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Stewart Calculus Early Transcendentals 7th txtbk.pdf
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CA L C U L U S
E A R L Y T R A N S C E N D E N T A L S
S E V E N T H E D I T I O N
JAMES STEWART
McMASTER UNIVERSITY
AND
UNIVERSITY OF TORONTO 2012
Contents
Preface xi
To the Student xxiii
Diagnostic Tests xxiv
A PREVIEW OF CALCULUS 1
1 Functions and Models 9
1.1 Four Ways to Represent a Function 10
1.2 Mathematical Models: A Catalog of Essential Functions 23
1.3 New Functions from Old Functions 36
1.4 Graphing Calculators and Computers 44
1.5 Exponential Functions 51
1.6 Inverse Functions and Logarithms 58
Review 72
Principles of Problem Solving 75
2 Limits and Derivatives 81
2.1 The Tangent and Velocity Problems 82
2.2 The Limit of a Function 87
2.3 Calculating Limits Using the Limit Laws 99
2.4 The Precise Definition of a Limit 108
2.5 Continuity 118
2.6 Limits at Infinity; Horizontal Asymptotes 130
2.7 Derivatives and Rates of Change 143
Writing Project N Early Methods for Finding Tangents 153
2.8 The Derivative as a Function 154
Review 165
Problems Plus 170
3.1 Derivatives of Polynomials and Exponential Functions 174
Applied Project N Building a Better Roller Coaster 184
3.2 The Product and Quotient Rules 184
3.3 Derivatives of Trigonometric Functions 191
3.4 The Chain Rule 198
Applied Project N Where Should a Pilot Start Descent? 208
3.5 Implicit Differentiation 209
Laboratory Project N Families of Implicit Curves 217
3.6 Derivatives of Logarithmic Functions 218
3.7 Rates of Change in the Natural and Social Sciences 224
3.8 Exponential Growth and Decay 237
3.9 Related Rates 244
3.10 Linear Approximations and Differentials 250
Laboratory Project N Taylor Polynomials 256
3.11 Hyperbolic Functions 257
Review 264
Problems Plus 268
4.1 Maximum and Minimum Values 274
Applied Project N The Calculus of Rainbows 282
4.2 The Mean Value Theorem 284
4.3 How Derivatives Affect the Shape of a Graph 290
4.4 Indeterminate Forms and l’Hospital’s Rule 301
Writing Project N The Origins of l’Hospital’s Rule 310
4.5 Summary of Curve Sketching 310
4.6 Graphing with Calculus and Calculators 318
4.7 Optimization Problems 325
Applied Project N The Shape of a Can 337
4.8 Newton’s Method 338
4.9 Antiderivatives 344
Review 351
Problems Plus 355
5.1 Areas and Distances 360
5.2 The Definite Integral 371
Discovery Project N Area Functions 385
5.3 The Fundamental Theorem of Calculus 386
5.4 Indefinite Integrals and the Net Change Theorem 397
Writing Project N Newton, Leibniz, and the Invention of Calculus 406
5.5 The Substitution Rule 407
Review 415
Problems Plus 419
6.1 Areas Between Curves 422
Applied Project N The Gini Index 429
6.2 Volumes 430
6.3 Volumes by Cylindrical Shells 441
6.4 Work 446
6.5 Average Value of a Function 451
Applied Project N Calculus and Baseball 455
Applied Project N Where to Sit at the Movies 456
Review 457
Problems Plus 459
7.1 Integration by Parts 464
7.2 Trigonometric Integrals 471
7.3 Trigonometric Substitution 478
7.4 Integration of Rational Functions by Partial Fractions 484
7.5 Strategy for Integration 494
7.6 Integration Using Tables and Computer Algebra Systems 500
Discovery Project N Patterns in Integrals 505
7.7 Approximate Integration 506
7.8 Improper Integrals 519
Review 529
Problems Plus 533
8.1 Arc Length 538
Discovery Project N Arc Length Contest 545
8.2 Area of a Surface of Revolution 545
Discovery Project N Rotating on a Slant 551
8.3 Applications to Physics and Engineering 552
Discovery Project N Complementary Coffee Cups 562
8.4 Applications to Economics and Biology 563
8.5 Probability 568
Review 575
Problems Plus 577
9.1 Modeling with Differential Equations 580
9.2 Direction Fields and Euler’s Method 585
9.3 Separable Equations 594
Applied Project N How Fast Does a Tank Drain? 603
Applied Project N Which Is Faster, Going Up or Coming Down? 604
9.4 Models for Population Growth 605
9.5 Linear Equations 616
9.6 Predator-Prey Systems 622
Review 629
Problems Plus 633
10.1 Curves Defined by Parametric Equations 636
Laboratory Project N Running Circles around Circles 644
10.2 Calculus with Parametric Curves 645
Laboratory Project N Bézier Curves 653
10.3 Polar Coordinates 654
Laboratory Project N Families of Polar Curves 664
10.4 Areas and Lengths in Polar Coordinates 665
10.5 Conic Sections 670
10.6 Conic Sections in Polar Coordinates 678
Review 685
Problems Plus 688
11.1 Sequences 690
Laboratory Project N Logistic Sequences 703
11.2 Series 703
11.3 The Integral Test and Estimates of Sums 714
11.4 The Comparison Tests 722
11.5 Alternating Series 727
11.6 Absolute Convergence and the Ratio and Root Tests 732
11.7 Strategy for Testing Series 739
11.8 Power Series 741
11.9 Representations of Functions as Power Series 746
11.10 Taylor and Maclaurin Series 753
Laboratory Project N An Elusive Limit 767
Writing Project N How Newton Discovered the Binomial Series 767
11.11 Applications of Taylor Polynomials 768
Applied Project N Radiation from the Stars 777
Review 778
Problems Plus 781
10 Parametric Equations and Polar Coordinates 635
11 Infinite Sequences and Series 689
12.1 Three-Dimensional Coordinate Systems 786
12.2 Vectors 791
12.3 The Dot Product 800
12.4 The Cross Product 808
Discovery Project N The Geometry of a Tetrahedron 816
12.5 Equations of Lines and Planes 816
Laboratory Project N Putting 3D in Perspective 826
12.6 Cylinders and Quadric Surfaces 827
Review 834
Problems Plus 837
13.1 Vector Functions and Space Curves 840
13.2 Derivatives and Integrals of Vector Functions 847
13.3 Arc Length and Curvature 853
13.4 Motion in Space: Velocity and Acceleration 862
Applied Project N Kepler’s Laws 872
Review 873
Problems Plus 876
14.1 Functions of Several Variables 878
14.2 Limits and Continuity 892
14.3 Partial Derivatives 900
14.4 Tangent Planes and Linear Approximations 915
14.5 The Chain Rule 924
14.6 Directional Derivatives and the Gradient Vector 933
14.7 Maximum and Minimum Values 946
Applied Project N Designing a Dumpster 956
Discovery Project N Quadratic Approximations and Critical Points 956
14.8 Lagrange Multipliers 957
Applied Project N Rocket Science 964
Applied Project N Hydro-Turbine Optimization 966
Review 967
Problems Plus 971
15.1 Double Integrals over Rectangles 974
15.2 Iterated Integrals 982
15.3 Double Integrals over General Regions 988
15.4 Double Integrals in Polar Coordinates 997
15.5 Applications of Double Integrals 1003
15.6 Surface Area 1013
15.7 Triple Integrals 1017
Discovery Project N Volumes of Hyperspheres 1027
15.8 Triple Integrals in Cylindrical Coordinates 1027
Discovery Project N The Intersection of Three Cylinders 1032
15.9 Triple Integrals in Spherical Coordinates 1033
Applied Project N Roller Derby 1039
15.10 Change of Variables in Multiple Integrals 1040
Review 1049
Problems Plus 1053
16.1 Vector Fields 1056
16.2 Line Integrals 1063
16.3 The Fundamental Theorem for Line Integrals 1075
16.4 Green’s Theorem 1084
16.5 Curl and Divergence 1091
16.6 Parametric Surfaces and Their Areas 1099
16.7 Surface Integrals 1110
16.8 Stokes’ Theorem 1122
Writing Project N Three Men and Two Theorems 1128
16.9 The Divergence Theorem 1128
16.10 Summary 1135
Review 1136
Problems Plus 1139
17.1 Second-Order Linear Equations 1142
17.2 Nonhomogeneous Linear Equations 1148
17.3 Applications of Second-Order Differential Equations 1156
17.4 Series Solutions 1164
Review 1169
A Numbers, Inequalities, and Absolute Values A2
B Coordinate Geometry and Lines A10
C Graphs of Second-Degree Equations A16
D Trigonometry A24
E Sigma Notation A34
F Proofs of Theorems A39
G The Logarithm Defined as an Integral A50
H Complex Numbers A57
I Answers to Odd-Numbered Exercises A65
Diagnostic Tests The book begins with four diagnostic tests, in Basic Algebra, Analytic Geometry, Functions,
and Trigonometry.
A Preview of Calculus This is an overview of the subject and includes a list of questions to motivate the study of
calculus.
1 Functions and Models From the beginning, multiple representations of functions are stressed: verbal, numerical,
visual, and algebraic. A discussion of mathematical models leads to a review of the standard
functions, including exponential and logarithmic functions, from these four points of
view.
2 Limits and Derivatives The material on limits is motivated by a prior discussion of the tangent and velocity problems.
Limits are treated from descriptive, graphical, numerical, and algebraic points of
view. Section 2.4, on the precise definition of a limit, is an optional section. Sections
2.7 and 2.8 deal with derivatives (especially with functions defined graphically and numerically)
before the differentiation rules are covered in Chapter 3. Here the examples and
exercises explore the meanings of derivatives in various contexts. Higher derivatives are
introduced in Section 2.8.
3 Differentiation Rules All the basic functions, including exponential, logarithmic, and inverse trigonometric functions,
are differentiated here. When derivatives are computed in applied situations, students
are asked to explain their meanings. Exponential growth and decay are covered in this
chapter.
4 Applications of Differentiation The basic facts concerning extreme values and shapes of curves are deduced from the
Mean Value Theorem. Graphing with technology emphasizes the interaction between calculus
and calculators and the analysis of families of curves. Some substantial optimization
problems are provided, including an explanation of why you need to raise your head 42°
to see the top of a rainbow.
5 Integrals The area problem and the distance problem serve to motivate the definite integral, with
sigma notation introduced as needed. (Full coverage of sigma notation is provided in
Appendix E.) Emphasis is placed on explaining the meanings of integrals in various contexts
and on estimating their values from graphs and tables.
6 Applications of Integration Here I present the applications of integration—area, volume, work, average value—that
can reasonably be done without specialized techniques of integration. General methods are
emphasized. The goal is for students to be able to divide a quantity into small pieces, estimate
with Riemann sums, and recognize the limit as an integral.
7 Techniques of Integration All the standard methods are covered but, of course, the real challenge is to be able to
recognize which technique is best used in a given situation. Accordingly, in Section 7.5, I
present a strategy for integration. The use of computer algebra systems is discussed in
Section 7.6.
8 Further Applications of Integration
Here are the applications of integration—arc length and surface area—for which it is useful
to have available all the techniques of integration, as well as applications to biology,
economics, and physics (hydrostatic force and centers of mass). I have also included a section
on probability. There are more applications here than can realistically be covered in a
given course. Instructors should select applications suitable for their students and for
which they themselves have enthusiasm.
9 Differential Equations Modeling is the theme that unifies this introductory treatment of differential equations.
Direction fields and Euler’s method are studied before separable and linear equations are
solved explicitly, so that qualitative, numerical, and analytic approaches are given equal
consideration. These methods are applied to the exponential, logistic, and other models for
population growth. The first four or five sections of this chapter serve as a good introduction
to first-order differential equations. An optional final section uses predator-prey models
to illustrate systems of differential equations.
10 Parametric Equations and Polar Coordinates
This chapter introduces parametric and polar curves and applies the methods of calculus
to them. Parametric curves are well suited to laboratory projects; the three presented here
involve families of curves and Bézier curves. A brief treatment of conic sections in polar
coordinates prepares the way for Kepler’s Laws in Chapter 13.
11 Infinite Sequences and Series The convergence tests have intuitive justifications (see page 714) as well as formal proofs.
Numerical estimates of sums of series are based on which test was used to prove convergence.
The emphasis is on Taylor series and polynomials and their applications to physics.
Error estimates include those from graphing devices.
12 Vectors and The Geometry of Space
The material on three-dimensional analytic geometry and vectors is divided into two chapters.
Chapter 12 deals with vectors, the dot and cross products, lines, planes, and surfaces.
13 Vector Functions This chapter covers vector-valued functions, their derivatives and integrals, the length and
curvature of space curves, and velocity and acceleration along space curves, culminating
in Kepler’s laws.
14 Partial Derivatives Functions of two or more variables are studied from verbal, numerical, visual, and algebraic
points of view. In particular, I introduce partial derivatives by looking at a specific
column in a table of values of the heat index (perceived air temperature) as a function of
the actual temperature and the relative humidity.
15 Multiple Integrals Contour maps and the Midpoint Rule are used to estimate the average snowfall and average
temperature in given regions. Double and triple integrals are used to compute probabilities,
surface areas, and (in projects) volumes of hyperspheres and volumes of intersections of
three cylinders. Cylindrical and spherical coordinates are introduced in the context of evaluating
triple integrals.
16 Vector Calculus Vector fields are introduced through pictures of velocity fields showing San Francisco Bay
wind patterns. The similarities among the Fundamental Theorem for line integrals, Green’s
Theorem, Stokes’ Theorem, and the Divergence Theorem are emphasized.
17 Second-Order Differential Equations
Since first-order differential equations are covered in Chapter 9, this final chapter deals
with second-order linear differential equations, their application to vibrating springs and
electric circuits, and series solutions.
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