A Preview of Calculus 

2  (8) 


10  (82) 

Four Ways to Represent a Function 


11  (14) 

Mathematical Models: A Catalog of Essential Functions 


25  (13) 

New Functions from Old Functions 


38  (11) 

Graphing Calculators and Computers 


49  (6) 


55  (8) 

Inverse Functions and Logarithms 


63  (11) 


74  (18) 

Laboratory Project: Running Circles around Circles 


82  (1) 


83  (3) 

Principles of Problem Solving 


86  (6) 


92  (90) 

The Tangent and Velocity Problems 


93  (5) 


98  (10) 

Calculating Limits Using the Limit Laws 


108  (9) 


117  (11) 

Limits Involving Infinity 


128  (11) 

Tangents, Velocities, and Other Rates of Change 


139  (9) 


148  (7) 

Writing Project: Early Methods for Finding Tangents 


155  (1) 

The Derivative as a Function 


155  (13) 

What Does f' Say about f? 


168  (14) 


175  (4) 


179  (3) 


182  (80) 

Derivatives of Polynomials and Exponential Functions 


183  (10) 

Applied Project: Building a Better Roller Coaster 


192  (1) 

The Product and Quotient Rules 


193  (7) 

Rates of Change in the Natural and Social Sciences 


200  (13) 

Derivatives of Trigonometric Functions 


213  (7) 


220  (12) 

Laboratory Project: Bezier Curves 


231  (1) 

Applied Project: Where Should a Pilot Start Descent? 


232  (1) 


232  (8) 

Derivatives of Logarithmic Functions 


240  (7) 

Discovery Project: Hyperbolic Functions 


246  (1) 

Linear Approximations and Differentials 


247  (15) 

Laboratory Project: Taylor Polynomials 


254  (1) 


255  (3) 


258  (4) 

Applications of Differentiation 


262  (80) 


263  (6) 

Maximum and Minimum Values 


269  (9) 

Applied Project: The Calculus of Rainbows 


277  (1) 

Derivatives and the Shapes of Curves 


278  (11) 

Graphing with Calculus and Calculators 


289  (8) 

Indeterminate Forms and I'Hospital's Rule 


297  (9) 

Writing Project: The Origins of I'Hospital's Rule 


305  (1) 


306  (11) 

Applied Project: The Shape of a Can 


316  (1) 

Applications to Business and Economics 


317  (5) 


322  (5) 


327  (15) 


335  (4) 


339  (3) 


342  (98) 


343  (11) 


354  (12) 

Evaluating Definite Integrals 


366  (11) 

Discovery Project: Area Functions 


376  (1) 

The Fundamental Theorem of Calculus 


377  (9) 

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


385  (1) 


386  (7) 


393  (7) 

Additional Techniques of Integration 


400  (5) 

Integration Using Tables and Computer Algebra Systems 


405  (7) 

Discovery Project: Patterns in Integrals 


411  (1) 


412  (11) 


423  (17) 


433  (4) 


437  (3) 

Applications of Integration 


440  (58) 


441  (6) 


447  (14) 

Discovery Project: Rotating on a Slant 


460  (1) 


461  (6) 

Discovery Project: Arc Length Contest 


466  (1) 

Average Value of a Function 


467  (4) 

Applied Project: Where to Sit at the Movies 


470  (1) 

Applications to Physics and Engineering 


471  (11) 

Applications to Economics and Biology 


482  (4) 


486  (12) 


493  (3) 


496  (2) 


498  (58) 

Modeling with Differential Equations 


499  (5) 

Direction Fields and Euler's Method 


504  (9) 


513  (11) 

Applied Project: How Fast Does a Tank Drain? 


521  (2) 

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


523  (1) 

Exponential Growth and Decay 


524  (11) 

Applied Project: Calculus and Baseball 


534  (1) 


535  (9) 


544  (12) 


551  (3) 


554  (2) 

Infinite Sequences and Series 


556  (80) 


557  (10) 

Laboratory Project: Logistic Sequences 


567  (1) 


567  (10) 

The Integral and Comparison Tests; Estimating Sums 


577  (9) 


586  (8) 


594  (5) 

Representations of Functions as Power Series 


599  (6) 

Taylor and Maclaurin Series 


605  (12) 

Laboratory Project: An Elusive Limit 


617  (1) 


617  (4) 

Writing Project: How Newton Discovered the Binomial Series 


621  (1) 

Applications of Taylor Polynomials 


621  (15) 

Applied Project: Radiation from the Stars 


630  (1) 


631  (3) 


634  (2) 

Vectors and the Geometry of Space 


636  (58) 

ThreeDimensional Coordinate Systems 


637  (5) 


642  (9) 


651  (6) 


657  (9) 

Discovery Project: The Geometry of a Tetrahedron 


665  (1) 

Equations of Lines and Planes 


666  (10) 

Laboratory Project: Putting 3D in Perspective 


675  (1) 


676  (9) 

Cylindrical and Spherical Coordinates 


685  (9) 

Laboratory Project: Families of Surfaces 


690  (1) 


690  (3) 


693  (1) 


694  (44) 

Vector Functions and Space Curves 


695  (7) 

Derivatives and Integrals of Vector Functions 


702  (6) 


708  (8) 

Motion in Space: Velocity and Acceleration 


716  (12) 

Applied Project: Kepler's Laws 


727  (1) 


728  (10) 


734  (2) 


736  (2) 


738  (90) 

Functions of Several Variables 


739  (11) 


750  (6) 


756  (13) 

Tangent Planes and Linear Approximations 


769  (11) 


780  (8) 

Directional Derivatives and the Gradient Vector 


788  (13) 

Maximum and Minimum Values 


801  (12) 

Applied Project: Designing a Dumpster 


811  (1) 

Discovery Project: Quadratic Approximations and Critical Points 


812  (1) 


813  (15) 

Applied Project: Rocket Science 


820  (1) 

Applied Project: HydroTurbine Optimization 


821  (1) 


822  (4) 


826  (2) 


828  (76) 

Double Integrals over Rectangles 


829  (8) 


837  (6) 

Double Integrals over General Regions 


843  (8) 

Double Integrals in Polar Coordinates 


851  (6) 

Applications of Double Integrals 


857  (10) 


867  (5) 


872  (10) 

Discovery Project: Volumes of Hyperspheres 


881  (1) 

Triple Integrals in Cylindrical and Spherical Coordinates 


882  (7) 

Applied Project: Roller Derby 


888  (1) 

Discovery Project: The Intersection of Three Cylinders 


889  (1) 

Change of Variables in Multiple Integrals 


889  (15) 


898  (4) 


902  (2) 


904  


905  (7) 


912  (12) 

The Fundamental Theorem for Line Integrals 


924  (9) 


933  (7) 


940  (8) 


948  (11) 


959  (7) 

Writing Project: Three Men and Two Theorems 


965  (1) 


966  (7) 


973  


974  (3) 


977  


1  (134) 

A Intervals, Inequalities, and Absolute Values 


2  (5) 


7  (11) 


18  (11) 

D Precise Definitions of Limits 


29  (10) 


39  (5) 


44  (6) 

G Integration of Rational Functions by Partial Fractions 


50  (8) 


58  (16) 


74  (9) 

J Answers to OddNumbered Exercises 


83  (52) 
Index 

135  