revision_final exam math 4

8/12/2019 Revision_Final Exam Math 4

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Semester I 2012/2013 Kohort 1

UNIVERSITI TUN HUSSEIN ONN MALAYSIA

BWM 21403 MATEMATIK IV

REVISION O! !INAL E"AM

Q1 #$% Sho& th$t x x x y 'os4s()2%# += (s $ *e)er$+ so+,t(o) o- the .(--ere)t($+

e,$t(o) ydx

yd

2

2

−=

#4 m$rs%

#% So+5e the -o++o&()* e,$t(o) ,s()* the se6$r$+e metho. $). *(5e 7o,r

$)s&er () the e86o)e)t -orm

32 x xdx

dy y x =−

#9 m$rs%

#9 m$rs%

#'% :eterm()e (- the -o++o&()* +()e$r e,$t(o)s $re homo*e)eo,s;

#(% x y y 309<9 =−−′′

#((%dx

dy y

dx

yd 4=

2

2

=+

#2 m$rs%

#.% !(). the *e)er$+ so+,t(o) -or the -o++o&()* homo*e)eo,s e,$t(o)

#(% 04 =−′−′′ y y y

#((% 0>2

2

=+dx

yd

#? m$rs%




Q2 !(). the root#s% o- -,)'t(o)3 2# % 1= 0?4 010? f x x x x= − + − th$t +(es () the *(5e)

()ter5$+ 0 0@ a b 7 ,s()* B(se't(o) metho. Iter$te ,)t(+ # % 000

i f c ε < = ;

#$% 0 0@ @ 002a b = − 801?

#% 0 0@ @020=a b = 809#'% 0 0

@ @0?14a b = 80?99

Q3 !(). the root#s% o- -,)'t(o)3 2# % 3 32? 0>24 f x x x x= − + − th$t +(es () the *(5e)

()ter5$+ 0 0@ a b 7 ,s()* B(se't(o) metho. Iter$te ,)t(+ # % 000

i f c ε < = ;

#$% 0 0@ @009a b = 809

#% 0 0@ @0991a b = 80=

#'% 0 0@ @12a b = 822

Q4 !(). the root#s% o- -,)'t(o)3# % 2 =4 0? f x x x= − + th$t +(es () the *(5e) ()ter5$+

0 0@ a b 7 ,s()* B(se't(o) metho. Iter$te ,)t(+ # % 000

i f c ε < = ;

#$% 0 0@ @009a b = 8010?

#% 0 0@ @12a b = 81?9=

#'% 0 0@ @ 3 1a b = − − 8 C 1>=

Q5 L$&)'o 6ro.,'es three *r$.es o- 'ommer'($+ -ert(+(Ders r$.e A r$.e B $).

r$.e F E$'h *r$.e o- -ert(+(Der 'o)t$()s three .(--ere)t ),tr(e)ts N(tro*e)

Ghos6h$te $). Got$ss(,m #$s sho&) () the $''om6$)7()* t$+e e+o&%

!ert(+(Ders/),tr(e)ts N(tro*e) #*% Ghos6h$te #*% Got$ss(,m #*%

r$.e A 1? 4

r$.e B 20 4 4

r$.e F 24 3 9

The ,$+(t7 s6e'(-('$t(o) re,(res 29400 * o- N(tro*e) 4>00 * o- Ghos6h$te $).

9200 * o- Got$ss(,m () the -ert(+(Ders Th(s m$),-$'t,r()* 6ro+em '$) e

re6rese)te. () the -o++o&()* s7stem o- +()e$r e,$t(o)s




1 2 3

1 2 3

1 2 3

1? 20 24 29400

4 4 3 4>00

4 9 9200

x x x

x x x

x x x

+ + =

+ + =

+ + =

&here 1 x 2 x $). 3 x $s the ),mer o- $*s o- r$.e A -ert(+(Der r$.e B -ert(+(Der$). r$.e F -ert(+(Der '$) e m$.e res6e't(5e+7

#$% !orm Ax b= $se. o) the $o5e 6ro+em

#% He)'e -(). ho& m$)7 $*s o- e$'h t76e o- -ert(+(Der '$) e 6ro.,'e. (- $++ the

),tr(e)ts $re ,se. 7 ,s()* Fro,t -$'tor(D$t(o) metho.

Q6 S,66ose th$t $ te$m o- three ,m6ers $re 'o))e'te. 7 ,)*ee 'or.s The7 $re e()*

he+. () 6+$'e 5ert('$++7 so th$t e$'h 'or. (s -,++7 e8te).e. ,t ,)stret'he. A-ter the7$re re+e$se. *r$5(t7 t$es ho+. $). the ,m6ers &(++ e5e)t,$++7 'ome to the

e,(+(r(,m 6os(t(o)s B7 the $ss,m6t(o) th$t e$'h 'or. eh$5es $s $ +()e$r s6r()* $).

-o++o&()* Hooes +$& th(s -reeCo.7 6ro+em '$) e -orm,+$te. $s

1 2 1 2 2 1

2 1 2 3 2 3 3 2

3 2 3 3 3

# %

# %

k k x k x m g

k x k k x k x m g

k x k x m g

+ − =

− + + − =

− + =

&here m = the m$ss o- ,m6er #*% k = the s6r()* 'o)st$)t -or 'or. #N/m% x = the

.(s6+$'eme)t o- ,m6er th$t me$s,re. .o&)&$r. -rom the e,(+(r(,m 6os(t(o) #m%

$). g = *r$5(t$t(o)$+ $''e+er$t(o) # 2m/s % (5e) th$t the 6$r$meter 5$+,es $re

190m = 2

=0m = 3?0m =

1 2 30 100 0k k k = = = $). >?1 g = 2m/s

#$% !orm Ax b= $se. o) the $o5e 6ro+em 7 s,st(t,t()* the *(5e) 6$r$meter

5$+,es

#% He)'e .eterm()e the three ,))o&) .(s6+$'eme)ts 7 ,s()* Fro,t metho.

Q7 A s6r()* m$ss s7stem 'o)s(sts o- three m$sses 1 2m = * 2 3m = * $). 3 2m = *

&h('h $re 'o))e'te. 7 s6r()*s &here s6r()* 'o)st$)ts $re < 10k s = N/m A-ter the

m$sses $re 6,++e. .o&)&$r. 7 the -or'e o- *r$5(t7 e$'h s6r()* &(++ res,+t to

.(--ere)t .(s6+$'eme)ts #m% Th(s -or'eC$+$)'e e,$t(o)s '$) e re6rese)te. $s

1 2 1

1 2 3 2

2 3 3

3 2

2 3

kx kx m g

kx kx kx m g

kx kx m g

− =

− + − =

− + =

&here *r$5(t$t(o)$+ $''e+er$t(o) >?1 g = 2

m/s

#$% !orm Ax b= $se. o) the $o5e 6ro+em 7 s,st(t,t()* the *(5e) 6$r$meter

5$+,es




#% He)'e .eterm()e the three ,))o&) .(s6+$'eme)ts 7 ,s()* Fro,t metho.

Q8 The $r' +e)*th o- the ',r5e %# x f y = o5er the ()ter5$+ b xa ≤≤ (s *(5e) 7

Ar' +e)*th ∫ +

−b

adx

x

x 2

2

1

1

#$% A66ro8(m$te the $r' +e)*th o- ',r5e %# x f () the ()ter5$+ @0 1 7

,s()* the tr$6eDo(.$+ r,+e &(th $ ste6 s(De o- h 01

#10 m$rs%

#% A66ro8(m$te the $r' +e)*th o- ',r5e %# x f () the ()ter5$+ @0 1 7

,s()* the tr$6eDo(.$+ r,+e &(th $ ste6 s(De o- h 02

# m$rs%

#'% Re-er to the Table Q8 e+o& -(). the 5$+,e o- A $). B (- the e8$'t

so+,t(o) (s C0=1 Wh$t (s 7o,r 'o)'+,s(o)J

Table Q8 : Different step size and absolute error for Trapezoidal Rule

Numerial !et"od #bsolute $rror

Tr$6eDo(.$+ r,+e #h 01% A

Tr$6eDo(.$+ r,+e #h 02?% B

# m$rs%

Q% #$% (5e) th$t




>?4$).11330?920 433=== eee

Use Ne&to) .(5(.e. .(--ere)'e metho. to est(m$te the 5$+,e o- 13e !().

the

$so+,te error -or th(s $66ro8(m$t(o)

#10 m$rs%

#% !(t $ Ne&to)s .(5(.e. .(--ere)'e ()ter6o+$t()* 6o+7)om($+ to est(m$te +o*

12 ,s()* the .$t$ e+o&

+o* 1 0 +o* 14 0149 $). +o* 1? 02

!(). the $so+,te error -or th(s $66ro8(m$t(o) &here the e8$'t 5$+,e '$) e-o,). -rom 7o,r '$+',+$tor

#10 m$rs%

Q1& So+5e the -o++o&()* ()(t($+ 5$+,e 6ro+em #IVG% 7 ,s()* 1st Or.er T$7+or ser(es

#E,+ers metho.%;

#$% 2#1 % dy x xydx

+ = &(th #2% 1 y = o5er ()ter5$+ 2 x≤ ≤ $). 03h =

#%2 2# %

dy x xy y xy

dx− = − &(th #2% 3 y = o5er ()ter5$+ 2 3 x≤ ≤ $). 02h =

#'%2 2# %

dy x y xy

dx+ = &(th #1% 0?4 y = o5er ()ter5$+ 1 1 x≤ ≤ $). 00h =

#.% # 1% dy

x xydx

+ + = &(th #0% 2 y = o5er ()ter5$+ 0 1 x≤ ≤ $). 01h =

#'' T($ )$*T+++

#ns,ers:




N-T$: #ns,ers for Q2.Q7 are in appro/imate 0alues adala" dalam nilai anaran

sa"aa #nda bole" mendapatan a,apan sama"ampir sama denanna

Q1 a sho&)

b A x x y ++= 232 323

(c) (i) Nonhomogen

(ii) Homogen

(d)

(i) 01<4M =−− y y y

04 ;e/,$t(o)st('Fh$r$'ter( 2 =−− mm

(m + 1)(m - 5)=0

m= -1 @ 5

;so+,t(o)*e)er$+ He)'e x x Be Ae y += −

(ii) 0>2

2

=+ ydx

yd

(iii) > ;e/,$t(o)st('Fh$r$'ter( 2 −=m 3 im ±=

x B x A y 3s()3'os ;so+,t(o)*e)er$+ He)'e +=

Q2 #$% x 01?

#% x 09

#'% x 0?99

Q3 #$% x 09

#% x 0=

#'% x 22




Q4 #$% x 010?

#% x 1?9=

#'% x 1>=

Q5 #$% !orm o- Ax b= (s;

1

2

3

1? 20 24 29400

4 4 3 4>00

4 9 9200

x

x

x

÷ ÷ ÷= ÷ ÷ ÷ ÷ ÷ ÷

#% 1 2400 900 x x= = $). 3

300 x =

There-ore L$&)'o &(++ 6ro.,'e 400 $*s o- r$.e A -ert(+(Der 900 $*s o-

r$.e B -ert(+(Der $). 300 $*s o- r$.e F -ert(+(Der

Q6 #$% S,st(t,t()* $++ 6$r$meter 5$+,es;

1 2

1 2 3

2 3

#0 100% 100 90#>?1%

100 #100 0% 0 =0#>?1%

0 0 ?0#>?1%

x x

x x x

x x

+ − =

− + + − = ⇒

− + =

1 2

1 2 3

2 3

10 100 ??9

100 10 0 9?9=

0 0 =?4?

x x

x x x

x x

− =

− + − =

− + =

!orm o- Ax b= (s;

1

2

3

10 100 0 ??9

100 10 0 9?9=

0 0 0 =?4?

x

x

x

− ÷ ÷ ÷− − = ÷ ÷ ÷ ÷ ÷ ÷−

#% 1 241202 >1= x x= = $). 3

=1913 x =

There-ore the ,))o&) .(s6+$'eme)ts $re 141202 x = m 2

>1= x = m $).

3=1913 x = m

Q7 #$% S,st(t,t()* $++ 6$r$meter 5$+,es;

1 2

1 2 3

2 3

3#10% 2#10% 2#>?1%

2#10% 3#10% 10 3#>?1%

10 10 2#>?1%

x x

x x x

x x

− =− + − = ⇒

− + =

1 2

1 2 3

2 3

30 20 1>92

20 30 10 2>43

10 10 242

x x

x x x

x x

− =− + − = ⇒

− + =

!orm o- Ax b= (s;




1

2

3

30 20 0 1>92

20 30 10 2>43

0 10 10 242

x

x

x

− ÷ ÷ ÷− − = ÷ ÷ ÷ ÷ ÷ ÷−

#% 1 2=3? 100 x x= = $). 3 120? x = #$% There-ore the ,))o&) .(s6+$'eme)ts $re 1

=3? x = m 2100 x = m $).

3120? x = m

Q8 #$% h 01 a 0 b 1

i i x 1

1%#

2

2

+

−=

x

x x f i

f 0 or f ) f i

0

1

2

3

4

9

=

?

>

10

0

01

02

03

04

0

09

0=

0?

0>

10

C1

C0>?

C0>23

C0?3

C0=24

C09

C04=1

C0342

C022

C010

0

Ro& o- i x 2m

Ro& o- i f 6m

Tot$+ C1 C2

B7 tr$6eDo(.$+ r,+e

≈+−∫ 11

1

0 2

2

dx x x ++ ∑=

13

1

0 22%10#

i

in f f f




( )%2#212

%10#−+− 1m

C0=0 1m

#% h 02 a 0 b 1

I i x 1

1%#

2

2

+

−=

x

x x f i

f 0 or f ) f i

0

1

2

3

4

0

02

04

09

0?

1

C1

C0>23

C0=24

C04=1

C022

0

Ro& o- i x 1m

Ro& o- i f 2m

Tot$+ C1 C233?

B7 tr$6eDo(.$+ r,+e

≈+

−∫

1

11

0 2

2

dx x

x

++ ∑

=

9

1

0 22

%20#

i

in f f f

( )%33?2#212

%20#−+− 1m

C09? 1m

#'% A C0=1 0=0 0001 2m




B C0=1 09? 0003 2m

9e an onlude t"at b usin smaller step size "&1 ,e an produe more aurate

result 1m

Q% #$% Wr(te () .(5(.e.C.(--ere)'e t$+e;

i i

x A0@

i f A1@

i f A2@

i f

0

1

2

3

3

4

200?9

3311

4>?

290?

42>99

19>0?

1m 1m 1m 1m

Wr(te Ne&to)s ()ter6o+$tor7 .(5(.e.C.(--ere)'e 6o+7)om($+

%%##%#%# 10

A2@

00

A1@

0

A0@

02 x x x x f x x f f x P −−+−+=

%3%#3#>0?19%3#0?290?920 −−+−+ x x x 1m

He)'e 13e f #31% P 22019 2m

Us()* '$+',+$tor the e8$'t 5$+,e -or 13e 221>? 1m

So $so+,te error -or th(s metho. 22019 221>? 01?2 1m

#$% Wr(te () .(5(.e.C.(--ere)'e t$+e;

i i

x A0@

i f A1@

i f A2@

i f

0

1

2

1

14

1?

0

0149

02

039

02=3

C011

1m 1m 1m 1m




Wr(te Ne&to)s ()ter6o+$tor7 .(5(.e.C.(--ere)'e 6o+7)om($+

%%##%#%# 10

A2@

00

A1@

0

A0@

02 x x x x f x x f f x P −−+−+=

%41%#1#110%1#3900 −−−−+ x x x 2m

He)'e +o* 12 f #12% P 00=? 2m

Us()* '$+',+$tor the e8$'t 5$+,e -or +o* 12 00=> 1m

So $so+,te error -or th(s metho. 00=> 00=? 0001 1m

Q1& #$%2#1 %

dy x xy

dx+ = &(th #2% 1 y = o5er ()ter5$+ 2 x≤ ≤ $). 03h =

A66+7 E,+ers Metho.;

( )

2

1 2

# %1

031

i i

i i i i i

i

dy xy f x ydx x

x y y y hf x y y

x+

= =+

= + = + ÷+

F$+',+$tor -orm,+$;203## % #1 %%Y XY X + ÷ +

i i x N,mer('$+ So+,t(o) i

y

0 2 1

1 23 112

2 29 1243

3 2> 139?

4 32 14>4

3 1922

9 3? 1=1

= 41 1??0




? 44 2010

> 4= 2140

10 22=1

#%2 2# %

dy x xy y xy

dx− = − &(th #2% 3 y = o5er ()ter5$+ 2 3 x≤ ≤ $). 02h =


( )

2

2

2

1 2

# %

02 i i i

i i i i i

i i i

dy y xy f x ydx x xy

y x y y y hf x y y

x x y+

−= =−

−= + = + ÷+

F$+',+$tor -orm,+$;2 202## % # %%Y Y XY X XY + − ÷ −


y

0 2 3

1 22 4

2 24 90?3

3 29 =>

4 2? 10204

3 12>1?




#'%2 2# %

dy x y xy

dx+ = &(th #1% 0?4 y = o5er ()ter5$+ 1 1 x≤ ≤ $). 00h =


( )

2 2

1 2 2

# %

00 i i

i i i i i

i i

dy xy f x y

dx x y

x y y y hf x y y

x y+

= =+

= + = + ÷+

F$+',+$tor -orm,+$;2 200## % # %%Y XY X Y + ÷ +

i i x N,mer('$+ So+,t(o) i y

0 1 0?4

1 10 0909

2 11 092?

3 11 090

4 12 09=1

12 09>2

9 13 0=13

= 13 0=34

? 14 0=

> 14 0==9

10 1 0=>=




#.% # 1% dy

x xydx

+ + = &(th #0% 2 y = o5er ()ter5$+ 0 1 x≤ ≤ $). 01h =


( )1

# %

1

011

i i

i i i i i

i

dy xy f x y

dx x x y

y y hf x y y x

+

= =

+ = + = + ÷+

F$+',+$tor -orm,+$; 01## % # 1%%Y XY X + ÷ +


y

0 0 2

1 01 2

2 02 201?

3 03 202

4 04 20>>

0 21>

9 09 2231

= 0= 231

? 0? 2410

> 0> 21=

10 1 2939

revision_final exam math 4

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