k, - uw-madison department of mathematicsterwilli/htmlfiles/hw8-3.pdfno. g" 3 date 0 ans...

26
No. J?,.3 Date / A VlS (0,)'( (10) N frob veclvr: 5 (el N )vn-t I-v L (g(l '( ( "3 x -= I -, Z -, 5" -::: 3 I .!.t I +,2 (c) 'Vq -=-,2'i 1X==-/ ( J) K, X 1- t- :; / x II -==/ 3 \ 01. C1 ::= b C -::;- ;) C\..- )0 u-l- Z-Ct -==/ 1c:z=:::-/ C( b =- -J; G -:::' Yz- x - h c d ] a. =/ e<.-= 2. c ol -=,Lj 10 ;. C f- o -; / )0 z. c.. .f- .tt ,?C =",C:, C. ..,,;. # L C{==-,9

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Page 1: K, - UW-Madison Department of Mathematicsterwilli/Htmlfiles/hw8-3.pdfNo. g" 3 Date 0 Ans ReC\50n \0 p::::: 2 I [; -'3 J[~ +J 5 --, , 2.-r 3-jj. l 2.5 2.5 \ f~ = --L 90 150 + Z t5 ]

No. J?,.3 Date /

A VlS

(0,)'(

(10) N frob veclvr: e."I-n~ 5 /'V'?Tn~~ (el N ~ )vn-t I-v L (g(l '(

( ~) "3 x -= I - , Z -, 5" -::: 3I

(~) .!.t I ~2')< +,2

(c) 'Vq -=-,2'i 1X==-/

( J) K, X (1k~ 1- ~ t- 2-+-~) :; /

x II -==/

3 \

01. 0.j-c~/

C1 ::= b C -::;- ;) C\..­

)0 u-l- ~t- Z-Ct -==/ 1c:z=:::-/

C( ~~ b =- -J; G -:::' Yz­

x - [~ h c d ]

a. +-b-i-C+~ =/

e<.-= 2. c

ol ~O

-=,Lj10 ;. C f- o -; /)0 z. c.. .f- .tt

,?C =",C:,

C. ..,,;. # L C{==-,9

Page 2: K, - UW-Madison Department of Mathematicsterwilli/Htmlfiles/hw8-3.pdfNo. g" 3 Date 0 Ans ReC\50n \0 p::::: 2 I [; -'3 J[~ +J 5 --, , 2.-r 3-jj. l 2.5 2.5 \ f~ = --L 90 150 + Z t5 ]

----------No.

Date

(J11~vn

~ ~) ]p -I

, 5 5

H L.

a. +-' ~ .::::= I

a. =2-b

6\ -=:; /) b -;;: 'IJ

\

J.. 100p~- [; ;][; : J

i 100~: ::]

- j~ 1 J, Cooe~<po 40

'2.. 2- ,'; ~"0= - Lc;>o

+ l. "3 looH-~""'1 '" X P 3

11f?-- -)0 0

r-i-~ 1- IpJ

[; ;][;: ;;]~, «: )

Page 3: K, - UW-Madison Department of Mathematicsterwilli/Htmlfiles/hw8-3.pdfNo. g" 3 Date 0 Ans ReC\50n \0 p::::: 2 I [; -'3 J[~ +J 5 --, , 2.-r 3-jj. l 2.5 2.5 \ f~ = --L 90 150 + Z t5 ]

No.

Date "3

(Jhl~V)1

51~ 1"86 27 2 + /2.0f'3 -=: fq~ .~ --L­

(000

+ 1 () 0 I gO + 200~20 J [ ~zg 3-12

-:::; J --.L (000

3 ezo"20

-'­"3 'O()O

If7-(,. 1. 000

FlY,). k 5v,A ~G (S J L<O\st , b

: ]

Page 4: K, - UW-Madison Department of Mathematicsterwilli/Htmlfiles/hw8-3.pdfNo. g" 3 Date 0 Ans ReC\50n \0 p::::: 2 I [; -'3 J[~ +J 5 --, , 2.-r 3-jj. l 2.5 2.5 \ f~ = --L 90 150 + Z t5 ]

1

No.

Date

h w

p= H['~ Y

W J

p'3-=::- S-­;2"1­z 'I[~ J[ /] L

- I~- [h ] i713 I~

M-~ 1 If'? IS

- '3 1'1

Lot ;;11.. +

J~ , q

Page 5: K, - UW-Madison Department of Mathematicsterwilli/Htmlfiles/hw8-3.pdfNo. g" 3 Date 0 Ans ReC\50n \0 p::::: 2 I [; -'3 J[~ +J 5 --, , 2.-r 3-jj. l 2.5 2.5 \ f~ = --L 90 150 + Z t5 ]

--

No.

Date 5

Re.&t.>Oh

x ~ [, I ~ '1-L

- I - t ­(0 ~ lC> 1

::; II)0

[+ +J 0.. 4-- 6 = I

WP=W o ::: IN ( P- I)

[c> ] = [- ~ 10 J r,3 ,3]0 L,S -,5

o -= -,3 C<.- t-, 5 j.

"5 A. -= S(f-4.)

84-==5

t\. -== 5h " -:;. 1?

Page 6: K, - UW-Madison Department of Mathematicsterwilli/Htmlfiles/hw8-3.pdfNo. g" 3 Date 0 Ans ReC\50n \0 p::::: 2 I [; -'3 J[~ +J 5 --, , 2.-r 3-jj. l 2.5 2.5 \ f~ = --L 90 150 + Z t5 ]

No. g" 3 Date 0

Ans ReC\50n

\0

2 I p::::: '3

-[; J[~ +J5

- 2. 3- , -r , jj.- l2.5 2.5

\ f~ =

- 90 150 + Z t5 ] --LL- lOb t (,zr

~O + 1/'1' 90 +- "3 '"

I '10 q 11) -- -'­[ ] r,2,r

11'~ q )/

f'i= Yf)][ IfD 0/3,] -'­[I~D f,ZS'z

I~'t tlfl 1":/-'1 95"/

~c.'OD -+- :rHtto f2C.'Ot /c,fs,/15- ::: Z"1t g1S­::::: Itl rj. 0 ---J-.,-z. ~'"\..V(¥<

Page 7: K, - UW-Madison Department of Mathematicsterwilli/Htmlfiles/hw8-3.pdfNo. g" 3 Date 0 Ans ReC\50n \0 p::::: 2 I [; -'3 J[~ +J 5 --, , 2.-r 3-jj. l 2.5 2.5 \ f~ = --L 90 150 + Z t5 ]

No.

Date

,~

PI Jbl~

[-f +] +]J

w =vvP

0= W(p-~)

'\ !

- 1. C\... 2.

[+ ~J[;

Page 8: K, - UW-Madison Department of Mathematicsterwilli/Htmlfiles/hw8-3.pdfNo. g" 3 Date 0 Ans ReC\50n \0 p::::: 2 I [; -'3 J[~ +J 5 --, , 2.-r 3-jj. l 2.5 2.5 \ f~ = --L 90 150 + Z t5 ]

1\10.

Date

I~

t\ (l 5

"ih.i.!.e v,

o ==

W=:;Lc~ {,

~04-c=

w(p-rJ

c

/

]

Re.q5crn

[0 o 1== [ ~ b c J -/

0

I

,5

-/

0 ~J 0

CJ

--

-

- b I­

a-C.

C

( re-f)

0 --

Page 9: K, - UW-Madison Department of Mathematicsterwilli/Htmlfiles/hw8-3.pdfNo. g" 3 Date 0 Ans ReC\50n \0 p::::: 2 I [; -'3 J[~ +J 5 --, , 2.-r 3-jj. l 2.5 2.5 \ f~ = --L 90 150 + Z t5 ]

I

No.

Date

[+ +J [~ .." ][::J

\

J

Page 10: K, - UW-Madison Department of Mathematicsterwilli/Htmlfiles/hw8-3.pdfNo. g" 3 Date 0 Ans ReC\50n \0 p::::: 2 I [; -'3 J[~ +J 5 --, , 2.-r 3-jj. l 2.5 2.5 \ f~ = --L 90 150 + Z t5 ]

No. ~;:3

Date to

/-'\

f.-..,bleln AnS

l'f

[-f

[. ~ c -/ / 0[0 0 o ] - b ] -I .LD -2­ "l..

I -I0 ~ J

0 - -t\...+- -'"3 c=3A..

10 -::: -z..c...:=: ..L60 a..- -z.,

_ c.. ::: I c;:;: .1­a.. b -- ~7: c.

Page 11: K, - UW-Madison Department of Mathematicsterwilli/Htmlfiles/hw8-3.pdfNo. g" 3 Date 0 Ans ReC\50n \0 p::::: 2 I [; -'3 J[~ +J 5 --, , 2.-r 3-jj. l 2.5 2.5 \ f~ = --L 90 150 + Z t5 ]

No.

Date tI

AnS

pic. LOiA./e-..- {Y 14A 7.A #t'\5 fr. ~ k.

0:::: w(p-z)

[OC'Oc>]=[~1o c otJ -I ;

() -, ,

0= -CL +-,1 d- a. - "Id.

o - O­ -I:, +­ , I ci.

o :::::.. b -c.

o :=: c - ,z.d2.. c= .. 2£'

(,,/ f­ ,,'Lt-,L +-1!c1=/

'] 01 -:::./ &>I:;":=:" z:­ J

J,-== 3­30 C ­- .!L­?a

z.o 50

Page 12: K, - UW-Madison Department of Mathematicsterwilli/Htmlfiles/hw8-3.pdfNo. g" 3 Date 0 Ans ReC\50n \0 p::::: 2 I [; -'3 J[~ +J 5 --, , 2.-r 3-jj. l 2.5 2.5 \ f~ = --L 90 150 + Z t5 ]

-- -

f,jNo.

Date 1:2

(]biew. AnS R<'e{5 or/

«e.a-Lll-4- rII ~~

tt /0 ~5

[ L~ 2, :, J ,5"Z~

< ~ ~c.

W=LCl-b c~]

[0 0 0] = rCc- ~ c ~ J0 -, ~ /~ 0 0 o -,,5 .... S 0

o 0 -,$',';'

\ ,!i " ~ S"" -/

... ~ a.= 15'A. 9~ = s! :::- ,~.b ~ t:i -:;: 7£•'I fA.

'Ie ::::; IOJ. , 'f c =- cJ.

b ­

( )" +- .Lt + I" + ..!t.. ) of = IT -, ~ y

$' 10c -Z) 2..3

~~

Page 13: K, - UW-Madison Department of Mathematicsterwilli/Htmlfiles/hw8-3.pdfNo. g" 3 Date 0 Ans ReC\50n \0 p::::: 2 I [; -'3 J[~ +J 5 --, , 2.-r 3-jj. l 2.5 2.5 \ f~ = --L 90 150 + Z t5 ]

K,3No.

Date (3

Q. M+-c+ol=1

W -:::: vvp J- .J- 0P -T = -I ... z.

0 J- j..-/ "Z.

-, -z..

j.. .J­0 '2. "Z.

0 0 -/

a :. tJ.

J... '\.. -b +- .L c -::. a '" ""L

-fc;..+~~-C=O

~b +-{:c':.JL

I -V

Page 14: K, - UW-Madison Department of Mathematicsterwilli/Htmlfiles/hw8-3.pdfNo. g" 3 Date 0 Ans ReC\50n \0 p::::: 2 I [; -'3 J[~ +J 5 --, , 2.-r 3-jj. l 2.5 2.5 \ f~ = --L 90 150 + Z t5 ]

19

No.

Date

[

,I)

,. ,

P:J.I<rY'\ Al\s

~h,!.t Hit MIS'S &J5

r"'(J~ .!t- I p= [5 5 , " H

#'~4.. =-,~b

tc;c... '= ~Ob

tlt., <t b \

/,::::: S ~ 'i.= f

G- !J&[, ,r]P () , g ,Z

P-X= [-'~ ''1], '/ -,2

,'-'&i - IV/'

qq = 1>(,

tL ~ ~j,

6~..L ]

Page 15: K, - UW-Madison Department of Mathematicsterwilli/Htmlfiles/hw8-3.pdfNo. g" 3 Date 0 Ans ReC\50n \0 p::::: 2 I [; -'3 J[~ +J 5 --, , 2.-r 3-jj. l 2.5 2.5 \ f~ = --L 90 150 + Z t5 ]

No.

Date

Ans

'2-1

\

.. "1-&i.-==- ,C;'

::r~ ~ {,j,

-1- C(. -==' " { /-4-1

,"$ &t. -z:=­ ~

"Z'l. ( q. ) R~"IIt....1' '1.

p==

;;:

(k Q NK~O~) ~ >.t )f ~ R >e

~ '* o ~)Y 0

( ~ ~ W)Jo )e 1lI

» ~ T(

(b) ~Qt ~ 2.­

(;P ~

(:::.

0 )(t

0

)c

)0

)c

0

~

0

K: ;)

0'

»c

o

kJ

~

JIi1

)

Page 16: K, - UW-Madison Department of Mathematicsterwilli/Htmlfiles/hw8-3.pdfNo. g" 3 Date 0 Ans ReC\50n \0 p::::: 2 I [; -'3 J[~ +J 5 --, , 2.-r 3-jj. l 2.5 2.5 \ f~ = --L 90 150 + Z t5 ]

No.

Date

AnS

(c)

Page 17: K, - UW-Madison Department of Mathematicsterwilli/Htmlfiles/hw8-3.pdfNo. g" 3 Date 0 Ans ReC\50n \0 p::::: 2 I [; -'3 J[~ +J 5 --, , 2.-r 3-jj. l 2.5 2.5 \ f~ = --L 90 150 + Z t5 ]

No.

Date '1"

2.l.t (,TfII. f~,,"tn f' l~ If ~

v .....NI rrtlh j.. t~

5/,

..L -z. 3 1f= j. J­

'3J

~ 0"3 z, z.. -, '1

_J.. .:z-P-I ::: ,J -' _.. --J

• '3

-)

( - cD ~ -,... -,

~dh W-;;-["t..6 CdJ

li rl f- c t-d ::::-/

_2- +-- .lJ:, ~ J-c.~t:;... ") , ~ ..j-­

- 'lcz - ~" "2.­

~ ---c'} et. J

I I I / -6 3 3 2­ -r;, 0

"Z­ o -(,

1­- t.-~ <j

0 j.. 1

..l­""? "1­

1

Z.-4j

0 _z­

'3

, ~

..l­1 ..i­"1

1­-"I I

J

.-l­:J

_L.

l

+--~ of -;::; 0

1:::- d -e­ ~ ~

+'~c{if

== 0

/ o o o

Page 18: K, - UW-Madison Department of Mathematicsterwilli/Htmlfiles/hw8-3.pdfNo. g" 3 Date 0 Ans ReC\50n \0 p::::: 2 I [; -'3 J[~ +J 5 --, , 2.-r 3-jj. l 2.5 2.5 \ f~ = --L 90 150 + Z t5 ]

~r31\10.

Date If

()blem A-Y\-s Re.q, $" C/'>1

2i1Jc~

j

Irz. '" ('1. + r;, (",?:>0 9 'I g

-2­ r;'-:. t'l-z..r,0 -g -L 0

-2­ ,.::I f« -2.',0 -2. ~-g 0

-I r: ~~ ~ '7'

,

b

rz C:-? I<t0 I 1 0 I 0 -8 -2- 0

0 't 9 f

, j rj

I ::::: r; -Iz.

0 / Lf a 0 -3 I 0

/ I6 i7 =- ~ -t-Pr;.0 0 30 0

0 0 -;)..1- g I'll ::::: f", - '1 rz.-3

f 0 -5 I 0

Lj 0 I0 .J­1. fi I

-= fJ ~'"5

0 0 -,21 g -3 a 0 I 0

/ 5 I

/ 0 0 I 1- r; -= /jr-1fJ

0 0 0 0 (2'= r-z. - ltr:jI

0 0 I 0

il- I0 0 0 f - r; -::: ~ + J.rrj

5 ,d:::: -l- c~ b-::: tiz:. ..l­

totD ~ 5

Page 19: K, - UW-Madison Department of Mathematicsterwilli/Htmlfiles/hw8-3.pdfNo. g" 3 Date 0 Ans ReC\50n \0 p::::: 2 I [; -'3 J[~ +J 5 --, , 2.-r 3-jj. l 2.5 2.5 \ f~ = --L 90 150 + Z t5 ]

g~] No.

Date 1.-'1

flbielYl A-ns 12~')(/}'t

'2-'f I tNt;;.

, 2- -z.S 1­

0 '3[1 3J '3 , ..1-to e,

')') 0 '$ "2- J'Z­2­

[-s 1..- ) J

P,L.S \

I

2. ~ r. t 'i ~.-

I -=t" g [0'"

roS~ ~

Page 20: K, - UW-Madison Department of Mathematicsterwilli/Htmlfiles/hw8-3.pdfNo. g" 3 Date 0 Ans ReC\50n \0 p::::: 2 I [; -'3 J[~ +J 5 --, , 2.-r 3-jj. l 2.5 2.5 \ f~ = --L 90 150 + Z t5 ]

20

No.

Date

.., (, '1-­t I ~. (,_., '-, I ~ I Cj _ •.

Page 21: K, - UW-Madison Department of Mathematicsterwilli/Htmlfiles/hw8-3.pdfNo. g" 3 Date 0 Ans ReC\50n \0 p::::: 2 I [; -'3 J[~ +J 5 --, , 2.-r 3-jj. l 2.5 2.5 \ f~ = --L 90 150 + Z t5 ]

Date 2-1

,AnS

(b) ~~4

eel GP 5tJh " "} vrU.

z.. 'fsJ.J:e., ~u.Le

fE4--cA /y-t:t,,,~, n~ ~....,~ ~t<-n 'd ef II..e ~ t-tit

)

I), " 1,1t, .. _

Page 22: K, - UW-Madison Department of Mathematicsterwilli/Htmlfiles/hw8-3.pdfNo. g" 3 Date 0 Ans ReC\50n \0 p::::: 2 I [; -'3 J[~ +J 5 --, , 2.-r 3-jj. l 2.5 2.5 \ f~ = --L 90 150 + Z t5 ]

No. F, '3 Date zz.

2.4

An5

[-a... J t-6

-

[ ,- et. ]

,0 [-+ ~J I,.. 1i' LCf T~ Yl= '(y b ;::; YJ

b -l I

= ..L - '}/y1 t + 31..f'1,-~

." + I -$

..!! 'I 9- - -.... ." ",-1 =1­41­ 1;'y

,­ ~-­1....-6-a.

.....-

I

--

I

Page 23: K, - UW-Madison Department of Mathematicsterwilli/Htmlfiles/hw8-3.pdfNo. g" 3 Date 0 Ans ReC\50n \0 p::::: 2 I [; -'3 J[~ +J 5 --, , 2.-r 3-jj. l 2.5 2.5 \ f~ = --L 90 150 + Z t5 ]

No.

Date Z3

, -J -~

p- -'­--6 J

o -;: W ( P-.z)

-)( .J- 6'7 z:. 0

.XI ~ t.? '1

XI -::: b (1->-.)

U4- b ) ~ ?t

Page 24: K, - UW-Madison Department of Mathematicsterwilli/Htmlfiles/hw8-3.pdfNo. g" 3 Date 0 Ans ReC\50n \0 p::::: 2 I [; -'3 J[~ +J 5 --, , 2.-r 3-jj. l 2.5 2.5 \ f~ = --L 90 150 + Z t5 ]

1\10.

Date

P-J- ~ /

b - ... -6

w;;- [)( J'1:C

w -::; Pw

-XT-~o-:;:O

X - '1 f'b c :;: 0

1 -{&t-H}l;=O

I

~J

Page 25: K, - UW-Madison Department of Mathematicsterwilli/Htmlfiles/hw8-3.pdfNo. g" 3 Date 0 Ans ReC\50n \0 p::::: 2 I [; -'3 J[~ +J 5 --, , 2.-r 3-jj. l 2.5 2.5 \ f~ = --L 90 150 + Z t5 ]

l\lo.

Date z.s

/\.,~ , )b

'to = [, 't , S­

[ , 7 ; S- , / J[ 0 , g , ZJ -= [5 ,yz ,.ili] ,. 2 0, i o 0{

, j .1.] :::: [/116 .- '13'1 ] ,'1... 0

o 0

Page 26: K, - UW-Madison Department of Mathematicsterwilli/Htmlfiles/hw8-3.pdfNo. g" 3 Date 0 Ans ReC\50n \0 p::::: 2 I [; -'3 J[~ +J 5 --, , 2.-r 3-jj. l 2.5 2.5 \ f~ = --L 90 150 + Z t5 ]

1'10.

Date

Rett5n,

HfL f 1'r;T b j.<; e..-/rp>o

P ::: u[ ,:'S(PIJl M /3

::: r ,2)]f-I ,zS­

-,JI J

, 2.> 4. ::: ,"1 b

'2.-) 'i ~Ja b

') ~ -:::- (, J,

) c;c. ~ " (L -r.,. )

Hct.-:::-- C