revision_final exam math 4
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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$rs%
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Q2 !(). the root#s% o- -,)'t(o)3 2# % 1= 0?4 010? f x x x x= − + − th$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
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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
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Semester I 2012/2013 Kohort 1
#% 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$t
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Semester I 2012/2013 Kohort 1
>?4$).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≤ ≤ $). 01h =
#'' T($ )$*T+++
#ns,ers:
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Semester I 2012/2013 Kohort 1
N-T$: #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
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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;
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Semester I 2012/2013 Kohort 1
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
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Semester I 2012/2013 Kohort 1
( )%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
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Semester I 2012/2013 Kohort 1
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$+e;
i i
x A0@
i f A1@
i f A2@
i f
0
1
2
1
14
1?
0
0149
02
039
02=3
C011
1m 1m 1m 1m
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Wr(te Ne&to)s ()ter6o+$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('$+ So+,t(o) i
y
0 2 1
1 23 112
2 29 1243
3 2> 139?
4 32 14>4
3 1922
9 3? 1=1
= 41 1??0
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Semester I 2012/2013 Kohort 1
? 44 2010
> 4= 2140
10 22=1
#%2 2# %
dy x xy y xy
dx− = − &(th #2% 3 y = o5er ()ter5$+ 2 3 x≤ ≤ $). 02h =
A66+7 E,+ers Metho.;
( )
2
2
2
1 2
# %
02 i i i
i i i i i
i i i
dy y xy f x ydx x xy
y x y y y hf x y y
x x y+
−= =−
−= + = + ÷+
F$+',+$tor -orm,+$;2 202## % # %%Y Y XY X XY + − ÷ −
i i x N,mer('$+ So+,t(o) i
y
0 2 3
1 22 4
2 24 90?3
3 29 =>
4 2? 10204
3 12>1?
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#'%2 2# %
dy x y xy
dx+ = &(th #1% 0?4 y = o5er ()ter5$+ 1 1 x≤ ≤ $). 00h =
A66+7 E,+ers Metho.;
( )
2 2
1 2 2
# %
00 i i
i i i i i
i i
dy xy f x y
dx x y
x y y y hf x y y
x y+
= =+
= + = + ÷+
F$+',+$tor -orm,+$;2 200## % # %%Y XY X Y + ÷ +
i i x N,mer('$+ So+,t(o) i y
0 1 0?4
1 10 0909
2 11 092?
3 11 090
4 12 09=1
12 09>2
9 13 0=13
= 13 0=34
? 14 0=
> 14 0==9
10 1 0=>=
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Semester I 2012/2013 Kohort 1
#.% # 1% dy
x xydx
+ + = &(th #0% 2 y = o5er ()ter5$+ 0 1 x≤ ≤ $). 01h =
A66+7 E,+ers Metho.;
( )1
# %
1
011
i i
i i i i i
i
dy xy f x y
dx x x y
y y hf x y y x
+
= =
+ = + = + ÷+
F$+',+$tor -orm,+$; 01## % # 1%%Y XY X + ÷ +
i i x N,mer('$+ So+,t(o) i
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