
Coordinates, Graphs, Lines 


1  (50) 

Real Numbers, Intervals, and Inequalities (A Review) 


1  (11) 


12  (6) 

Coordinate Planes and Graphs 


18  (9) 


27  (11) 

Distance; Circles; Equations of the Form y = ax2 + bx + c 


38  (13) 


51  (70) 


51  (10) 


61  (6) 


67  (11) 

Limits (An Intuitive Introduction) 


78  (9) 

Limits (Computational Techniques) 


87  (11) 

Limits: A Rigorous Approach 


98  (6) 


104  (7) 

Limits and Continuity of Trigonometric Functions 


111  (10) 


121  (58) 

Tangent Lines and Rates of Change 


121  (8) 


129  (11) 

Techniques of Differentiation 


140  (12) 

Derivatives of Trigonometric Functions 


152  (4) 


156  (7) 


163  (6) 

ΔNotation; Differentials 


169  (10) 

Applications of Differentiation 


179  (68) 


179  (6) 

Intervals of Increase and Decrease; Concavity 


185  (6) 

Relative Extrema; First and Second Derivative Tests 


191  (6) 

Graphs of Polynomials and Rational Functions 


197  (8) 


205  (2) 

Maximum and Minimum Values of a Function 


207  (8) 

Applied Maximum and Minimum Problems 


215  (14) 


229  (4) 

Rolle's Theorem; MeanValue Theorem 


233  (5) 

Motion Along a Line (Rectilinear Motion) 


238  (9) 


247  (60) 


247  (3) 

Antiderivatives; The Indefinite Integral 


250  (7) 

Integration by Substitution 


257  (5) 


262  (7) 


269  (6) 


275  (11) 

The First Fundamental Theorem of Calculus 


286  (7) 

Evaluating Definite Integrals by Substitution; Approximation by Riemann Sums 


293  (5) 

The Second Fundamental Theorem of Calculus 


298  (9) 

Applications of the Definite Integral 


307  (42) 


307  (6) 

Volumes by Slicing; Disks and Washers 


313  (8) 

Volumes by Cylindrical Shells 


321  (5) 


326  (3) 

Area of a Surface of Revolution 


329  (4) 

Application of Integration to Rectilinear Motion 


333  (5) 


338  (4) 


342  (7) 

Logarithmic and Exponential Functions 


349  (64) 

Logarithms and Exponents (An Overview) 


349  (10) 

Derivatives and Integrals of Logarithmic and Exponential Functions 


359  (9) 

Graphs Involving Exponentials and Logarithms 


368  (6) 


374  (9) 

Logarithmic and Exponential Functions (Rigorous Approach) 


383  (9) 


392  (5) 

FirstOrder Differential Equations and Applications 


397  (16) 

Inverse Trigonometric and Hyperbolic Functions 


413  (22) 

Inverse Trigonometric Functions 


413  (8) 

Derivatives and Integrals Involving Inverse Trigonometric Functions 


421  (6) 

Inverse Hyperbolic Functions 


427  (8) 

Techniques of Integration 


435  (50) 

Review; Integration Using Tables and Computer Algebra Systems 


435  (6) 


441  (6) 

Integrating Powers of Sine and Cosine 


447  (4) 

Integrating Powers of Secant and Tangent 


451  (3) 

Trigonometric Substitutions 


454  (5) 

Integrating Rational Functions; Partial Fractions 


459  (9) 

Miscellaneous Substitutions 


468  (4) 

Numerical Integration; Simpson's Rule 


472  (13) 

Improper Integrals; L'Hopital's Rule 


485  (24) 


485  (7) 

L'Hopital's Rule (Indeterminate Forms of Type 0/0) 


492  (6) 

Other Indeterminate Forms (∞/∞, 0.&ingin;, 0°, ∞°, 1∞, &infin  ∞) 


498  (11) 


509  (82) 


509  (7) 


516  (7) 


523  (8) 


531  (6) 

Additional Convergence Tests 


537  (5) 

The Limit Comparison Test 


542  (6) 

Alternating Series; Conditional Convergence 


548  (8) 


556  (5) 

Taylor and Maclaurin Series 


561  (7) 

Taylor Formula with Remainder; Convergence of Taylor Series 


568  (9) 

Computations Using Taylor Series 


577  (5) 

Differentiation and Integration of Power Series 


582  (9) 

Topics in Analytic Geometry 


591  (32) 

Introduction to the Conic Sections 


591  (2) 

The Parabola; Translation of Coordinate Axes 


593  (6) 


599  (7) 


606  (8) 

Rotation of Axes; SecondDegree Equations 


614  (9) 

Polar Coordinates and Parametric Equations 


623  (34) 


623  (4) 

Graphs in Polar Coordinates 


627  (9) 

Area in Polar Coordinates 


636  (5) 


641  (10) 

Tangent Lines and Arc Length in Polar Coordinates 


651  (6) 

ThreeDimensional Space; Vectors 


657  (58) 

Rectangular Coordinates in 3Space; Spheres; Cylindrical Surfaces 


657  (6) 


663  (9) 


672  (7) 


679  (7) 

Parametric Equations of Lines 


686  (5) 


691  (7) 


698  (10) 

Cylindrical and Spherical Coordinates 


708  (7) 


715  (54) 

Introduction of VectorValued Functions 


715  (5) 

Calculus of VectorValued Functions 


720  (8) 

Change of Parameter; Arc Length 


728  (8) 

Unit Tangent and Normal Vectors 


736  (4) 


740  (7) 


747  (13) 

Kepler's Laws of Planetary Motion 


760  (9) 


769  (82) 

Functions of Two or More Variables 


769  (14) 


783  (7) 


790  (9) 

Differentiability and Chain Rules for Functions of Two Variables 


799  (10) 

Tangent Planes; Total Differentials for Functions of Two Variables 


809  (6) 

Directional Derivatives and Gradients for Functions of Two Variables 


815  (7) 

Differentiability, Directional Derivatives, and Gradients for Functions of Three Variables 


822  (6) 

Functions of n Variables; More on the Chain Rule 


828  (5) 

Maxima and Minima of Functions of Two Variables 


833  (11) 


844  (7) 


851  (68) 


851  (7) 

Double Integrals Over Nonrectangular Regions 


858  (8) 

Double Integrals in Polar Coordinates 


866  (6) 


872  (6) 


878  (7) 

Centroid, Center of Gravity, Theorem of Pappus 


885  (9) 

Triple Integrals in Cylindrical and Spherical Coordinates 


894  (11) 

Change of Variables in Multiple Integrals; Jacobians 


905  (14) 

Topics in Vector Calculus 


919  (64) 


919  (8) 


927  (11) 

Independence of Path; Conservative Vector Fields 


938  (8) 


946  (6) 

Introduction to Surface Integrals 


952  (7) 

Surface Integrals of Vector Fields; Flux 


959  (8) 


967  (8) 


975  (8) 

SecondOrder Differential Equations 


983  (14) 

SecondOrder Linear Homogeneous Differential Equations with Constant Coefficients 


983  (6) 

SecondOrder Linear Nonhomogeneous Differential Equations with Constant Coefficients; Undetermined Coefficients 


989  (6) 


995  (2) 


997  
Appendix A. Review of Sets 

A1  
Appendix B. Trigonometry Review 

A3  
Appendix C. Proofs 

A19  
Appendix D. Cramer's Rule 

A30  
Appendix E. Complex Numbers 

A34  
Answers 

A45  
Index 

I1  
Photo Credits 

P1  