APEX Calculus, Version 2.0, Volume 3, in b&w

360 Pages·2014·5.77 MB·English

C(cid:131)(cid:189)(cid:145)(cid:231)(cid:189)(cid:231)(cid:221) III
APX
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Version2.0
Authors
Gregory Hartman, Ph.D.
DepartmentofAppliedMathema(cid:415)cs
VirginiaMilitaryIns(cid:415)tute
Brian Heinold, Ph.D.
DepartmentofMathema(cid:415)csandComputerScience
MountSaintMary’sUniversity
Troy Siemers, Ph.D.
DepartmentofAppliedMathema(cid:415)cs
VirginiaMilitaryIns(cid:415)tute
Dimplekumar Chalishajar, Ph.D.
DepartmentofAppliedMathema(cid:415)cs
VirginiaMilitaryIns(cid:415)tute
Editor
Jennifer Bowen, Ph.D.
DepartmentofMathema(cid:415)csandComputerScience
TheCollegeofWooster
Copyright©2014GregoryHartman
Licensed to the public under Crea(cid:415)ve Commons
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Contents
Preface iii
TableofContents v
9 CurvesinthePlane 469
9.1 ConicSec(cid:415)ons . . . . . . . . . . . . . . . . . . . . . . . . . . . 469
9.2 ParametricEqua(cid:415)ons . . . . . . . . . . . . . . . . . . . . . . . 483
9.3 CalculusandParametricEqua(cid:415)ons . . . . . . . . . . . . . . . . 493
9.4 Introduc(cid:415)ontoPolarCoordinates . . . . . . . . . . . . . . . . 503
9.5 CalculusandPolarFunc(cid:415)ons . . . . . . . . . . . . . . . . . . . 516
10 Vectors 529
10.1 Introduc(cid:415)ontoCartesianCoordinatesinSpace . . . . . . . . . 529
10.2 AnIntroduc(cid:415)ontoVectors . . . . . . . . . . . . . . . . . . . . 543
10.3 TheDotProduct . . . . . . . . . . . . . . . . . . . . . . . . . . 557
10.4 TheCrossProduct . . . . . . . . . . . . . . . . . . . . . . . . . 570
10.5 Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580
10.6 Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590
11 VectorValuedFunc(cid:415)ons 599
11.1 Vector–ValuedFunc(cid:415)ons . . . . . . . . . . . . . . . . . . . . . 599
11.2 CalculusandVector–ValuedFunc(cid:415)ons . . . . . . . . . . . . . . 605
11.3 TheCalculusofMo(cid:415)on . . . . . . . . . . . . . . . . . . . . . . 618
11.4 UnitTangentandNormalVectors . . . . . . . . . . . . . . . . . 631
11.5 TheArcLengthParameterandCurvature . . . . . . . . . . . . 640
12 Func(cid:415)onsofSeveralVariables 651
12.1 Introduc(cid:415)ontoMul(cid:415)variableFunc(cid:415)ons . . . . . . . . . . . . . 651
12.2 LimitsandCon(cid:415)nuityofMul(cid:415)variableFunc(cid:415)ons . . . . . . . . . 658
12.3 Par(cid:415)alDeriva(cid:415)ves . . . . . . . . . . . . . . . . . . . . . . . . . 668
12.4 Differen(cid:415)abilityandtheTotalDifferen(cid:415)al . . . . . . . . . . . . 680
12.5 TheMul(cid:415)variableChainRule . . . . . . . . . . . . . . . . . . . 689
12.6 Direc(cid:415)onalDeriva(cid:415)ves . . . . . . . . . . . . . . . . . . . . . . 696
12.7 TangentLines,NormalLines,andTangentPlanes . . . . . . . . 705
12.8 ExtremeValues . . . . . . . . . . . . . . . . . . . . . . . . . . 715
13 Mul(cid:415)pleIntegra(cid:415)on 725
13.1 IteratedIntegralsandArea . . . . . . . . . . . . . . . . . . . . 725
13.2 DoubleIntegra(cid:415)onandVolume . . . . . . . . . . . . . . . . . . 735
13.3 DoubleIntegra(cid:415)onwithPolarCoordinates . . . . . . . . . . . . 746
13.4 CenterofMass . . . . . . . . . . . . . . . . . . . . . . . . . . 753
13.5 SurfaceArea . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765
13.6 VolumeBetweenSurfacesandTripleIntegra(cid:415)on. . . . . . . . . 772
A Solu(cid:415)onsToSelectedProblems A.1
Index A.17
9: C(cid:231)(cid:217)(cid:242)(cid:155)(cid:221) (cid:174)(cid:196) (cid:227)(cid:171)(cid:155) P(cid:189)(cid:131)(cid:196)(cid:155)
9.1 Conic Sec(cid:415)ons
TheancientGreeksrecognizedthatinteres(cid:415)ngshapescanbeformedbyinter-
sec(cid:415)ngaplanewithadoublenappedcone(i.e.,twoiden(cid:415)calconesplaced(cid:415)p–
to–(cid:415)passhowninthefollowingfigures).Astheseshapesareformedassec(cid:415)ons
ofconics,theyhaveearnedtheofficialname“conicsec(cid:415)ons.”
Thethree“mostinteres(cid:415)ng”conicsec(cid:415)onsaregiveninthetoprowofFigure
9.1. Theyaretheparabola, theellipse(whichincludescircles)andthehyper-
bola. Ineachofthesecases,theplanedoesnotintersectthe(cid:415)psofthecones
(usuallytakentobetheorigin).
Parabola Ellipse Circle Hyperbola
Point Line CrossedLines
Figure9.1:NondegenerateConicSec(cid:415)ons
When the plane does contain the origin, three degenerate cones can be
formed as shown the bo(cid:425)om row of Figure 9.1: a point, a line, and crossed
lines.Wefocushereonthenondegeneratecases.
Whiletheabovegeometricconstructsdefinetheconicsinanintui(cid:415)ve,visual
way, these constructs are not very helpful when trying to analyze the shapes
Chapter9 CurvesinthePlane
algebraicallyorconsiderthemasthegraphofafunc(cid:415)on. Itcanbeshownthat
allconicscanbedefinedbythegeneralsecond–degreeequa(cid:415)on
Ax2+Bxy+Cy2+Dx+Ey+F=0:
While this algebraic defini(cid:415)on has its uses, most find another geometric per-
spec(cid:415)veoftheconicsmorebeneficial.
Eachnondegenerateconiccanbedefinedasthelocus,orset,ofpointsthat
sa(cid:415)sfy a certain distance property. These distance proper(cid:415)es can be used to
generateanalgebraicformula,allowingustostudyeachconicasthegraphofa
func(cid:415)on.
Parabolas
..
Defini(cid:415)on40 Parabola
A parabola is the locus of all points equidistant from a point (called a
focus)andaline(calledthedirectrix)thatdoesnotcontainthefocus.
y
of etr Figure 9.2 illustratesthis defini(cid:415)on. The pointhalfwaybetweenthe focus
Axis ymm andthedirectrixisthevertex. Thelinethroughthefocus,perpendiculartothe
S directrix,istheaxisofsymmetry,asthepor(cid:415)onoftheparabolaononesideof
d (x;y)
Focus } thislineisthemirror–imageofthepor(cid:415)onontheoppositeside.
The defini(cid:415)on leads us to an algebraic formula for the parabola. Let P =
p
...........} d (x;y)beapointonaparabolawhosefocusisatF =(0;p)andwhosedirectrix
Vertex p isaty=(cid:0)p. (We’llassumefornowthatthefocusliesonthey-axis;byplacing
Directrix
thefocuspunitsabovethex-axisandthedirectrixpunitsbelowthisaxis,the
vertexwillbeat(0;0).)
WeusetheDistanceFormulatofindthedistanced betweenFandP:
Figure 9.2: Illustra(cid:415)ng the defini(cid:415)on of 1
√
the parabola and establishing an alge-
braicformula. d1 = (x(cid:0)0)2+(y(cid:0)p)2:
Thedistanced fromPtothedirectrixismorestraigh(cid:414)orward:
2
d =y(cid:0)((cid:0)p)=y+p:
2
Thesetwodistancesareequal.Se(cid:427)ngd =d ,wecansolveforyintermsofx:
1 2
d =d
√ 1 2
x2+(y(cid:0)p)2 =y+p
Notes:
470
9.1 ConicSec(cid:415)ons
Nowsquarebothsides.
x2+(y(cid:0)p)2 =(y+p)2
x2+y2(cid:0)2yp+p2 =y2+2yp+p2
x2 =4yp
1
y= x2:
4p
The geometric defini(cid:415)on of the parabola has led us to the familiar quadra(cid:415)c
func(cid:415)onwhosegraphisaparabolawithvertexattheorigin.Whenweallowthe
vertextonotbeat(0;0),wegetthefollowingstandardformoftheparabola.
..
KeyIdea33 GeneralEqua(cid:415)onofaParabola
1. Ver(cid:415)calAxisofSymmetry:Theequa(cid:415)onoftheparabolawithver-
texat(h;k)anddirectrixy=k(cid:0)pinstandardformis
1
y= (x(cid:0)h)2+k:
4p
Thefocusisat(h;k+p).
2. HorizontalAxisofSymmetry: Theequa(cid:415)onoftheparabolawith
vertexat(h;k)anddirectrixx=h(cid:0)pinstandardformis
1
x= (y(cid:0)k)2+h:
4p
Thefocusisat(h+p;k).
Note:pisnotnecessarilyaposi(cid:415)venumber.
y
2
..Example273 Findingtheequa(cid:415)onofaparabola
Givetheequa(cid:415)onoftheparabolawithfocusat(1;2)anddirectrixaty=3. x
(cid:0)2 2 4
S(cid:202)(cid:189)(cid:231)(cid:227)(cid:174)(cid:202)(cid:196) Thevertexislocatedhalfwaybetweenthefocusanddirec- (cid:0)2
trix,so(h;k) = (1;2:5). Thisgivesp = (cid:0)0:5. UsingKeyIdea33wehavethe
(cid:0)4
equa(cid:415)onoftheparabolaas
1 1 ............. (cid:0)6
y= (x(cid:0)1)2+2:5=(cid:0) (x(cid:0)1)2+2:5:
4((cid:0)0:5) 2
Figure9.3: TheparaboladescribedinEx-
ample273.
TheparabolaissketchedinFigure9.3. ..
Notes:
471
Chapter9 CurvesinthePlane
..Example274 Findingthefocusanddirectrixofaparabola
Findthefocusanddirectrixoftheparabolax = 1y2(cid:0)y+1. Thepoint(7;12)
8
liesonthegraphofthisparabola;verifythatitisequidistantfromthefocusand
directrix.
S(cid:202)(cid:189)(cid:231)(cid:227)(cid:174)(cid:202)(cid:196) Weneedtoputtheequa(cid:415)onoftheparabolainitsgeneral
y form.Thisrequiresustocompletethesquare:
10
10 x= 1y2(cid:0)y+1
10 8
( )
1
5 = y2(cid:0)8y+8
8
( )
1
x = y2(cid:0)8y+16(cid:0)16+8
(cid:0)10 (cid:0)5 5 10 8
( )
1
............... (cid:0)5 = (y(cid:0)4)2(cid:0)8
8
1
Figure9.4: TheparaboladescribedinEx- = (y(cid:0)4)2(cid:0)1:
ample274.Thedistancesfromapointon 8
theparabolatothefocusanddirectrixis
given. Hencethevertexislocatedat((cid:0)1;4).Wehave 1 = 1,sop=2.Weconclude
8 4p
thatthefocusislocatedat(1;4)andthedirectrixisx = (cid:0)3. Theparabolais
graphedinFigure9.4,alongwithitsfocusanddirectrix.
The point (7;12) lies on the graph and is 7 (cid:0) ((cid:0)3) = 10 units from the
directrix.Thedistancefrom(7;12)tothefocusis:
√ p
(7(cid:0)1)2+(12(cid:0)4)2 = 100=10:
Indeed,thepointontheparabolaisequidistantfromthefocusanddirectrix. ..
Reflec(cid:415)veProperty
Oneofthefascina(cid:415)ngthingsaboutthenondegenerateconicsec(cid:415)onsistheir
reflec(cid:415)veproper(cid:415)es.Parabolashavethefollowingreflec(cid:415)veproperty:
Anyrayemana(cid:415)ngfromthefocusthatintersectstheparabola
.... reflectsoffalongalineperpendiculartothedirectrix.
ThisisillustratedinFigure9.5. Thefollowingtheoremstatesthismorerig-
Figure9.5: Illustra(cid:415)ngtheparabola’sre-
flec(cid:415)veproperty. orously.
Notes:
472