A Preview of Calculus 

2  (8) 


10  (74) 

Four Ways to Represent a Function 


11  (13) 


24  (14) 

New Functions from Old Functions 


38  (12) 

Graphing Calculators and Computers 


50  (6) 


56  (8) 

Inverse Functions and Logarithms 


64  (14) 


75  (3) 

Principles of Problem Solving 


78  (6) 


84  (96) 

The Tangent and Velocity Problems 


85  (5) 


90  (12) 

Calculating Limits Using the Limit Laws 


102  (10) 

The Precise Definition of a Limit 


112  (10) 


122  (11) 

Limits at Infinity; Horizontal Asymptotes 


133  (14) 

Tangents, Velocities, and Other Rates of Change 


147  (9) 


156  (7) 

Writing Project &box; Early Methods for Finding Tangents 


162  (1) 

The Derivative as a Function 


163  (15) 


174  (4) 


178  (2) 


180  (96) 

Derivatives of Polynomials and Exponential Functions 


181  (9) 

The Product and Quotient Rules 


190  (6) 

Rates of Change in the Natural and Social Sciences 


196  (12) 

Derivatives of Trigonometric Functions 


208  (7) 


215  (9) 


224  (9) 


233  (7) 

Applied Project &box; Where Should a Pilot Start Descent? 


240  (1) 

Derivatives of Logarithmic Functions 


240  (6) 


246  (7) 


253  (6) 

Linear Approximations and Differentials 


259  (12) 

Laboratory Project &box; Taylor Polynomials 


266  (1) 


267  (4) 


271  (5) 

Applications of Differentiation 


276  (90) 

Maximum and Minimum Values 


277  (11) 

Applied Project &box; The Calculus of Rainbows 


286  (2) 


288  (6) 

How Derivatives Affect the Shape of a Graph 


294  (11) 

Indeterminate Forms and L'Hospital's Rule 


305  (9) 

Writing Project &box; The Origins of L'Hospital's Rule 


313  (1) 

Summary of Curve Sketching 


314  (8) 

Graphing with Calculus and Calculators 


322  (7) 


329  (11) 

Applied Project &box; The Shape of a Can 


339  (1) 

Applications to Economics 


340  (5) 


345  (6) 


351  (12) 


359  (4) 


363  (3) 


366  (66) 


367  (11) 


378  (13) 

Discovery Project &box; Area Functions 


390  (1) 

The Fundamental Theorem of Calculus 


391  (10) 

Indefinite Integrals and the Total Change Theorem 


401  (9) 

Writing Project &box; Newton, Leibniz, and the Invention of Calculus 


409  (1) 


410  (8) 

The Logarithm Defined as an Integral 


418  (11) 


425  (4) 


429  (3) 

Applications of Integration 


432  (36) 


433  (7) 


440  (11) 

Volumes by Cylindrical Shells 


451  (5) 


456  (4) 

Average Value of a Function 


460  (5) 

Applied Project &box; Where to Sit at the Movies 


463  (1) 


463  (2) 


465  (3) 

Techniques of Integration 


468  (72) 


469  (7) 


476  (7) 

Trigonometric Substitution 


483  (7) 

Integration of Rational Functions by Partial Fractions 


490  (9) 


499  (6) 

Integration Using Tables and Computer Algebra Systems 


505  (7) 

Discovery Project &box; Patterns in Integrals 


511  (1) 


512  (11) 


523  (14) 


534  (3) 


537  (3) 

Further Applications of Integration 


540  (40) 


541  (7) 

Area of a Surface of Revolution 


548  (7) 

Discovery Project &box; Rotating on a Slant 


554  (1) 

Applications to Physics and Engineering 


555  (9) 

Applications to Economics and Biology 


564  (5) 


569  (9) 


576  (2) 


578  (2) 


580  (60) 

Modeling with Differential Equations 


581  (5) 

Direction Fields and Euler's Method 


586  (9) 


595  (8) 

Applied Project &box; Which Is Faster, Going Up or Coming Down? 


602  (1) 

Exponential Growth and Decay 


603  (10) 

Applied Project &box; Calculus and Baseball 


612  (1) 


613  (9) 


622  (6) 


628  (10) 


634  (4) 


638  (2) 

Parametric Equations and Polar Coordinates 


640  (52) 

Curves Defined by Parametric Equations 


641  (7) 

Laboratory Project &box; Families of Hypocycloids 


648  (1) 


648  (7) 

Laboratory Project &box; Bezier Curves 


655  (1) 

Arc Length and Surface Area 


655  (13) 


668  (102) 

Areas and Lengths in Polar Coordinates 


770  (5) 


775  (7) 

Conic Sections in Polar Coordinates 


782  (8) 


788  (2) 


790  

Infinite Sequences and Series 


692  (90) 


693  (11) 

Laboratory Project &box; Logistic Sequences 


704  (1) 


704  (10) 

The Integral Test and Estimates of Sums 


714  (7) 


721  (5) 


726  (5) 

Absolute Convergence and the Ratio and Root Tests 


731  (7) 

Strategy for Testing Series 


738  (2) 


740  (5) 

Representations of Functions as Power Series 


745  (6) 

Taylor and Maclaurin Series 


751  (11) 


762  (4) 

Writing Project &box; How Newton Discovered the Binomial Series 


765  (1) 

Applications of Taylor Polynomials 


766  (12) 

Applied Project &box; Radiation from the Stars 


774  (1) 


775  (3) 


778  (4) 

Vectors and the Geometry of Space 


782  (54) 

ThreeDimensional Coordinate Systems 


783  (5) 


788  (8) 


796  (7) 


803  (9) 

Discovery Project &box; The Geometry of a Tetrahedron 


811  (1) 

Equations of Lines and Planes 


812  (9) 

Cylinders and Quadric Surfaces 


821  (6) 

Cylindrical and Spherical Coordinates 


827  (8) 

Laboratory Project &box; Families of Surfaces 


832  (1) 


832  (3) 


835  (1) 


836  (36) 

Vector Functions and Space Curves 


837  (6) 

Derivatives and Integrals of Vector Functions 


843  (6) 


849  (8) 

Motion in Space: Velocity and Acceleration 


857  (13) 

Applied Project &box; Kepler's Laws 


866  (1) 


867  (3) 


870  (2) 


872  (94) 

Functions of Several Variables 


873  (14) 


887  (8) 


895  (13) 

Tangent Planes and Linear Approximations 


908  (9) 


917  (9) 

Directional Derivatives and the Gradient Vector 


926  (13) 

Maximum and Minimum Values 


939  (12) 

Applied Project &box; Designing a Dumpster 


949  (1) 

Discovery Project &box; Quadratic Approximations and Critical Points 


950  (1) 


951  (13) 

Applied Project &box; Rocket Science 


958  (1) 

Applied Project &box; HydroTurbine Optimization 


959  (1) 


960  (4) 


964  (2) 


966  (74) 

Double Integrals over Rectangles 


967  (9) 


976  (5) 

Double Integrals over General Regions 


981  (8) 

Double Integrals in Polar Coordinates 


989  (6) 

Applications of Double Integrals 


995  (10) 


1005  (3) 


1008  (10) 

Discovery Project &box; Volumes of Hyperspheres 


1018  (1) 

Triple Integrals in Cylindrical and Spherical Coordinates 


1018  (8) 

Applied Project &box; Roller Derby 


1024  (1) 

Discovery Project &box; The Intersection of Three Cylinders 


1025  (1) 

Change of Variables in Multiple Integrals 


1026  (12) 


1034  (4) 


1038  (2) 


1040  (84) 


1041  (6) 


1047  (12) 

The Fundamental Theorem for Line Integrals 


1059  (9) 


1068  (7) 


1075  (8) 

Parametric Surfaces and Their Areas 


1083  (10) 


1093  (12) 


1105  (6) 

Writing Project &box; Three Men and Two Theorems 


1110  (1) 


1111  (7) 


1118  (4) 


1119  (3) 


1122  (2) 

SecondOrder Differential Equations 


1124  (1) 

SecondOrder Linear Equations 


1125  (6) 

Nonhomogeneous Linear Equations 


1131  (7) 

Applications of SecondOrder Differential Equations 


1138  (8) 


1146  


1150  
Appendixes 

A1  (1) 

A Intervals, Inequalities, and Absolute Values 


A2  (1) 

B Coordinate Geometry and Lines 


A10  (1) 

C Graphs of SecondDegree Equations 


A16  (1) 


A24  (1) 


A37  (1) 


A42  (1) 


A52  (1) 

H Answers to OddNumbered Exercises 


A60  (1) 
Index 

A121  