8/13/2019 AMAT223 Final Exam Revision Exercises

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FINAL EXAM REVISION EXERCISES

QUESTION 1

(a) Consider the points )3,1,2( A , )2,1,(B , and )1,3,3( C ! Let ABu = and !ACv =

(i) Find the "e#tors u and v , the $en%th && u and the 'nit "e#tor !u

(ii) Find the dot prod'#t vu , and the #ross prod'#t !vu (iii) Find the e'ation o* the p$ane #ontainin% the points ,A ,B and !C

(+) Find the e'ation o* the p$ane hi#h #ontains the point )2,1,( and is para$$e$ to the p$ane !1-.32 =+ zyx

(#) Let !)$n1

())2(#os()(2

.. jtt

eitttr t ++++=

Finddt

rd!

(d) Consider the "e#tor *ie$d !)1/()20(),,( .3.232 kyzzxxjxzyiyzxzyxF ++++=

Find the di"er%en#e F and the #'r$ !F

QUESTION 2

(a) Consider the rea$"a$'ed *'n#tion !2),,( 323 xyzxyzzyxf ++= (i) Find ,xf ,yf ,

z

f and the %radient "e#tor !f

(ii) Consider the s'r*a#e ith e'ation ,1-),,( =zyxf

i!e! ith e'ation 1-2 323 =++ xyzxyz

Find the e'ation o* the tan%ent p$ane to this s'r*a#e at the point )!1,2,/( P

(+) Consider the e'ation !1.23=+++ zxyzxy Find

y

z

+3 i4p$i#it di**erentiation!

(#) Consider the *'n#tion !0

3),(2

22 xyxyyxf =

(i) Find ,xf ,yf ,xxf ,yyf ,xyf and sho that !.)3(3

2),( 2yxyx =

(ii) Find and #$assi*3 the #riti#a$ point(s) o* the *'n#tion !f

(d) 5se the Method o* La%ran%e M'$tip$iers to *ind the 4a6i4'4 and7or 4ini4'4 "a$'e(s) o* the

*'n#tion 223),( yxyxyxf ++= s'+8e#t to the #onstraint !.22 =+ yx QUESTION 3

(a) Consider the s'r*a#e ith e'ation !322 =+ xyz Find and na4e its tra#e in the *o$$oin%

p$anes9

(i) !3=z A$so, s:et#h the "erti#a$ pro8e#tion o* this tra#e on the xy p$ane!(ii) !/=y A$so, s:et#h the "erti#a$ pro8e#tion o* this tra#e on the xz p$ane!

(iii) 3=

x A$so, s:et#h the "erti#a$ pro8e#tion o* this tra#e on theyz

p$ane!(+) Consider the s'r*a#e ith e'ation !222=+ yx

(i) Na4e the s'r*a#e!

Find and na4e its tra#e in the *o$$oin% p$anes9

(ii) !=x A$so, s:et#h the "erti#a$ pro8e#tion o* this tra#e on the yz p$ane!(iii) -=z A$so, s:et#h the "erti#a$ pro8e#tion o* this tra#e on the xy p$ane!(i") !0=y A$so, s:et#h the "erti#a$ pro8e#tion o* this tra#e on the xz p$ane!

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8/13/2019 AMAT223 Final Exam Revision Exercises

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QUESTION 4(a) Rerite the do'+$e inte%ra$ ith the order o* inte%ration re"ersed!

.

2

2

/

),(y

dydxyxf

(+) Consider the do'+$e inte%ra$ 22

/

/

!

y

dydxx

(i) S:et#h the re%ion o* inte%ration!

(ii) Rerite the do'+$e inte%ra$ in po$ar #oordinates and e"a$'ate it!

(#) Rerite, +'t do not e"a$'ate, the trip$e inte%ra$ in #3$indri#a$ #oordinates!

-

/

3

/

en#e, *ind a rea$"a$'ed *'n#tion ),( yxf s'#h that !fF =

(iii) >en#e, e"a$'ate the $ine inte%ra$ +2

,).2( .23

C

dyyxdxxyx

here 2C is the #'r"e 23+= xy *ro4 the point )2,/( to the point )!3,1(

;a%e 2 o* 2