SOLUTIONbMANUAL
, Contents
Chapterb0. BeforebCalculus..............................................................................................................................1
Chapterb1. LimitsbandbContinuity ............................................................................................. 39
Chapterb2. ThebDerivative......................................................................................................... 71
Chapterb3. TopicsbinbDifferentiation ........................................................................................ 109
Chapterb4. ThebDerivativebinbGraphingbandbApplications ...................................................... 153
Chapterb5. Integration............................................................................................................... 243
Chapterb6.
ApplicationsbofbthebDefinitebIntegralbinbGeometry,bScience,bandbEngineering…b30
5bChapterb7. PrincipalsbofbIntegralbEvaluation ...........................................................................................363
Chapterb8. MathematicalbModelingbwithbDifferentialbEquations.....................................................413
Chapterb9. InfinitebSeries .................................................................................................................................437
Chapterb10. ParametricbandbPolarbCurves;bConicbSections ..................................................................485
Chapterb11. Three-DimensionalbSpace;bVectors ....................................................................... 545
Chapterb12. Vector-ValuedbFunctions........................................................................................ 589
Chapterb13. PartialbDerivatives........................................................................................................................627
Chapterb14. MultiplebIntegrals .........................................................................................................................675
Chapterb15. TopicsbinbVectorbCalculus ..........................................................................................................713
AppendixbA. GraphingbFunctionsbUsingbCalculatorsbandbComputerbAlgebrabSystems ..............745
AppendixbB. TrigonometrybReview.................................................................................................................753
AppendixbC. SolvingbPolynomialbEquations.................................................................................................759
,Before Calculus b
Exercise Set 0.1
b b
1. (a)b —2.9,b—2.0,b2.35,b2.9 (b)b b None (c)b b yb =b 0 (d)b —1.75b≤bxb≤b2.15,b xb=b—3,b xb=b3
(e) ymaxb=b2.8b atb xb=b—2.6;b yminb=b—2.2b atb xb=b1.2
2. (a)b xb=b—1,b4 (b)b None (c)b yb=b—1 (d)b xb=b0,b3,b5
(e) ymaxb=b9b atb xb=b6;b yminb=b—2b atb xb=b0
3. (a)b b Yes (b)b b Yes (c)b Nob(verticalblinebtestbfails) (d)b Nob(verticalblinebtestbfails)
4. (a)bThebnaturalbdomainbofbfb isbxb/=
1,bandbforbgbitbisbthebsetbofballbx.b fb(x)b=bg(x)bonbthebintersectionbofbthe
irbdomains. —
(b)bTheb domainbofbfb isb thebsetbofb allbxb≥b0;btheb domainbofbgb isb theb same,bandbfb(x)b=bg(x).
5. (a)b 1999,b $47,700 (b)b 1993,b $41,600
(c)bbThebslopebbetweenb2000bandb2001bisbsteeperbthanbthebslopebbetweenb2001bandb2002,bsobthebmedianbincomebw
asbdecliningbmorebrapidlybduringbthebfirstbyearbofbtheb2-yearbperiod.
47.7b—b41.6b 6.1b
6. (a)bInbthousands,bapproximatelyb =b perbyr,borb$1017/yr.
6 6
(b) Fromb1993btob1996bthebmedianbincomebincreasedbfromb$41.6Kbtob$44Kb(Kbforb‘kilodollars’;ballbfiguresbappro
x- —
—bwasb(44b 41.6)/3bK/yrb=b2.4/3
imate);bthebaveragebratebofbincreasebduringbthisbtime ≈ bK/yrb=b$800/year.b Fromb
1996btob1999bthebaveragebratebofbincreasebwasb(47.7b 44)/3bK/yrb=b3.7/3bK/yrb $1233/year.bThebincreasebwas
blargerbduringbtheblastb3byearsbofbthebperiod.
(c) 1994b andb 2005.
√
7. ( a √) b fb(0)b=b3(0)2b—b2b=b—2;bfb(2)b=b3(2)2b—b2b=b10;bfb(—2)b=b3(—2)2b—b2b=b10;bfb(3)b=b3(3)2b—b2b=b25;bfb( 2)b=
3(b b 2)2b —b2b=b4;b fb(3t)b=b3(3t)2b —b2b=b27t2b —b2.
√ √
(b)b fb(0)b=b2(0)b=b0;bfb(2)b=b2(2)b=b4;bfb(—2)b=b2(—2)b=b—4;bfb(3)b=b2(3)b=b6;bfb( 2)b=b2 2;b fb(3t)b=b1/(3t)bfor
tb>b1b andbfb(3t)b=b6tb forb tb≤b1.
3b+b =b 2;b g(—1)b =b — =b 0;b g(π)b =b
πb+b1b
;b g(—1.1)b =b
— —0.1 =b b1b ;b g(t2b —b 1)b =
1 =b
8. (a)b g(3)b = 1b+b1 1.1b+b1
3b—1 —1b—b1 πb —b1 —1.1b—b1 —2.1 21
t2b— t2
b1b+b1
=b 2b .
t2b—b1b—b1b t —b2b
, √ √
(b)
√ g(3)b =b 3b+b1b =b 2;b g(—1)b =b 3;b g(π)b =b πb+b1;b g(—1.1)b =b 3;b g(t2b —b 1)b =b 3b ifb t2b <b 2b andb g(t2b —b 1)b =
t2b —b1b+b1b=b|t|b ifb t2b≥b2.
1
, Contents
Chapterb0. BeforebCalculus..............................................................................................................................1
Chapterb1. LimitsbandbContinuity ............................................................................................. 39
Chapterb2. ThebDerivative......................................................................................................... 71
Chapterb3. TopicsbinbDifferentiation ........................................................................................ 109
Chapterb4. ThebDerivativebinbGraphingbandbApplications ...................................................... 153
Chapterb5. Integration............................................................................................................... 243
Chapterb6.
ApplicationsbofbthebDefinitebIntegralbinbGeometry,bScience,bandbEngineering…b30
5bChapterb7. PrincipalsbofbIntegralbEvaluation ...........................................................................................363
Chapterb8. MathematicalbModelingbwithbDifferentialbEquations.....................................................413
Chapterb9. InfinitebSeries .................................................................................................................................437
Chapterb10. ParametricbandbPolarbCurves;bConicbSections ..................................................................485
Chapterb11. Three-DimensionalbSpace;bVectors ....................................................................... 545
Chapterb12. Vector-ValuedbFunctions........................................................................................ 589
Chapterb13. PartialbDerivatives........................................................................................................................627
Chapterb14. MultiplebIntegrals .........................................................................................................................675
Chapterb15. TopicsbinbVectorbCalculus ..........................................................................................................713
AppendixbA. GraphingbFunctionsbUsingbCalculatorsbandbComputerbAlgebrabSystems ..............745
AppendixbB. TrigonometrybReview.................................................................................................................753
AppendixbC. SolvingbPolynomialbEquations.................................................................................................759
,Before Calculus b
Exercise Set 0.1
b b
1. (a)b —2.9,b—2.0,b2.35,b2.9 (b)b b None (c)b b yb =b 0 (d)b —1.75b≤bxb≤b2.15,b xb=b—3,b xb=b3
(e) ymaxb=b2.8b atb xb=b—2.6;b yminb=b—2.2b atb xb=b1.2
2. (a)b xb=b—1,b4 (b)b None (c)b yb=b—1 (d)b xb=b0,b3,b5
(e) ymaxb=b9b atb xb=b6;b yminb=b—2b atb xb=b0
3. (a)b b Yes (b)b b Yes (c)b Nob(verticalblinebtestbfails) (d)b Nob(verticalblinebtestbfails)
4. (a)bThebnaturalbdomainbofbfb isbxb/=
1,bandbforbgbitbisbthebsetbofballbx.b fb(x)b=bg(x)bonbthebintersectionbofbthe
irbdomains. —
(b)bTheb domainbofbfb isb thebsetbofb allbxb≥b0;btheb domainbofbgb isb theb same,bandbfb(x)b=bg(x).
5. (a)b 1999,b $47,700 (b)b 1993,b $41,600
(c)bbThebslopebbetweenb2000bandb2001bisbsteeperbthanbthebslopebbetweenb2001bandb2002,bsobthebmedianbincomebw
asbdecliningbmorebrapidlybduringbthebfirstbyearbofbtheb2-yearbperiod.
47.7b—b41.6b 6.1b
6. (a)bInbthousands,bapproximatelyb =b perbyr,borb$1017/yr.
6 6
(b) Fromb1993btob1996bthebmedianbincomebincreasedbfromb$41.6Kbtob$44Kb(Kbforb‘kilodollars’;ballbfiguresbappro
x- —
—bwasb(44b 41.6)/3bK/yrb=b2.4/3
imate);bthebaveragebratebofbincreasebduringbthisbtime ≈ bK/yrb=b$800/year.b Fromb
1996btob1999bthebaveragebratebofbincreasebwasb(47.7b 44)/3bK/yrb=b3.7/3bK/yrb $1233/year.bThebincreasebwas
blargerbduringbtheblastb3byearsbofbthebperiod.
(c) 1994b andb 2005.
√
7. ( a √) b fb(0)b=b3(0)2b—b2b=b—2;bfb(2)b=b3(2)2b—b2b=b10;bfb(—2)b=b3(—2)2b—b2b=b10;bfb(3)b=b3(3)2b—b2b=b25;bfb( 2)b=
3(b b 2)2b —b2b=b4;b fb(3t)b=b3(3t)2b —b2b=b27t2b —b2.
√ √
(b)b fb(0)b=b2(0)b=b0;bfb(2)b=b2(2)b=b4;bfb(—2)b=b2(—2)b=b—4;bfb(3)b=b2(3)b=b6;bfb( 2)b=b2 2;b fb(3t)b=b1/(3t)bfor
tb>b1b andbfb(3t)b=b6tb forb tb≤b1.
3b+b =b 2;b g(—1)b =b — =b 0;b g(π)b =b
πb+b1b
;b g(—1.1)b =b
— —0.1 =b b1b ;b g(t2b —b 1)b =
1 =b
8. (a)b g(3)b = 1b+b1 1.1b+b1
3b—1 —1b—b1 πb —b1 —1.1b—b1 —2.1 21
t2b— t2
b1b+b1
=b 2b .
t2b—b1b—b1b t —b2b
, √ √
(b)
√ g(3)b =b 3b+b1b =b 2;b g(—1)b =b 3;b g(π)b =b πb+b1;b g(—1.1)b =b 3;b g(t2b —b 1)b =b 3b ifb t2b <b 2b andb g(t2b —b 1)b =
t2b —b1b+b1b=b|t|b ifb t2b≥b2.
1
