1
CHAPTERV18
18.1 (a)
1.0791812V−V0.90309V
fV(10)V=V0.90309V+V (10V−V8)V=V0.991136
1
12V−V8
1−V0.991136V
V =V 100%V=V0.886%
t
1
(b)
1.0413927V−V0.9542425V
fV(10)V=V0.9542425V+V (10V−V9)V=V0.997818
1
11−V9
1−V0.997818V
V =V 100%V=V0.218%
t
1
18.2 First,VorderVtheVpoints
x0V=V9 f(x0)V=V0.9542425
x1V=V11V f(x1)V=V1.0413927
x2V=V8 f(x2)V=V0.9030900
ApplyingVEq.V(18.4)
b0V=V0.9542425
EquationV(18.5)Vyields
1.0413927V−V0.9542425V
b1V =V =V0.0435751
11−V9
EquationV(18.6)Vgives
0.9030900V−1.0413927V
−V0.0435751
8V−11 0.0461009V−V0.0435751V
b2V =V V
=V =V−0.0025258
8V−V9 8V−V9
SubstitutingVtheseVvaluesVintoVEq.V(18.3)VyieldsVtheVquadraticVformula
f2V(x)V=V0.9542425V+V0.0435751(xV−V9)V−V0.0025258(xV−V9)(xV−11)
whichVcanVbeVevaluatedVatVxV=V10Vfor
f2V(10)V=V0.9542425V+V0.0435751(10V−V9)V−V0.0025258(10V−V9)(10V−11)V=V1.0003434
18.3 First,VorderVtheVpoints
x0V=V9 f(x0)V=V0.9542425
x1V=V11V f(x1)V=V1.0413927
x2V=V8 f(x2)V=V0.9030900
x3V=V12V f(x3)V=V1.0791812
TheVfirstVdividedVdifferencesVcanVbeVcomputedVas
PROPRIETARYVMATERIAL.V ©VTheVMcGraw-
HillVCompanies,VInc.V AllVrightsVreserved.V NoVpartVofVthisVManualVmayVbeVdisplayed,VreproducedVorVdistri
butedVinVanyVformVorVbyVanyVmeans,VwithoutVtheVpriorVwrittenVpermissionVofVtheVpublisher,VorVusedVbey
ondVtheVlimitedVdistributionVtoVteachersVandVeducatorsVpermittedVbyVMcGraw-
HillVforVtheirVindividualVcourseVpreparation.V IfVyouVareVaVstudentVusingVthisVManual,VyouVareVusingVitVwi
thoutVpermission.
, 2
1.0413927V−V0.9542425V
fV[xV ,VxV ]V=V =V0.0435751
1
VV0 11−V9
0.9030900V−1.0413927V
fV[xV ,VxV ]V=V =V0.0461009
2
VV1 8V−V11
1.0791812V−V0.9030900V
fV[xV ,VxV ]V=V =V0.0440228
3
VV2 12V−V8
TheVsecondVdividedVdifferencesVare
0.0461009V−V0.0435751V
fV[xV ,VxV ,VxV ]V=V =V−0.0025258
2V V 1
VV0 8V−V9
0.0440228V−V0.0461009V
fV[xV ,VxV ,VxV ]V=V =V−0.0020781
3V V 2
VV1 12V−V11
TheVthirdVdividedVdifferenceVis
−0.0020781−V(−0.0025258)V
fV[x3VV,VVxV2V,VVxV ,VxV ]V=V =V0.00014924
1V V 0 12V−V9
SubstitutingVtheVappropriateVvaluesVintoVEq.V(18.7)Vgives
f3V(x)V=V0.9542425V+V0.0435751(xV−V9)V−V0.0025258(xV−V9)(xV−11)
+V0.00014924(xV−V9)(xV−11)(xV−V8)
whichVcanVbeVevaluatedVatVxV=V10Vfor
f3V(x)V=V0.9542425V+V0.0435751(10V−V9)V−V0.0025258(10V−V9)(10V−11)
+V0.00014924(10V−V9)(10V−11)(10V−V8)V=V1.0000449
18.4
18.1 V(a):
x0V=V8 f(x0)V=V0.9030900
x1V=V12 f(x1)V=V1.0791812
10V−12V 10V−V8V
fV (10)V=V 0.9030900V+V 1.0791812V=V0.991136
1
8V−12 12V−V8
18.1 (b):
x0V=V9 f(x0)V=V0.9542425
x1V=V11 f(x1)V=V1.0413927
10V−11V 10V−V9V
fV (10)V=V 0.9542425V+V 1.0413927V=V0.997818
1
9V−11 11−V9
18.2 :
x0V=V8 f(x0)V=V0.9030900
x1V=V9 f(x1)V=V0.9542425
x2V=V11V f(x2)V=V1.0413927
PROPRIETARYVMATERIAL.V ©VTheVMcGraw-
HillVCompanies,VInc.V AllVrightsVreserved.V NoVpartVofVthisVManualVmayVbeVdisplayed,VreproducedVorVdistri
butedVinVanyVformVorVbyVanyVmeans,VwithoutVtheVpriorVwrittenVpermissionVofVtheVpublisher,VorVusedVbey
ondVtheVlimitedVdistributionVtoVteachersVandVeducatorsVpermittedVbyVMcGraw-
HillVforVtheirVindividualVcourseVpreparation.V IfVyouVareVaVstudentVusingVthisVManual,VyouVareVusingVitVwi
thoutVpermission.
, 3
(10V−V9)(10V−11)V (10V−V8)(10V−11)V
fV2 (10)V=V 0.9030900V+V 0.9542425
(8V−V9)(8V−11) (9V−V8)(9V−11)
(10V−V8)(10V−V9)V
+V 1.0413927V=V1.0003434
(11−V8)(11−V9)
18.3 :
x0V=V8 f(x0)V=V0.9030900
x1V=V9 f(x1)V=V0.9542425
x2V=V11V f(x2)V=V1.0413927
x3V=V12V f(x3)V=V1.0791812
(10V−V9)(10V−11)(10V−12)V (10V−V8)(10V−11)(10V−12)V
fV3 (10)V=V 0.9030900V+V 0.9542425
(8V−V9)(8V−11)(8V−12) (9V−V8)(9V−11)(9V−12)
(10V−V8)(10V−V9)(10V−12)V (10V−V8)(10V−V9)(10V−11)V
+V 1.0413927V+V 1.0791812V=V1.0000449
(11−V8)(11−V9)(11−12) (12V−V8)(12V−V9)(12V−11)
18.5 First,VorderVtheVpointsVsoVthatVtheyVareVasVcloseVtoVandVasVcenteredVaboutVtheVunknownVasVpossible
x0V=V2.5V f(x0)V=V14
x1V=V3.2V f(x1)V=V15
x2V=V2 f(x2)V=V8
x3V=V4 f(x3)V=V8
x4V=V1.6V f(x4)V=V2
Next,VtheVdividedVdifferencesVcanVbeVcomputedVandVdisplayedVinVtheVformatVofVFig.V18.5,
i xi f(xi) f[xi+1,xi] f[xi+2,xi+1,xi] f[xi+3,xi+2,xi+1,xi] f[xi+4,xi+3,xi+2,xi+1,xi]
0 2.5 14 1.428571 -8.809524 1.011905 1.847718
1 3.2 15 5.833333 -7.291667 -0.651042
2 2 8 0 -6.25
3 4 8 2.5
4 1.6 2
TheVfirstVthroughVthird-orderVinterpolationsVcanVthenVbeVimplementedVas
f1(2.8)V=V14V+1.428571(2.8V−V2.5)V=V14.428571
f2V(2.8)V=V14V+1.428571(2.8V−V2.5)V−V8.809524(2.8V−V2.5)(2.8V−V3.2)V=V15.485714
f3V(2.8)V=V14V+1.428571(2.8V−V2.5)V−V8.809524(2.8V−V2.5)(2.8V−V3.2)
+V1.011905(2.8V−V2.5)(2.8V−V3.2)(2.8V−V2.)V=V15.388571
TheVerrorVestimatesVforVtheVfirstVandVsecond-orderVpredictionsVcanVbeVcomputedVwithVEq.V18.19Vas
R1V=V15.485714V−14.428571V=V1.057143
R2V =V15.388571−15.485714V=V−0.097143
TheVerrorVforVtheVthird-orderVpredictionVcanVbeVcomputedVwithVEq.V18.18Vas
R3V =V1.847718(2.8V−V2.5)(2.8V−V3.2)(2.8V−V2)(2.8V−V4)V=V0.212857
18.6 First,VorderVtheVpointsVsoVthatVtheyVareVasVcloseVtoVandVasVcenteredVaboutVtheVunknownVasVpossible
PROPRIETARYVMATERIAL.V ©VTheVMcGraw-
HillVCompanies,VInc.V AllVrightsVreserved.V NoVpartVofVthisVManualVmayVbeVdisplayed,VreproducedVorVdistri
butedVinVanyVformVorVbyVanyVmeans,VwithoutVtheVpriorVwrittenVpermissionVofVtheVpublisher,VorVusedVbey
ondVtheVlimitedVdistributionVtoVteachersVandVeducatorsVpermittedVbyVMcGraw-
HillVforVtheirVindividualVcourseVpreparation.V IfVyouVareVaVstudentVusingVthisVManual,VyouVareVusingVitVwi
thoutVpermission.
, 4
x0V=V3 f(x0)V=V19
x1V=V5 f(x1)V=V99
x2V=V2 f(x2)V=V6
x3V=V7 f(x3)V=V291
x4V=V1 f(x4)V=V3
Next,VtheVdividedVdifferencesVcanVbeVcomputedVandVdisplayedVinVtheVformatVofVFig.V18.5,
i xi f(xi) f[xi+1,xi] f[xi+2,xi+1,xi] f[xi+3,xi+2,xi+1,xi] f[xi+4,xi+3,xi+2,xi+1,xi]
0 3 19 40 9 1 0
1 5 99 31 13 1
2 2 6 57 9
3 7 291 48
4 1 3
TheVfirstVthroughVfourth-orderVinterpolationsVcanVthenVbeVimplementedVas
f1(4)V=V19V+V40(4V−V3)V=V59
f2V(4)V=V59V+V9(4V−V3)(4V−V5)V=V50
f3V(4)V=V50V +1(4V−V3)(4V−V5)(4V−V2)V=V48
f4V(4)V=V48V+V0(4V−V3)(4V−V5)(4V−V2)(4V−V7)V=V48
ClearlyVthisVdataVwasVgeneratedVwithVaVcubicVpolynomialVsinceVtheVdifferenceVbetweenVthe
V4thVandVtheV3rd-orderVversionsVisVzero.
18.7
FirstVorder:
x0V=V3 f(x0)V=V19
x1V=V5 f(x1)V=V99
4V−V5 4V−V3V
fV (10)V=V 19V+V 99V=V59
1
3V−V5 5V−V3
SecondVorder:
x0V=V3 f(x0)V=V19
x1V=V5 f(x1)V=V99
x2V=V2 f(x2)V=V6
(4V−V5)(4V−V2)V (4V−V3)(4V−V2)V (4V−V3)(4V−V5)V
fV2 (10)V=V 19V+V 99V +V 6V=V50
(3V−V5)(3V−V2) (5V−V3)(5V−V2) (2V−V3)(2V−V5)
ThirdVorder:
x0V=V3 f(x0)V=V19
x1V=V5 f(x1)V=V99
x2V=V2 f(x2)V=V6
x3V=V7 f(x3)V=V291
(4V−V5)(4V−V2)(4V−V7)V (4V−V3)(4V−V2)(4V−V7)V
fV (10)V=V 19V+V 99
3
(3V−V5)(3V−V2)(3V−V7) (5V−V3)(5V−V2)(5V−V7)
(4V−V3)(4V−V5)(4V−V7)V (4V−V3)(4V−V5)(4V−V2)V
+V 6V+V 291V=V48
(2V−V3)(2V−V5)(2V−V7) (7V−V3)(7V−V5)(7V−V2)
PROPRIETARYVMATERIAL.V ©VTheVMcGraw-
HillVCompanies,VInc.V AllVrightsVreserved.V NoVpartVofVthisVManualVmayVbeVdisplayed,VreproducedVorVdistri
butedVinVanyVformVorVbyVanyVmeans,VwithoutVtheVpriorVwrittenVpermissionVofVtheVpublisher,VorVusedVbey
ondVtheVlimitedVdistributionVtoVteachersVandVeducatorsVpermittedVbyVMcGraw-
HillVforVtheirVindividualVcourseVpreparation.V IfVyouVareVaVstudentVusingVthisVManual,VyouVareVusingVitVwi
thoutVpermission.
CHAPTERV18
18.1 (a)
1.0791812V−V0.90309V
fV(10)V=V0.90309V+V (10V−V8)V=V0.991136
1
12V−V8
1−V0.991136V
V =V 100%V=V0.886%
t
1
(b)
1.0413927V−V0.9542425V
fV(10)V=V0.9542425V+V (10V−V9)V=V0.997818
1
11−V9
1−V0.997818V
V =V 100%V=V0.218%
t
1
18.2 First,VorderVtheVpoints
x0V=V9 f(x0)V=V0.9542425
x1V=V11V f(x1)V=V1.0413927
x2V=V8 f(x2)V=V0.9030900
ApplyingVEq.V(18.4)
b0V=V0.9542425
EquationV(18.5)Vyields
1.0413927V−V0.9542425V
b1V =V =V0.0435751
11−V9
EquationV(18.6)Vgives
0.9030900V−1.0413927V
−V0.0435751
8V−11 0.0461009V−V0.0435751V
b2V =V V
=V =V−0.0025258
8V−V9 8V−V9
SubstitutingVtheseVvaluesVintoVEq.V(18.3)VyieldsVtheVquadraticVformula
f2V(x)V=V0.9542425V+V0.0435751(xV−V9)V−V0.0025258(xV−V9)(xV−11)
whichVcanVbeVevaluatedVatVxV=V10Vfor
f2V(10)V=V0.9542425V+V0.0435751(10V−V9)V−V0.0025258(10V−V9)(10V−11)V=V1.0003434
18.3 First,VorderVtheVpoints
x0V=V9 f(x0)V=V0.9542425
x1V=V11V f(x1)V=V1.0413927
x2V=V8 f(x2)V=V0.9030900
x3V=V12V f(x3)V=V1.0791812
TheVfirstVdividedVdifferencesVcanVbeVcomputedVas
PROPRIETARYVMATERIAL.V ©VTheVMcGraw-
HillVCompanies,VInc.V AllVrightsVreserved.V NoVpartVofVthisVManualVmayVbeVdisplayed,VreproducedVorVdistri
butedVinVanyVformVorVbyVanyVmeans,VwithoutVtheVpriorVwrittenVpermissionVofVtheVpublisher,VorVusedVbey
ondVtheVlimitedVdistributionVtoVteachersVandVeducatorsVpermittedVbyVMcGraw-
HillVforVtheirVindividualVcourseVpreparation.V IfVyouVareVaVstudentVusingVthisVManual,VyouVareVusingVitVwi
thoutVpermission.
, 2
1.0413927V−V0.9542425V
fV[xV ,VxV ]V=V =V0.0435751
1
VV0 11−V9
0.9030900V−1.0413927V
fV[xV ,VxV ]V=V =V0.0461009
2
VV1 8V−V11
1.0791812V−V0.9030900V
fV[xV ,VxV ]V=V =V0.0440228
3
VV2 12V−V8
TheVsecondVdividedVdifferencesVare
0.0461009V−V0.0435751V
fV[xV ,VxV ,VxV ]V=V =V−0.0025258
2V V 1
VV0 8V−V9
0.0440228V−V0.0461009V
fV[xV ,VxV ,VxV ]V=V =V−0.0020781
3V V 2
VV1 12V−V11
TheVthirdVdividedVdifferenceVis
−0.0020781−V(−0.0025258)V
fV[x3VV,VVxV2V,VVxV ,VxV ]V=V =V0.00014924
1V V 0 12V−V9
SubstitutingVtheVappropriateVvaluesVintoVEq.V(18.7)Vgives
f3V(x)V=V0.9542425V+V0.0435751(xV−V9)V−V0.0025258(xV−V9)(xV−11)
+V0.00014924(xV−V9)(xV−11)(xV−V8)
whichVcanVbeVevaluatedVatVxV=V10Vfor
f3V(x)V=V0.9542425V+V0.0435751(10V−V9)V−V0.0025258(10V−V9)(10V−11)
+V0.00014924(10V−V9)(10V−11)(10V−V8)V=V1.0000449
18.4
18.1 V(a):
x0V=V8 f(x0)V=V0.9030900
x1V=V12 f(x1)V=V1.0791812
10V−12V 10V−V8V
fV (10)V=V 0.9030900V+V 1.0791812V=V0.991136
1
8V−12 12V−V8
18.1 (b):
x0V=V9 f(x0)V=V0.9542425
x1V=V11 f(x1)V=V1.0413927
10V−11V 10V−V9V
fV (10)V=V 0.9542425V+V 1.0413927V=V0.997818
1
9V−11 11−V9
18.2 :
x0V=V8 f(x0)V=V0.9030900
x1V=V9 f(x1)V=V0.9542425
x2V=V11V f(x2)V=V1.0413927
PROPRIETARYVMATERIAL.V ©VTheVMcGraw-
HillVCompanies,VInc.V AllVrightsVreserved.V NoVpartVofVthisVManualVmayVbeVdisplayed,VreproducedVorVdistri
butedVinVanyVformVorVbyVanyVmeans,VwithoutVtheVpriorVwrittenVpermissionVofVtheVpublisher,VorVusedVbey
ondVtheVlimitedVdistributionVtoVteachersVandVeducatorsVpermittedVbyVMcGraw-
HillVforVtheirVindividualVcourseVpreparation.V IfVyouVareVaVstudentVusingVthisVManual,VyouVareVusingVitVwi
thoutVpermission.
, 3
(10V−V9)(10V−11)V (10V−V8)(10V−11)V
fV2 (10)V=V 0.9030900V+V 0.9542425
(8V−V9)(8V−11) (9V−V8)(9V−11)
(10V−V8)(10V−V9)V
+V 1.0413927V=V1.0003434
(11−V8)(11−V9)
18.3 :
x0V=V8 f(x0)V=V0.9030900
x1V=V9 f(x1)V=V0.9542425
x2V=V11V f(x2)V=V1.0413927
x3V=V12V f(x3)V=V1.0791812
(10V−V9)(10V−11)(10V−12)V (10V−V8)(10V−11)(10V−12)V
fV3 (10)V=V 0.9030900V+V 0.9542425
(8V−V9)(8V−11)(8V−12) (9V−V8)(9V−11)(9V−12)
(10V−V8)(10V−V9)(10V−12)V (10V−V8)(10V−V9)(10V−11)V
+V 1.0413927V+V 1.0791812V=V1.0000449
(11−V8)(11−V9)(11−12) (12V−V8)(12V−V9)(12V−11)
18.5 First,VorderVtheVpointsVsoVthatVtheyVareVasVcloseVtoVandVasVcenteredVaboutVtheVunknownVasVpossible
x0V=V2.5V f(x0)V=V14
x1V=V3.2V f(x1)V=V15
x2V=V2 f(x2)V=V8
x3V=V4 f(x3)V=V8
x4V=V1.6V f(x4)V=V2
Next,VtheVdividedVdifferencesVcanVbeVcomputedVandVdisplayedVinVtheVformatVofVFig.V18.5,
i xi f(xi) f[xi+1,xi] f[xi+2,xi+1,xi] f[xi+3,xi+2,xi+1,xi] f[xi+4,xi+3,xi+2,xi+1,xi]
0 2.5 14 1.428571 -8.809524 1.011905 1.847718
1 3.2 15 5.833333 -7.291667 -0.651042
2 2 8 0 -6.25
3 4 8 2.5
4 1.6 2
TheVfirstVthroughVthird-orderVinterpolationsVcanVthenVbeVimplementedVas
f1(2.8)V=V14V+1.428571(2.8V−V2.5)V=V14.428571
f2V(2.8)V=V14V+1.428571(2.8V−V2.5)V−V8.809524(2.8V−V2.5)(2.8V−V3.2)V=V15.485714
f3V(2.8)V=V14V+1.428571(2.8V−V2.5)V−V8.809524(2.8V−V2.5)(2.8V−V3.2)
+V1.011905(2.8V−V2.5)(2.8V−V3.2)(2.8V−V2.)V=V15.388571
TheVerrorVestimatesVforVtheVfirstVandVsecond-orderVpredictionsVcanVbeVcomputedVwithVEq.V18.19Vas
R1V=V15.485714V−14.428571V=V1.057143
R2V =V15.388571−15.485714V=V−0.097143
TheVerrorVforVtheVthird-orderVpredictionVcanVbeVcomputedVwithVEq.V18.18Vas
R3V =V1.847718(2.8V−V2.5)(2.8V−V3.2)(2.8V−V2)(2.8V−V4)V=V0.212857
18.6 First,VorderVtheVpointsVsoVthatVtheyVareVasVcloseVtoVandVasVcenteredVaboutVtheVunknownVasVpossible
PROPRIETARYVMATERIAL.V ©VTheVMcGraw-
HillVCompanies,VInc.V AllVrightsVreserved.V NoVpartVofVthisVManualVmayVbeVdisplayed,VreproducedVorVdistri
butedVinVanyVformVorVbyVanyVmeans,VwithoutVtheVpriorVwrittenVpermissionVofVtheVpublisher,VorVusedVbey
ondVtheVlimitedVdistributionVtoVteachersVandVeducatorsVpermittedVbyVMcGraw-
HillVforVtheirVindividualVcourseVpreparation.V IfVyouVareVaVstudentVusingVthisVManual,VyouVareVusingVitVwi
thoutVpermission.
, 4
x0V=V3 f(x0)V=V19
x1V=V5 f(x1)V=V99
x2V=V2 f(x2)V=V6
x3V=V7 f(x3)V=V291
x4V=V1 f(x4)V=V3
Next,VtheVdividedVdifferencesVcanVbeVcomputedVandVdisplayedVinVtheVformatVofVFig.V18.5,
i xi f(xi) f[xi+1,xi] f[xi+2,xi+1,xi] f[xi+3,xi+2,xi+1,xi] f[xi+4,xi+3,xi+2,xi+1,xi]
0 3 19 40 9 1 0
1 5 99 31 13 1
2 2 6 57 9
3 7 291 48
4 1 3
TheVfirstVthroughVfourth-orderVinterpolationsVcanVthenVbeVimplementedVas
f1(4)V=V19V+V40(4V−V3)V=V59
f2V(4)V=V59V+V9(4V−V3)(4V−V5)V=V50
f3V(4)V=V50V +1(4V−V3)(4V−V5)(4V−V2)V=V48
f4V(4)V=V48V+V0(4V−V3)(4V−V5)(4V−V2)(4V−V7)V=V48
ClearlyVthisVdataVwasVgeneratedVwithVaVcubicVpolynomialVsinceVtheVdifferenceVbetweenVthe
V4thVandVtheV3rd-orderVversionsVisVzero.
18.7
FirstVorder:
x0V=V3 f(x0)V=V19
x1V=V5 f(x1)V=V99
4V−V5 4V−V3V
fV (10)V=V 19V+V 99V=V59
1
3V−V5 5V−V3
SecondVorder:
x0V=V3 f(x0)V=V19
x1V=V5 f(x1)V=V99
x2V=V2 f(x2)V=V6
(4V−V5)(4V−V2)V (4V−V3)(4V−V2)V (4V−V3)(4V−V5)V
fV2 (10)V=V 19V+V 99V +V 6V=V50
(3V−V5)(3V−V2) (5V−V3)(5V−V2) (2V−V3)(2V−V5)
ThirdVorder:
x0V=V3 f(x0)V=V19
x1V=V5 f(x1)V=V99
x2V=V2 f(x2)V=V6
x3V=V7 f(x3)V=V291
(4V−V5)(4V−V2)(4V−V7)V (4V−V3)(4V−V2)(4V−V7)V
fV (10)V=V 19V+V 99
3
(3V−V5)(3V−V2)(3V−V7) (5V−V3)(5V−V2)(5V−V7)
(4V−V3)(4V−V5)(4V−V7)V (4V−V3)(4V−V5)(4V−V2)V
+V 6V+V 291V=V48
(2V−V3)(2V−V5)(2V−V7) (7V−V3)(7V−V5)(7V−V2)
PROPRIETARYVMATERIAL.V ©VTheVMcGraw-
HillVCompanies,VInc.V AllVrightsVreserved.V NoVpartVofVthisVManualVmayVbeVdisplayed,VreproducedVorVdistri
butedVinVanyVformVorVbyVanyVmeans,VwithoutVtheVpriorVwrittenVpermissionVofVtheVpublisher,VorVusedVbey
ondVtheVlimitedVdistributionVtoVteachersVandVeducatorsVpermittedVbyVMcGraw-
HillVforVtheirVindividualVcourseVpreparation.V IfVyouVareVaVstudentVusingVthisVManual,VyouVareVusingVitVwi
thoutVpermission.
