Calculus of a Single Variable

870 Pages·2013·46.711 MB·English

by  Ron Larson, Bruce H. Edwards

9781285060286_FES.qxp 10/1/12 1:48 PM Page 2
Index of Applications
Engineering and Physical Fluid force,541 Power,169
Sciences on a circular plate,502 Projectile motion,237,675,705
of gasoline,501,502 Radioactive decay,356,409,413,421,431
Acceleration,124,128,156,158,176,253 on a stern of a boat,502 Refrigeration,158
Air pressure,431 in a swimming pool,504,506 Ripples in a pond,149
Air traffic control,154,745,650 on a tank wall,501,502 Rolling a ball bearing,185
Aircraft glide path,193 of water,501 Satellite antenna,742
Angle of elevation,151,155,156 Force,289,501 Satellite orbit,694
Angular rate of change,374 Free-falling object,69,82,91 Satellites,127
Architecture,694 Gravitational force,577 Sending a space module into orbit,480,571
Area,116,126,153,256,603,674 Halley’s comet,694,737 Solar collector,693
Asteroid Apollo,738 Harmonic motion,36,38,138,353 Sound intensity,40,327,414
Atmospheric pressure and altitude,327,353 Heat transfer,336 Speed,29,175
Automobile aerodynamics,30 Height of sound,282
Average speed,40,89 of a baseball,29 Statics problems,494
Average velocity,112 of a basketball,32 Stopping distance,117,128,237
Beam deflection,693 Highway design,169,193 Surface area,153,158
Beam strength,35,222 Honeycomb,169 of an oil spill,443
Boiling temperature,35 Horizontal motion,355 of a pond,503
Boyle’s Law,485,504 Hyperbolic detection system,691 of a satellite-signal receiving dish,694
Breaking strength of a steel cable,364 Hyperbolic mirror,695 Surveying,311
Bridge design,694 Illumination,222,241 Suspension bridge,476
Building design,445,556 Inflating balloon,150 Temperature,18,176,204,344,405
Buoyant force,501 Kepler’s Laws,737,738 at which water boils,327
Car performance,35 Lawn sprinkler,169 Velocity,117,176,289,312
Carbon dating,413 Length,603 of a diver,113
Center of mass,of glass,496 of a catenary,473,503 of a piston,152
Centroid,494,495,502,519 of pursuit,476 of a rocket,582
Chemical mixture problem,427,429 of a stream,475 Velocity and acceleration,312,316,423
Chemical reaction,391,422,550 Linear and angular velocity,158 on the moon,160
Comet Hale-Bopp,741 Linear vs. angular speed,156 Velocity in a resisting medium,566
Construction,154 Lunar gravity,253 Vertical motion,116,157,174,175,250,
Depth Mass on the surface of Earth,486 252,382,392
of gasoline in a tank,503 Maximum area,219,220,221,222,224, Vibrating spring,157
of water in a swimming pool,153 240,242 Vibrating string,523
of water in a vase,29 Maximum cross-sectional area of an Volume,82,116,126,153
Distance,241 irrigation canal,223
of a box,30
Einstein’s Special Theory of Relativity Maximum volume,221,222,223
of fluid in a storage tank,540
and Newton’s First Law of Motion, of a box,215,216,220,222
of a pond,465
204 of a package,222
of a pontoon,461
Electric circuit,406,426,429 Minimum length,218,221,222,240
of a pyramid,452
Electric force,485 Minimum surface area,222
of a shampoo bottle,222
Electrical resistance,185 Minimum time,222,230
of a spherical ring,505
Electricity,155,303 Motion of a particle,712
of water in a conical tank,148
Electromagnetic theory,577 Moving ladder,154
Water flow,291
Emptying a tank of oil,481 Moving shadow,156,158,160
Error Navigation,695 Water running into a vase,193
in volume of a ball bearing,233 Orbit of Earth,708 Water tower,455
in volume and surface area of a cube, Parabolic reflector,684 Wheelchair ramp,12
236 Particle motion,128,287,290 Work,311,504
Explorer 18,694,741 Path of a projectile,182,712 done by an expanding gas,482
Explorer 55,694 Pendulum,138,237 done by a hydraulic cylinder,556
Falling object,34,315,426,429 Planetary motion,741 done in lifting a chain,482,484,504
Flow rate,286,355 Planetary orbits,687 done in splitting a piece of wood,485
(continued on back inside cover)
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DERIVATIVES AND INTEGRALS
s.
s Basic Differentiation Rules
e
c
Suc 1. d (cid:2)cu(cid:3)(cid:3)cu(cid:4) 2. d (cid:2)u ± v(cid:3)(cid:3)u(cid:4)± v(cid:4) 3. d (cid:2)uv(cid:3)(cid:3) uv(cid:4)(cid:5) vu(cid:4)
k dx dx dx
or d (cid:8)u(cid:9) vu(cid:4)(cid:2)uv(cid:4) d d
w 4. (cid:3) 5. (cid:2)c(cid:3)(cid:3)0 6. (cid:2)un(cid:3)(cid:3) nun(cid:2)1u(cid:4)
e dx v v2 dx dx
m
Ho 7. d (cid:2)x(cid:3)(cid:3)1 8. d (cid:2)(cid:4)u(cid:4)(cid:3)(cid:3) (cid:4)u(cid:4)(cid:6)u(cid:4)(cid:7), u(cid:6)0 9. d (cid:2)ln u(cid:3)(cid:3) u(cid:4)
r dx dx u dx u
o
ds f 10. d (cid:2)eu(cid:3)(cid:3)euu(cid:4) 11. d (cid:2)log u(cid:3)(cid:3) u(cid:4) 12. d (cid:2)au(cid:3)(cid:3)(cid:6)ln a(cid:7)auu(cid:4)
ar dx dx a (cid:6)ln a(cid:7)u dx
C
a 13. d (cid:2)sin u(cid:3)(cid:3)(cid:6)cos u(cid:7)u(cid:4) 14. d (cid:2)cos u(cid:3)(cid:3)(cid:2)(cid:6)sin u(cid:7)u(cid:4) 15. d (cid:2)tan u(cid:3)(cid:3)(cid:6)sec2 u(cid:7)u(cid:4)
ul dx dx dx
m
r d d d
Fo 16. (cid:2)cot u(cid:3)(cid:3)(cid:2)(cid:6)csc2 u(cid:7)u(cid:4) 17. (cid:2)sec u(cid:3)(cid:3)(cid:6)sec u tan u(cid:7)u(cid:4) 18. (cid:2)csc u(cid:3)(cid:3)(cid:2)(cid:6)csc u cot u(cid:7)u(cid:4)
ut dx dx dx
o d u(cid:4) d (cid:2)u(cid:4) d u(cid:4)
r 19. (cid:2)arcsin u(cid:3)(cid:3) 20. (cid:2)arccos u(cid:3)(cid:3) 21. (cid:2)arctan u(cid:3)(cid:3)
ea dx (cid:5)1(cid:2)u2 dx (cid:5)1(cid:2)u2 dx 1(cid:5)u2
T
d (cid:2)u(cid:4) d u(cid:4) d (cid:2)u(cid:4)
22. dx(cid:2)arccot u(cid:3)(cid:3) 1(cid:5)u2 23. dx(cid:2)arcsec u(cid:3)(cid:3) (cid:4)u(cid:4)(cid:5)u2(cid:2)1 24. dx(cid:2)arccsc u(cid:3)(cid:3) (cid:4)u(cid:4)(cid:5)u2(cid:2)1
d d d
25. (cid:2)sinh u(cid:3)(cid:3)(cid:6)cosh u(cid:7)u(cid:4) 26. (cid:2)cosh u(cid:3)(cid:3)(cid:6)sinh u(cid:7)u(cid:4) 27. (cid:2)tanh u(cid:3)(cid:3)(cid:6)sech2 u(cid:7)u(cid:4)
dx dx dx
d d d
28. (cid:2)coth u(cid:3)(cid:3)(cid:2)(cid:6)csch2 u(cid:7)u(cid:4) 29. (cid:2)sech u(cid:3)(cid:3)(cid:2)(cid:6)sech u tanh u(cid:7)u(cid:4) 30. (cid:2)csch u(cid:3)(cid:3)(cid:2)(cid:6)csch u coth u(cid:7)u(cid:4)
dx dx dx
d u(cid:4) d u(cid:4) d u(cid:4)
31. (cid:2)sinh(cid:2)1 u(cid:3)(cid:3) 32. (cid:2)cosh(cid:2)1 u(cid:3)(cid:3) 33. (cid:2)tanh(cid:2)1 u(cid:3)(cid:3)
dx (cid:5)u2(cid:5)1 dx (cid:5)u2(cid:2)1 dx 1(cid:2)u2
d u(cid:4) d (cid:2)u(cid:4) d (cid:2)u(cid:4)
34. dx(cid:2)coth(cid:2)1 u(cid:3)(cid:3) 1(cid:2)u2 35. dx(cid:2)sech(cid:2)1 u(cid:3)(cid:3) u(cid:5)1(cid:2)u2 36. dx(cid:2)csch(cid:2)1 u(cid:3)(cid:3) (cid:4)u(cid:4)(cid:5)1(cid:5)u2
Basic Integration Formulas
(cid:10) (cid:10) (cid:10) (cid:10) (cid:10)
1. kf(cid:6)u(cid:7) du(cid:3)k f(cid:6)u(cid:7) du 2. (cid:2)f(cid:6)u(cid:7) ± g(cid:6)u(cid:7)(cid:3) du(cid:3) f(cid:6)u(cid:7) du ± g(cid:6)u(cid:7) du
(cid:10) (cid:10)
un(cid:5)1
3. du(cid:3)u(cid:5)C 4. un du(cid:3) (cid:5)C, n(cid:6)(cid:2)1
(cid:10) (cid:10) n(cid:5)1
du (cid:4) (cid:4)
5. (cid:3)ln u (cid:5)C 6. eu du(cid:3)eu(cid:5)C
(cid:10)u (cid:10)
(cid:11) 1 (cid:12)
7. au du(cid:3) au(cid:5)C 8. sin u du (cid:3)(cid:2)cos u (cid:5)C
(cid:10) ln a (cid:10)
(cid:4) (cid:4)
9. cos u du (cid:3)sin u (cid:5)C 10. tan u du (cid:3)(cid:2)ln cos u (cid:5) C
(cid:10) (cid:10)
(cid:4) (cid:4) (cid:4) (cid:4)
11. cot u du (cid:3)ln sin u (cid:5)C 12. sec u du (cid:3)ln sec u (cid:5)tan u (cid:5) C
g (cid:10) (cid:10)
n
ni (cid:4) (cid:4)
ar 13. csc u du (cid:3)(cid:2)ln csc u (cid:5)cot u (cid:5)C 14. sec2 u du(cid:3)tan u (cid:5)C
e
L (cid:10) (cid:10)
e
g
ga 15. csc2 u du(cid:3)(cid:2)cot u (cid:5)C 16. sec u tan u du(cid:3)sec u (cid:5)C
en (cid:10) (cid:10)
C
Cole, 17. (cid:10)csc u cot u du(cid:3)(cid:2)csc u (cid:5)C 18. (cid:10)(cid:5)a2du(cid:2)u2 (cid:3)arcsin au (cid:5) C
/ (cid:4) (cid:4)
ks du 1 u du 1 u
o 19. (cid:3) arctan (cid:5)C 20. (cid:3) arcsec (cid:5) C
ro a2(cid:5)u2 a a u(cid:5)u2(cid:2)a2 a a
B
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9781285060286_FES.qxp 10/1/12 1:49 PM Page 4
TRIGONOMETRY
Definition of the Six Trigonometric Functions
Right triangle definitions,where 0 < (cid:7)< (cid:8)(cid:13)2. y
HAθy pdojtaecneunste Opposite c t saoinns (cid:7)(cid:7)(cid:7)(cid:3)(cid:3)(cid:3) hhooayyppdppppj scc esocct (cid:7)(cid:7)(cid:7)(cid:3)(cid:3)(cid:3) ohhaapyyddpppjj (−(−232,2,12)(−2212),56π23341π)51023°3π5°120°π2 9(00,° 610)°4π35°3(0π124°,π62(3)22(,2322, )21)
adj opp
0° 0
Circular function definitions,where (cid:7)is any angle. (−1, 0) π 180° 360° 2π (1, 0) x
y y r
(x, y) r = x2 + y2 sin (cid:7)(cid:3) xr csc (cid:7)(cid:3) yr (− 23, −12) 76π52π1202°5°240° 300°31353°70π°116π ( 23, −21)
y xr θ x c taons (cid:7)(cid:7)(cid:3)(cid:3) yr sc eoct (cid:7)(cid:7)(cid:3)(cid:3) xx (− 22,−(−212,)− 34) 43π 270° 3(20π, −1)53π(41, −( 322), − 22)
x y 2 2 2 2
Reciprocal Identities Double-Angle Formulas
1 1 1 sin 2u (cid:3)2 sin u cos u
sin x(cid:3) sec x(cid:3) tan x(cid:3)
csc x cos x cot x cos 2u (cid:3)cos2 u(cid:2)sin2 u(cid:3)2 cos2 u(cid:2) 1(cid:3) 1(cid:2) 2 sin2 u
1 1 1 2 tan u
csc x(cid:3) cos x(cid:3) cot x (cid:3) tan 2u (cid:3)
sin x sec x tan x 1(cid:2)tan2 u
Quotient Identities Power-Reducing Formulas
tan x (cid:3) sin x cot x (cid:3) cos x sin2 u(cid:3) 1(cid:2)cos 2u
cos x sin x 2
1(cid:5)cos 2u
Pythagorean Identities cos2 u(cid:3)
2
sin2 x(cid:5) cos2 x(cid:3) 1 1(cid:2)cos 2u
tan2 u(cid:3)
1(cid:5)tan2 x(cid:3) sec2 x 1(cid:5)cot2 x(cid:3)csc2 x 1(cid:5)cos 2u
Cofunction Identities Sum-to-Product Formulas
(cid:11)(cid:8) (cid:12) (cid:11)(cid:8) (cid:12) (cid:11)u(cid:5)v(cid:12) (cid:11)u(cid:2)v(cid:12)
sin (cid:2) x (cid:3) cos x cos (cid:2)x (cid:3)sin x sin u (cid:5)sin v (cid:3)2 sin cos
2 2 2 2
(cid:11)(cid:8) (cid:12) (cid:11)(cid:8) (cid:12) (cid:11)u(cid:5)v(cid:12) (cid:11)u(cid:2)v(cid:12)
csc (cid:2)x (cid:3) sec x tan (cid:2)x (cid:3)cot x sin u (cid:2)sin v (cid:3)2 cos sin
2 2 2 2
sec(cid:11)(cid:8)(cid:2)x(cid:12) (cid:3) csc x cot(cid:11)(cid:8)(cid:2)x(cid:12) (cid:3)tan x cos u (cid:5)cos v (cid:3)2 cos(cid:11)u(cid:5)2 v(cid:12) cos(cid:11)u(cid:2)2 v(cid:12)
2 2
(cid:11)u(cid:5)v(cid:12) (cid:11)u(cid:2)v(cid:12)
Even/Odd Identities cos u (cid:2)cos v (cid:3)(cid:2)2 sin sin
2 2
sin(cid:6)(cid:2)x(cid:7)(cid:3)(cid:2)sin x cos(cid:6)(cid:2)x(cid:7)(cid:3)cos x Product-to-Sum Formulas ning
csc(cid:6)(cid:2)x(cid:7)(cid:3)(cid:2)csc x tan(cid:6)(cid:2)x(cid:7)(cid:3)(cid:2)tan x ar
1 e
sec(cid:6)(cid:2)x(cid:7)(cid:3) sec x cot(cid:6)(cid:2)x(cid:7)(cid:3)(cid:2)cot x sin u sin v(cid:3) 2(cid:2)cos(cid:6)u(cid:2)v(cid:7)(cid:2)cos(cid:6)u(cid:5)v(cid:7)(cid:3) ge L
a
Sum and Difference Formulas cos u cos v(cid:3) 1(cid:2)cos(cid:6)u(cid:2)v(cid:7)(cid:5)cos(cid:6)u(cid:5) v(cid:7)(cid:3) eng
sin(cid:6)u ± v(cid:7)(cid:3) sin u cos v ± cos u sin v 2 e, C
cos(cid:6)u ± v(cid:7)(cid:3) cos u cos v (cid:9) sin u sin v sin u cos v(cid:3) 1(cid:2)sin(cid:6)u(cid:5)v(cid:7)(cid:5)sin(cid:6)u(cid:2)v(cid:7)(cid:3) Col
2 /
tan u ± tan v ks
tan(cid:6)u ± v(cid:7)(cid:3) 1 o
1 (cid:9) tan u tan v cos u sin v(cid:3) (cid:2)sin(cid:6)u(cid:5)v(cid:7)(cid:2)sin(cid:6)u(cid:2)v(cid:7)(cid:3) ro
2 B
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Calculus of a
Single Variable
10e
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Calculus of a
Single Variable
10e
Ron Larson
The Pennsylvania State University
The Behrend College
Bruce Edwards
University of Florida
Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States
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Calculus of a Single Variable © 2014, 2010, 2006 Brooks/Cole, Cengage Learning
Tenth Edition
ALL RIGHTS RESERVED. No part of this work covered by the copyright
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Contents
P
Preparation for Calculus 1
P.1 Graphs and Models 2
P.2 Linear Models and Rates of Change 10
P.3 Functions and Their Graphs 19
P.4 Fitting Models to Data 31
Review Exercises 37
P.S.Problem Solving 39
1
Limits and Their Properties 41
1.1 A Preview of Calculus 42
1.2 Finding Limits Graphically and Numerically 48
1.3 Evaluating Limits Analytically 59
1.4 Continuity and One-Sided Limits 70
1.5 Infinite Limits 83
Section Project:Graphs and Limits of
Trigonometric Functions 90
Review Exercises 91
P.S.Problem Solving 93
2
Differentiation 95
2.1 The Derivative and the Tangent Line Problem 96
2.2 Basic Differentiation Rules and Rates of Change 106
2.3 Product and Quotient Rules and Higher-Order
Derivatives 118
2.4 The Chain Rule 129
2.5 Implicit Differentiation 140
Section Project:Optical Illusions 147
2.6 Related Rates 148
Review Exercises 157
P.S.Problem Solving 159
v
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