1 LIMITS AND CONTINUITY

1 LIMITS (An Intuitive Approach)

13 Computing Limits

22 Limits at Infinity , End Behavior of a function

31 Limits (Discussed more Rigorously)

40 Continuity

51 Continuity of Trigonometric Functions

56 Inverse Trigonometric Functions

63 Exponential and Logarithmic Functions

79 THE DERIATIVE

79 Tangent Lines and Rates of Changes

89 The Derivation function

100 Introduction to Techniques of Differentiation

113 Derivatives of Trigonometric Functions

118 The Chain Rule

129 TOPICS IN DIFFERENTIATION

129 Implicit Differentiation

135 Derivatives of Logarithmic functions

140 Derivatives of exponential and inverse Trigonometric Functions

Related Rates 146

Local Linear Approximation Differentials 153

L'Hopital's Rule: Indeterminate Forms 160

THE DIRIVATIVE IN GRAPHING AND APPLICATIONS 172

Analysis of Functions I: Increase, Decrease, and Concavity 172

Analysis of Functions II: Relative Extreme Graphing Polynomials 183

Analysis of Functions III: Rational Functions, Cusps and Vertical Tangents 193

Absolute Maxima and Minima 204

Applied Maximum and Minimum Problem 212

Rectilinear motion 226

Newton's Methods 233

Rolle's Theorem ; Mean-Value Theorem 239

INTEGRATION 253

An Overview of the Area Problem 253

The Indefinite Integral 268

The Fundamental Theorem of calculus 294

Rectilinear Motion Revisited Using Integration 306

Average Value of a Function and its Application 314

Evaluating Definite Integrals by Substitution 319

Logarithmic and other Functions Defined by integrals 325

APPLICATION OF THE DEFINITE INTEGRAL IN GEOMETRY SCIENCE AND ENGINEERING 341

Area Between Two Curves 341

Volumes by Slicing : Disks and Washes 349

Volumes by Cylindrical Shells 358

Length of a Plane Curve 364

Area of a Surface of Revolution 370

Work 375

Moments, Centers of Gravity and Centroids 363

Fluid pressure and force 392

Hyperbolic Functions and hanging cables 398

PRINCIPLES OF INTEGRAL EVALUATION 412

An overview of Integration methods 412

Integration by parts 415

Integrating Trigonometric Functions 423

Trigonometric Substitution 431

Integration Rational Functions by partial fractions 437

Using Computer Algebra Systems and Tables of Integrals 445

Numerical Integration Simpsons Rule 454

Improper integrals 467

MATHEMATICAL MODELING WITH DIFFERENTIAL EQUATIONS 481

Modeling with Differential Equations 481

Separation of Variables 487

Slope Fields Euler's Method 498

First Order Differential Equations and Applications 504

INFINITE SERIES 514

Sequences 514

Monotone sequences 524

Infinite Series 531

Convergence Test 539

The Comparison, Ratio and root Tests 547

Alternating Series: Absolute and Conditional Convergence 553

Maclaurin and Taylor Polynomials 563

Maclaurin and Taylor Series: Power Series 573

Convergence of Taylor Series 582

Differentiating and Integrating Power Series Modeling with Taylor Series 591

PARAMETRIC AND POLAR CURVES: CONIC SECTIONS 605

Polar Coordinates 617

Tangent Lines, Arc Length and Area for Polar Curves 630

Conic Sections 640

657 Rotation of Axes: Second-Degree Equations

662 Conic Sections in Polar Coordinates

674 THREE-DIMENSIONAL SPACE: VECTORS

674 Rectangular Coordinates in 3-Space: Spheres: Cylindrical Surfaces

680 Vectors

691 Dot Product: Projections

700 Cross Product

710 Parametric Equations of Lines

717 Planes in 3-Space

725 Quadric Surfaces

735 Cylindrical and Spherical Coordinates

744 VECTOR-VALUED FUNCTIONS

744 Introduction to Vector-Valued Functions

750 Calculus of Vector - Valued Functions

759 Change of Parameter: Arc Length

768 Unit Tangent. Normal, and Binormal Vectors

773 Curvature

781 Motion Along a Curve

794 Kepler's Laws of Planetary Motion

805 PARATIAL DERIVATIVES

805 Functions of Two or More Variables

815 Limits and Continuity

825 Partial Derivatives

837 Differentiability, Differentials, and Local Linearity

845 The Chain Rule

855 Directional Derivatives and Gradients

866 Tangent Planes and Normal Vectors

872 Maxima and Minima of Functions of Two Variables

883 Lagrange Multipliers

894 MULTIPLE INTEGRALS

894 Double Integrals

902 Double Integrals over Nonrectangular Regions

910 Double Integral in Polar Coordinates

918 Surface Area: Parametric Surfaces

930 Triple Integral

938 Triple Integral in Cylindrical and Spherical Coordinates

947 Change of Variables in Multiple Integrals: Jacobians

959 Centers of Gravity Using Multiple Integrals

971 TOPICS IN VECTOR CALCULUS

971 Vector Fields

980 Line Integrals

995 Independence of Path: Conservative Vector Fields

1005 Green's Theorem

1013 Surface Integrals

1021 Applications of Surface Integrals: Flux

1030 The Divergence Theorem

1039 Stokes' Theorem

APPENDICES

A1 TRIGONOMETRY REVIEW (SUMMARY)

A8 FUNCTIONS (SUMMARY)

A11 NEW FUNCTION FROM OLD (SUMMARY)

A16 FAMILIES OF FUNCTIONS (SUMMARY)

A23 INVERSE FUNCTIONS (SUMMARY)

A28 ANSWERS TO ODD- NUMBERED EXCERCISES

I-1 INDEX