CONTENTS
Introduction .....................................................................................................................1
Chapterb b 1. Functions .................................................................................................2
Chapterb b 2. LimitsbandbContinuity ............................................................................43
Chapterb b 3. ThebDerivative .......................................................................................65
Chapterb b 4. LogarithmicbandbExponentialbFunctions ................................................99
Chapterb b 5. AnalysisbofbFunctionsbandbTheirbGraphs ..............................................139
Chapterb b 6. ApplicationsbofbthebDerivative .............................................................177
Chapterb b 7. Integration ...........................................................................................209
Chapterb b 8. ApplicationsbofbthebDefinitebIntegral
inbGeometry,bScience,bandbEngineering ...............................................256
Chapterb b 9. PrinciplesbofbIntegralbEvaluation .........................................................292
Chapterb10. MathematicalbModelingbwithbDifferentialbEquations ...........................343
Chapterb11. InfinitebSeries ......................................................................................361
Chapterb12. AnalyticbGeometrybinbCalculus............................................................408
Chapterb13. Three-DimensionalbSpace;bVectors .....................................................448
Chapterb14. Vector-ValuedbFunctions ......................................................................490
Chapterb15. PartialbDerivatives ...............................................................................524
Chapterb16. MultiplebIntegrals ................................................................................573
Chapterb17. TopicsbinbVectorbCalculus ....................................................................608
AppendixbA.b b RealbNumbers,bIntervals,bandbInequalities ............................................640
AppendixbB. AbsolutebValue .................................................................................... 647
AppendixbC. CoordinatebPlanesbandbLines ...............................................................650
AppendixbD.b b Distance,bCircles,bandbQuadraticbEquations..........................................658
AppendixbE. TrigonometrybReview..........................................................................668
AppendixbF. SolvingbPolynomialbEquations ............................................................674
, CALCULUS:
A New Horizon from Ancient Roots
b b b b b
EXERCISEbSETbFORbINTRODUCTION
123 41
1. (a)b b xb=b0.123123123b.. b.;b 1000xb=b123.123123123b. . . b=b123b+bx;b 999xb=b123;b xb =
= 999 333
115
xb=b12.7777b.. b.;b 10xb=b127.7777b.. b.,b sob 9xb=b10xb−bxb=b115;b xb=b b
(b) 9
(c) xb=b38.07818181b.. b.;b100xb=b3807.818181b.. b.;b99xb=b3769.74;
3769.74 41886 20943
xb= = 376974 = =
99 9900 1100 550
4296 537
(d) 0.4296000b. . . b =b 0.4296b = =
10000 1250
repeats
22b ¸ x ` ˛b
2. (a)b πbisbirrational,bandbthusbhasbabnonrepeatingbdecimalbexpansion,bwhereasb =b3.b1 4 2 8 5 7 ...b
7
22
(b) b > b πb
7
bb √ ! bb √ !
223 333 63 17+ b15 5 b 355 22 63 17+ b15 5b
3. (a) < < √ < < (b) √
71 106 25 7 + b 15 5 113 7 25 7 + b 15 5
bb
√ !
333 63 17+ b15 5b
(c) (d) √
106 25 7 + b 15 5
b
8 b2 b 16 b2 256 2
4. (a)b b Ifbrb isbthebradius,bthenbDb=b2rb s D = r = r .b Theb areab ofb ab circleb ofb radiusb rb is
9 9 81
o
πr2b sob 256/81b wasb theb approximationb usedb forb π.
(b) 256/81b ≈b 3.16049,b 22/7b ≈b 3.14268,b andb πb ≈b 3.14159b sob 256/81b isb worseb thanb 22/7.
5. Theb firstb series,b takenb tob tenb terms,b addsb tob 3.0418;b theb second,b asb printed,b addsb tob 3.1416.
1 1 1 1 1 1 1
6. (a) =b0.111111b... b=b 10b+ 100 + 1000 + 10000 + 100000 + 1000000 +b... b
b
9
2 1 8 5 1 8 5
(b) b+ + + + + +b... b
b=b0.185185b... b=b2 10 100 1000 10000 100000 1000000
7 3 1 1 1 1 1
(c) 14 b+ + + + + +b... b
b=b0.311111b... b=b4 10 100 1000 10000 100000 1000000
5
6 3 6 3 6 3
7. (a) 7 b+ + + + + +b... b
b=b0.636363b... b=b1 10 100 1000 10000 100000 1000000
1 2 4 2 4 2 4
(b) 8 b+ + + + + +b... b
b=b0.242424b... b=b3
10 100 1000 10000 100000 1000000
3 4 1 6 6 6 6
(c) 5 b+ + + + + +b... b
10 100 1000 10000 100000 1000000
b=b0.416666b... b=b1
2
8. (a)b b 1,b 2,b 1.75,b 1.7321 (b)b b 1,b 3,b 2.33,b 2.238,b 2.2361
9. (a)b b 1,b 4,b 2.875,b 2.6549,b 2.6458 (b)b b 1,b 25.5,b 13.7,b 8.69,b 7.22,b 7.0726,b 7.0711
10. (a)b b Letb x1b =b2 1b(ab+bb),b x2b =2b 1b(ab+bx1),b x3b =2b 1b(ab+bx2),b etc.b Thenb b b>bx1b >bx2b >b· · · b>bxn−1b >bxnb > bab
sob allb theb xi’sb areb distinct,b thereb areb infinitelyb manyb ofb themb andb theyb allb lieb betweenb ab andb b.
(b) xb=b0.99999b.. b.,b 10xb=b9.99999b.. b.,b 9xb=b9,b xb= b1b
(c) (1.999999b.. b.)/2b=b0.999999b. . . b=b1;byesbitbisbconsistent,basballbthreebarebequal.
(d) 10xb= b9b+bx,b sob xb=b9/9b=b1.b Theyb areb equal.
,1
, CHAPTER 1
Functions
EXERCISEbSETb1.1
1. (a)b b aroundb 1943 (b)b b 1960;b 4200
(c)b b no;b youb needb theb year’sb population (d)b b war;b marketingb techniques
(e) newsb ofb healthb risk;b socialb pressure,b antismokingb campaigns,b increasedb taxation
2. (a)b b 1989;b $35,600 (b)b b 1983;b $32,000 (c)b b theb firstb twob years;b theb curveb isb steeperb (downhill)
3. (a)b b −2.9,b−2.0,b2.35,b2.9 (b)b b none (c)b b yb = b0b
(d)b b −1.75b≤bxb≤b2.15 (e)b b ymaxb=b2.8b atb xb=b−2.6;byminb=b−2.2b atb xb=b1.2
4. (a)b b xb =b −1,b4 (b)b b none (c)b b yb =b−1
(d)b b xb=b0,b3,b5 (e)b b ymaxb = b9b atb xb=b6;b yminb =b−2b atb xb= b0b
5. (a)b b xb=b2,b4 (b)b b none (c)b b xb≤b2;b 4b ≤bx (d)b b yminb=b−1;bnobmaximumbvalue
6. (a)b b xb= b9 (b)b b none (c)b b xb ≥b 25 (d)b yminb=b1;bnobmaximumbvalue
7. (a)b b b Breaksb couldb beb causedb byb war,b pestilence,b flood,b earthquakes,b forb example.
(b) Cb decreasesb forb eightb hours,b takesb ab jumpb upwards,b andb thenb repeats.
8. (a)b b Yes,b ifb theb thermometerb isb notb nearb ab windowb orb doorb orb otherb sourceb ofb suddenb temperatur
ebchange.
(b) Theb numberb isb alwaysb anb integer,b sob theb changesb areb inb movementsb (jumps)b ofb atb leastb oneb unit.
9. (a)b b Ifb theb sideb adjacentb tob theb buildingb hasb lengthb xb thenb Lb =b xb+b2y.b Sinceb Ab =b xyb =b 1000,
Lb=bxb+b2000/x.
(b) x b>b0bandbxbmustbbebsmallerbthanbthebwidthbofbthebbuilding,bwhichbwasbnotbgiven.
(c) 120 (d) Lminb ≈b 89.44
20 80
80
10. (a)b b Vb =blwhb=b(6b−b2x)(6b−b2x)x (b)b Frombthebfigurebitbisbclearbthatb0b< bx b<b3.
(c) 20 (d)b b Vmaxb ≈b16
0 3
0
2
Introduction .....................................................................................................................1
Chapterb b 1. Functions .................................................................................................2
Chapterb b 2. LimitsbandbContinuity ............................................................................43
Chapterb b 3. ThebDerivative .......................................................................................65
Chapterb b 4. LogarithmicbandbExponentialbFunctions ................................................99
Chapterb b 5. AnalysisbofbFunctionsbandbTheirbGraphs ..............................................139
Chapterb b 6. ApplicationsbofbthebDerivative .............................................................177
Chapterb b 7. Integration ...........................................................................................209
Chapterb b 8. ApplicationsbofbthebDefinitebIntegral
inbGeometry,bScience,bandbEngineering ...............................................256
Chapterb b 9. PrinciplesbofbIntegralbEvaluation .........................................................292
Chapterb10. MathematicalbModelingbwithbDifferentialbEquations ...........................343
Chapterb11. InfinitebSeries ......................................................................................361
Chapterb12. AnalyticbGeometrybinbCalculus............................................................408
Chapterb13. Three-DimensionalbSpace;bVectors .....................................................448
Chapterb14. Vector-ValuedbFunctions ......................................................................490
Chapterb15. PartialbDerivatives ...............................................................................524
Chapterb16. MultiplebIntegrals ................................................................................573
Chapterb17. TopicsbinbVectorbCalculus ....................................................................608
AppendixbA.b b RealbNumbers,bIntervals,bandbInequalities ............................................640
AppendixbB. AbsolutebValue .................................................................................... 647
AppendixbC. CoordinatebPlanesbandbLines ...............................................................650
AppendixbD.b b Distance,bCircles,bandbQuadraticbEquations..........................................658
AppendixbE. TrigonometrybReview..........................................................................668
AppendixbF. SolvingbPolynomialbEquations ............................................................674
, CALCULUS:
A New Horizon from Ancient Roots
b b b b b
EXERCISEbSETbFORbINTRODUCTION
123 41
1. (a)b b xb=b0.123123123b.. b.;b 1000xb=b123.123123123b. . . b=b123b+bx;b 999xb=b123;b xb =
= 999 333
115
xb=b12.7777b.. b.;b 10xb=b127.7777b.. b.,b sob 9xb=b10xb−bxb=b115;b xb=b b
(b) 9
(c) xb=b38.07818181b.. b.;b100xb=b3807.818181b.. b.;b99xb=b3769.74;
3769.74 41886 20943
xb= = 376974 = =
99 9900 1100 550
4296 537
(d) 0.4296000b. . . b =b 0.4296b = =
10000 1250
repeats
22b ¸ x ` ˛b
2. (a)b πbisbirrational,bandbthusbhasbabnonrepeatingbdecimalbexpansion,bwhereasb =b3.b1 4 2 8 5 7 ...b
7
22
(b) b > b πb
7
bb √ ! bb √ !
223 333 63 17+ b15 5 b 355 22 63 17+ b15 5b
3. (a) < < √ < < (b) √
71 106 25 7 + b 15 5 113 7 25 7 + b 15 5
bb
√ !
333 63 17+ b15 5b
(c) (d) √
106 25 7 + b 15 5
b
8 b2 b 16 b2 256 2
4. (a)b b Ifbrb isbthebradius,bthenbDb=b2rb s D = r = r .b Theb areab ofb ab circleb ofb radiusb rb is
9 9 81
o
πr2b sob 256/81b wasb theb approximationb usedb forb π.
(b) 256/81b ≈b 3.16049,b 22/7b ≈b 3.14268,b andb πb ≈b 3.14159b sob 256/81b isb worseb thanb 22/7.
5. Theb firstb series,b takenb tob tenb terms,b addsb tob 3.0418;b theb second,b asb printed,b addsb tob 3.1416.
1 1 1 1 1 1 1
6. (a) =b0.111111b... b=b 10b+ 100 + 1000 + 10000 + 100000 + 1000000 +b... b
b
9
2 1 8 5 1 8 5
(b) b+ + + + + +b... b
b=b0.185185b... b=b2 10 100 1000 10000 100000 1000000
7 3 1 1 1 1 1
(c) 14 b+ + + + + +b... b
b=b0.311111b... b=b4 10 100 1000 10000 100000 1000000
5
6 3 6 3 6 3
7. (a) 7 b+ + + + + +b... b
b=b0.636363b... b=b1 10 100 1000 10000 100000 1000000
1 2 4 2 4 2 4
(b) 8 b+ + + + + +b... b
b=b0.242424b... b=b3
10 100 1000 10000 100000 1000000
3 4 1 6 6 6 6
(c) 5 b+ + + + + +b... b
10 100 1000 10000 100000 1000000
b=b0.416666b... b=b1
2
8. (a)b b 1,b 2,b 1.75,b 1.7321 (b)b b 1,b 3,b 2.33,b 2.238,b 2.2361
9. (a)b b 1,b 4,b 2.875,b 2.6549,b 2.6458 (b)b b 1,b 25.5,b 13.7,b 8.69,b 7.22,b 7.0726,b 7.0711
10. (a)b b Letb x1b =b2 1b(ab+bb),b x2b =2b 1b(ab+bx1),b x3b =2b 1b(ab+bx2),b etc.b Thenb b b>bx1b >bx2b >b· · · b>bxn−1b >bxnb > bab
sob allb theb xi’sb areb distinct,b thereb areb infinitelyb manyb ofb themb andb theyb allb lieb betweenb ab andb b.
(b) xb=b0.99999b.. b.,b 10xb=b9.99999b.. b.,b 9xb=b9,b xb= b1b
(c) (1.999999b.. b.)/2b=b0.999999b. . . b=b1;byesbitbisbconsistent,basballbthreebarebequal.
(d) 10xb= b9b+bx,b sob xb=b9/9b=b1.b Theyb areb equal.
,1
, CHAPTER 1
Functions
EXERCISEbSETb1.1
1. (a)b b aroundb 1943 (b)b b 1960;b 4200
(c)b b no;b youb needb theb year’sb population (d)b b war;b marketingb techniques
(e) newsb ofb healthb risk;b socialb pressure,b antismokingb campaigns,b increasedb taxation
2. (a)b b 1989;b $35,600 (b)b b 1983;b $32,000 (c)b b theb firstb twob years;b theb curveb isb steeperb (downhill)
3. (a)b b −2.9,b−2.0,b2.35,b2.9 (b)b b none (c)b b yb = b0b
(d)b b −1.75b≤bxb≤b2.15 (e)b b ymaxb=b2.8b atb xb=b−2.6;byminb=b−2.2b atb xb=b1.2
4. (a)b b xb =b −1,b4 (b)b b none (c)b b yb =b−1
(d)b b xb=b0,b3,b5 (e)b b ymaxb = b9b atb xb=b6;b yminb =b−2b atb xb= b0b
5. (a)b b xb=b2,b4 (b)b b none (c)b b xb≤b2;b 4b ≤bx (d)b b yminb=b−1;bnobmaximumbvalue
6. (a)b b xb= b9 (b)b b none (c)b b xb ≥b 25 (d)b yminb=b1;bnobmaximumbvalue
7. (a)b b b Breaksb couldb beb causedb byb war,b pestilence,b flood,b earthquakes,b forb example.
(b) Cb decreasesb forb eightb hours,b takesb ab jumpb upwards,b andb thenb repeats.
8. (a)b b Yes,b ifb theb thermometerb isb notb nearb ab windowb orb doorb orb otherb sourceb ofb suddenb temperatur
ebchange.
(b) Theb numberb isb alwaysb anb integer,b sob theb changesb areb inb movementsb (jumps)b ofb atb leastb oneb unit.
9. (a)b b Ifb theb sideb adjacentb tob theb buildingb hasb lengthb xb thenb Lb =b xb+b2y.b Sinceb Ab =b xyb =b 1000,
Lb=bxb+b2000/x.
(b) x b>b0bandbxbmustbbebsmallerbthanbthebwidthbofbthebbuilding,bwhichbwasbnotbgiven.
(c) 120 (d) Lminb ≈b 89.44
20 80
80
10. (a)b b Vb =blwhb=b(6b−b2x)(6b−b2x)x (b)b Frombthebfigurebitbisbclearbthatb0b< bx b<b3.
(c) 20 (d)b b Vmaxb ≈b16
0 3
0
2
