A Preview of Calculus 

2  (8) 


10  (56) 

Four Ways to Represent a Function 


11  (13) 


24  (14) 

New Functions from Old Functions 


38  (12) 

Graphing Calculators and Computers 


50  (9) 


56  (3) 

Principles of Problem Solving 


59  (7) 

Limits and Rates of Change 


66  (62) 

The Tangent and Velocity Problems 


67  (5) 


72  (12) 

Calculating Limits Using the Limit Laws 


84  (10) 

The Precise Definition of a Limit 


94  (10) 


104  (10) 

Tangents, Velocities, and Other Rates of Change 


114  (12) 


124  (2) 


126  (2) 


128  (94) 


129  (7) 

Writing Project Early Methods for Finding Tangents 


135  (1) 

The Derivative as a Function 


136  (11) 


147  (11) 

Rates of Change in the Natural and Social Sciences 


158  (12) 

Derivatives of Trigonometric Functions 


170  (7) 


177  (8) 


185  (7) 


192  (7) 

Applied Project *** Where Should a Pilot Start Descent? 


199  (1) 


199  (6) 

Linear Approximations and Differentials 


205  (13) 

Laboratory Project *** Taylor Polynomials 


213  (1) 


214  (4) 


218  (4) 

Applications of Differentiation 


222  (90) 

Maximum and Minimum Values 


223  (11) 

Applied Project *** The Calculus of Rainbows 


232  (2) 


234  (6) 

How Derivatives Affect the Shape of a Graph 


240  (9) 

Limits at Infinity; Horizontal Asymptotes 


249  (14) 

Summary of Curve Sketching 


263  (8) 

Graphing with Calculus and Calculators 


271  (6) 


277  (11) 

Applied Project *** The Shape of a Can 


287  (1) 

Applications to Economics 


288  (5) 


293  (6) 


299  (11) 


306  (4) 


310  (2) 


312  (58) 


313  (11) 


324  (13) 

Discovery Project *** Area Functions 


336  (1) 

The Fundamental Theorem of Calculus 


337  (9) 

Indefinite Integrals and the Total Change Theorem 


346  (10) 

Writing Project *** Newton, Leibniz, and the Invention of Calculus 


355  (1) 


356  (11) 


363  (4) 


367  (3) 

Applications of Integration 


370  (36) 


371  (7) 


378  (11) 

Volumes by Cylindrical Shells 


389  (5) 


394  (4) 

Average Value of a Function 


398  (5) 


401  (2) 


403  (3) 

Inverse Functions: Exponential, Logarithmic, and Inverse Trigonometric Functions 


406  (96) 


407  (9) 

Instructors may cover either Sections 7.27.4 or Sections 7.2*7.4*. See the Preface. 



Exponential Functions and Their Derivatives 


416  (12) 


428  (7) 

Derivatives of Logarithmic Functions 


435  (10) 

The Natural Logarithmic Function 


445  (8) 

The Natural Exponential Function 


453  (7) 

General Logarithmic and Exponential Functions 


460  (9) 

Inverse Trigonometric Functions 


469  (9) 

Applied Project *** Where to Sit at the Movies 


478  (1) 


478  (7) 

Indeterminate Forms and L'Hospital's Rule 


485  (15) 

Writing Project *** The Origins of L'Hospital's Rule 


496  (1) 


496  (4) 


500  (2) 

Techniques of Integration 


502  (72) 


503  (7) 


510  (7) 

Trigonometric Substitution 


517  (7) 

Integration of Rational Functions by Partial Fractions 


524  (9) 


533  (6) 

Integration Using Tables and Computer Algebra Systems 


539  (7) 

Discovery Project *** Patterns in Integrals 


545  (1) 


546  (11) 


557  (14) 


568  (3) 


571  (3) 

Further Applications of Integration 


574  (40) 


575  (7) 

Area of a Surface of Revolution 


582  (7) 

Discovery Project *** Rotating on a Slant 


588  (1) 

Applications to Physics and Engineering 


589  (9) 

Applications to Economics and Biology 


598  (5) 


603  (9) 


610  (2) 


612  (2) 


614  (60) 

Modeling with Differential Equations 


615  (5) 

Direction Fields and Euler's Method 


620  (9) 


629  (8) 

Applied Project *** Which Is Faster, Going Up or Coming Down? 


636  (1) 

Exponential Growth and Decay 


637  (10) 

Applied Project *** Calculus and Baseball 


646  (1) 


647  (9) 


656  (6) 


662  (10) 


668  (4) 


672  (2) 

Parametric Equations and Polar Coordinates 


674  (52) 

Curves Defined by Parametric Equations 


675  (7) 

Laboratory Project *** Families of Hypocycloids 


682  (1) 


682  (7) 

Laboratory Project *** Bezier Curves 


689  (1) 

Arc Length and Surface Area 


689  (5) 


694  (10) 

Areas and Lengths in Polar Coordinates 


704  (5) 


709  (7) 

Conic Sections in Polar Coordinates 


716  (8) 


722  (2) 


724  (2) 

Infinite Sequences and Series 


726  (90) 


727  (11) 

Laboratory Project *** Logistic Sequences 


738  (1) 


738  (10) 

The Integral Test and Estimates of Sums 


748  (7) 


755  (5) 


760  (5) 

Absolute Convergence and the Ratio and Root Tests 


765  (7) 

Strategy for Testing Series 


772  (2) 


774  (5) 

Representations of Functions as Power Series 


779  (6) 

Taylor and Maclaurin Series 


785  (11) 


796  (4) 

Writing Project *** How Newton Discovered the Binomial Series 


799  (1) 

Applications of Taylor Polynomials 


800  (12) 

Applied Project *** Radiation from the Stars 


808  (1) 


809  (3) 


812  (4) 

Vectors and the Geometry of Space 


816  (54) 

ThreeDimensional Coordinate Systems 


817  (5) 


822  (8) 


830  (7) 


837  (9) 

Discovery Project *** The Geometry of a Tetrahedron 


845  (1) 

Equations of Lines and Planes 


846  (9) 

Cylinders and Quadric Surfaces 


855  (6) 

Cylindrical and Spherical Coordinates 


861  (8) 

Laboratory Project *** Families of Surfaces 


866  (1) 


866  (3) 


869  (1) 


870  (36) 

Vector Functions and Space Curves 


871  (6) 

Derivatives and Integrals of Vector Functions 


877  (6) 


883  (8) 

Motion in Space: Velocity and Acceleration 


891  (13) 

Applied Project *** Kepler's Laws 


900  (1) 


901  (3) 


904  (2) 


906  (94) 

Functions of Several Variables 


907  (14) 


921  (8) 


929  (13) 

Tangent Planes and Linear Approximations 


942  (9) 


951  (9) 

Directional Derivatives and the Gradient Vector 


960  (13) 

Maximum and Minimum Values 


973  (12) 

Applied Project *** Designing a Dumpster 


983  (1) 

Discovery Project *** Quadratic Approximations and Critical Points 


984  (1) 


985  (13) 

Applied Project *** Rocket Science 


992  (1) 

Applied Project *** HydroTurbine Optimization 


993  (1) 


994  (4) 


998  (2) 


1000  (74) 

Double Integrals over Rectangles 


1001  (9) 


1010  (5) 

Double Integrals over General Regions 


1015  (8) 

Double Integrals in Polar Coordinates 


1023  (6) 

Applications of Double Integrals 


1029  (10) 


1039  (3) 


1042  (10) 

Discovery Project *** Volumes of Hyperspheres 


1052  (1) 

Triple Integrals in Cylindrical and Spherical Coordinates 


1052  (8) 

Applied Project *** Roller Derby 


1058  (1) 

Discovery Project *** The Intersection of Three Cylinders 


1059  (1) 

Change of Variables in Multiple Integrals 


1060  (12) 


1068  (4) 


1072  (2) 


1074  (84) 


1075  (6) 


1081  (12) 

The Fundamental Theorem for Line Integrals 


1093  (9) 


1102  (7) 


1109  (8) 

Parametric Surfaces and Their Areas 


1117  (10) 


1127  (12) 


1139  (6) 

Writing Project *** Three Men and Two Theorems 


1144  (1) 


1145  (7) 


1152  (4) 


1153  (3) 


1156  (2) 

SecondOrder Differential Equations 


1158  (26) 

SecondOrder Linear Equations 


1159  (6) 

Nonhomogeneous Linear Equations 


1165  (7) 

Applications of SecondOrder Differential Equations 


1172  (8) 


1180  (4) 


1184  
Appendixes 

1  (118) 

A Intervals, Inequalities, and Absolute Values 


2  (8) 

B Coordinate Geometry and Lines 


10  (6) 

C Graphs of SecondDegree Equations 


16  (8) 


24  (10) 


34  (5) 


39  (9) 


48  (8) 

H Answers to OddNumbered Exercises 


56  (63) 
Index 

119  