A Preview of Calculus 

2  (8) 


10  (84) 

Four Ways to Represent a Function 


11  (13) 


24  (14) 

New Functions from Old Functions 


38  (11) 

Graphing Calculators and Computers 


49  (7) 


56  (8) 

Inverse Functions and Logarithms 


64  (11) 


75  (19) 

Laboratory Project Running Circles around Circles 


83  (1) 


84  (4) 

Priciples of Problem Solving 


88  (6) 


94  (94) 

The Tangent and Velocity Problems 


95  (5) 


100  (10) 

Calculating Limits Using the Limit Laws 


110  (9) 


119  (11) 

Limits Involving Infinity 


130  (12) 

Tangents, Velocities, and Other Rates of Change 


142  (8) 


150  (7) 

Writing Project Early Methods for Finding Tangents 


157  (1) 

The Derivative as a Function 


157  (14) 


171  (4) 

What Does f' Say about f? 


175  (13) 


181  (4) 


185  (3) 


188  (76) 

Derivatives of Polynomials and Exponential Functions 


189  (10) 

Applied Project Building a Better Roller Coaster 


198  (1) 

The Product and Quotient Rules 


199  (7) 

Rates of Change in the Natural and Social Sciences 


206  (12) 

Derivatives of Trigonometric Functions 


218  (7) 


225  (12) 

Laboratory Project Bezier Curves 


236  (1) 

Applied Project Where Should a Pilot Start Descent? 


237  (1) 


237  (8) 

Derivatives of Logarithmic Functions 


245  (7) 

Discovery Project Hyperbolic Functions 


251  (1) 

Linear Approximations and Differentials 


252  (12) 

Laboratory Project Taylor Polynomials 


257  (1) 


258  (3) 


261  (3) 

Applications of Differentiation 


264  (80) 


265  (6) 

Maximum and Minimum Values 


271  (9) 

Applied Project The Calculus of Rainbows 


279  (1) 

Derivatives and the Shapes of Curves 


280  (11) 

Graphing with Calculus and Calculators 


291  (7) 

Indeterminate Forms and 1'Hospital's Rule 


298  (9) 

Writing Project The Origins of l'Hospital's Rule 


307  (1) 


307  (12) 

Applied Project The Shape of a Can 


318  (1) 

Applications to Economics 


319  (5) 


324  (5) 


329  (15) 


336  (4) 


340  (4) 


344  (102) 


345  (12) 


357  (12) 

Evaluating Definite Integrals 


369  (11) 

Discovery Project Area Functions 


379  (1) 

The Fundamental Theorem of Calculus 


380  (9) 

Writing Project Newton, Leibniz, and the Invention of Calculus 


388  (1) 


389  (7) 


396  (7) 

Additional Techniques of Integration 


403  (6) 

Integration Using Tables and Computer Algebra Systems 


409  (7) 

Discovery Project Patterns in Integrals 


415  (1) 


416  (12) 


428  (18) 


438  (4) 


442  (4) 

Applications of Integration 


446  (60) 


447  (6) 


453  (14) 

Discovery Project Rotating on a Slant 


466  (1) 


467  (6) 

Discovery Project Arc Length Contest 


472  (1) 

Average Value of a Function 


473  (3) 

Applied Project Where to Sit at the Movies 


476  (1) 

Applications to Physics and Engineering 


476  (11) 

Applications to Economics and Biology 


487  (5) 


492  (14) 


499  (3) 


502  (4) 


506  (56) 

Modeling with Differential Equations 


507  (5) 

Direction Fields and Euler's Method 


512  (10) 


522  (9) 

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


530  (1) 

Exponential Growth and Decay 


531  (10) 

Applied Project Calculus and Baseball 


540  (1) 


541  (9) 


550  (12) 


557  (3) 


560  (2) 

Infinite Sequences and Series 


562  (84) 


563  (10) 

Laboratory Project Logistic Sequences 


573  (1) 


573  (10) 

The Integral and Comparison Tests; Estimating Sums 


583  (9) 


592  (8) 


600  (5) 

Representations of Functions as Power Series 


605  (6) 

Taylor and Maclaurin Series 


611  (11) 


622  (4) 

Writing Project How Newton Discovered the Binomial Series 


626  (1) 

Applications of Taylor Polynomials 


626  (9) 

Applied Project Radiation from the Stars 


634  (1) 

Using Series to Solve Differential Equations 


635  (11) 


640  (3) 


643  (3) 

Vectors and the Geometry of Space 


646  (58) 

ThreeDimensional Coordinate Systems 


647  (5) 


652  (9) 


661  (6) 


667  (9) 

Discovery Project The Geometry of a Tetrahedron 


675  (1) 

Equations of Lines and Planes 


676  (9) 


685  (9) 

Cylindrical and Spherical Coordinates 


694  (10) 

Laboratory Project Families of Surfaces 


699  (1) 


700  (3) 


703  (1) 


704  (44) 

Vector Functions and Space Curves 


705  (6) 

Derivatives and Integrals of Vector Functions 


711  (6) 


717  (8) 


725  (11) 

Applied Project Kepler's Laws 


735  (1) 


736  (12) 


742  (3) 


745  (3) 


748  (90) 

Functions of Several Variables 


749  (11) 


760  (6) 


766  (13) 

Tangent Planes and Linear Approximations 


779  (11) 


790  (8) 

Directional Derivatives and the Gradient Vector 


798  (13) 

Maximum and Minimum Values 


811  (11) 

Applied Project Designing a Dumpster 


820  (1) 

Discovery Project Quadratic Approximations and Critical Points 


821  (1) 


822  (16) 

Applied Project Rocket Science 


829  (1) 

Applied Project HydroTurbine Optimization 


830  (1) 


831  (5) 


836  (2) 


838  (78) 

Double Integrals over Rectangles 


839  (10) 


849  (5) 

Double Integrals over General Regions 


854  (9) 

Double Integrals in Polar Coordinates 


863  (5) 

Applications of Double Integrals 


868  (10) 


878  (5) 


883  (10) 

Discovery Project Volumes of Hyperspheres 


893  (1) 

Triple Integrals in Cylindrical and Spherical Coordinates 


893  (8) 

Applied Project Roller Derby 


900  (1) 

Discovery Project The Intersection of Three Cylinders 


901  (1) 

Change of Variables in Multiple Integrals 


901  (15) 


910  (4) 


914  (2) 


916  


917  


924  

The Fundamental Theorem for Line Integrals 


936  


945  


952  


960  


971  

Writing Project Three Men and Two Theorems 


977  


978  


985  


986  


989  
Appendixes 

A1  

A Intervals, Inequalities, and Absolute Values 


A2  


A7  


A18  

D Precise Definitions of Limits 


A29  


A39  


A44  

G Integration of Rational Functions by Partial Fractions 


A50  


A58  


A74  

J Answers to OddNumbered Exercises 


A83  
Index 

A135  