"\

I

so r»4"s

-"

1;;

1 1 *1»,

~;

2”?“ 3» "‘

ll

All

4 1 " ,1,,.’Y%1“‘sl1

- |

I'-I ‘_ ____.

c

1|,

,.. r"\

_{_..} _J _"5.

» t

_l/.,,i.:

d

_I

‘J4

_A

’i‘J~1

_, I, Z} , '12 {/1 1 2

‘ -"F 1. "~_ -' ;

‘-I .;_

.‘ Z Y 73 3&1 _,_ “U1 1' 3'

. . ___,r 4 ,..- . ,5 /I/| /

'1/I

,

1»

.2.

r __ Z? :2,

F

_{Z 7'}

3 Z 5.3!? -~ "') H

_"1’

hula

_J

1

16. (3pts] Suppose r(t) is a smooth vector function, with unit tangent T(t) and unit normal N We know that T(t) is orthogonal to N (t) because:

[A] T(t) is smooth

»'B“~ Mixed second partials are equal

has constant length

[D] Both T[t) and N (2%) lie in the osculating plane at If [E] r'(t) traces out a circle

T

I

-I---__-'1

1

1: 1 ,l

i

'| 1. 5|

(9)

t, where n is a positive integer and f has continuous second-order partials. Show that if f is homogeneous of degree rt, then

tr

r.-M‘ V _,__,____,_ I; - ,

111% +1,/~25 =ﬂf(w,s)

Hint: Set 'u. : tzr, '0 = ty, and use the Chain Rule to differentiate both sides of the equation ﬂu, 1)) == 15"’ f (rrz, jg) tvith respect to t. Then set t : 1.

|' J

‘H/Am fl

yﬁln

,,

Q1;

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9’

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needed

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l

our S6. '

’}Vi'l '/1"

< 1, 1, 1 >. Find the maximum possible rate of change at the given point, and the direction in which it occurs.

--~' -‘>4 {Iii '

V F '-

1" ¢ ——--*

as

‘J-”~'~”Z1‘-1"":

»~ *-#1

(A912) J‘1/><1+ji*?l J W11?" /

é -2.

VF-(?)fé_!~"2) -3 fé) I

=‘

)

I11

ova} »m1’=><,\/1’!

913 QQ

QQ

QQ?

1’

‘I

' 1

tel/lfcl/1

W5-5

1*? as 5.. 55; I 5'

18.(6pts) Find the directional derivative git F(a:,y,z) = 1/$2 +y2 +z2 at the point (3,6,—2) in the direction

><M

4|--Qgb, ixﬁ“‘~‘-‘=1-._.'.‘?-H, ":!T_H-.--"

_-I-"'

iii?/?"£”5.J»,"tZ)Jl 1'-"

"-11

lJZ.l1'rl>ll1‘ F? ,

£0 lmléé

51 4,23‘, ,)"‘

71.6.».

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“lg '1 F1 l hr 2 3,,

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am 0 2' 3 2 1|

(10)

.:_- --i — —————-— ———a—— -—--—- |: |§;

1|--I

J

19. (Gpts) (No work required) Qonsider the function f = cos ac, O < m < Sketch: » (i) the '2r—periodic even extension of f

(ii) the rr-periodic odd extension of f

‘FT

(iii) tl1e

Make s

Ln

_.——

Cii\ 1 _

L~;a\

,,

,

..

~,.

5 -periodic extension of f

ure to clearly indicate convergence at points of discontinuity.

. ,-1

1 on

L®,j;l,

YB)

10014;. lilif

I 5

____,_._--— _ ..-,— ,»’

J-l /v

1 " \

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