A Preview of Calculus 

2  (8) 


10  (74) 

1.1 Four Ways to Represent a Function 


11  (13) 


24  (14) 

1.3 New Functions from Old Functions 


38  (12) 

1.4 Graphing Calculators and Computers 


50  (6) 

1.5 Exponential Functions 


56  (8) 

1.6 Inverse Functions and Logarithms 


64  (11) 


75  (3) 

Principles of Problem Solving 


78  (6) 


84  (96) 

2.1 The Tangent and Velocity Problems 


85  (5) 

2.2 The Limit of a Function 


90  (12) 

2.3 Calculating Limits Using the Limit Laws 


102  (10) 

2.4 The Precise Definition of a Limit 


112  (10) 


122  (11) 

2.6 Limits at Infinity; Horizontal Asymptotes 


133  (14) 

2.7 Tangents, Velocities, and Other Rates of Change 


147  (9) 


156  (7) 

Writing Project Early Methods for Finding Tangents 


162  (1) 

2.9 The Derivative as a Function 


163  (11) 


174  (4) 


178  (2) 


180  (96) 

3.1 Derivatives of Polynomials and Exponential Functions 


181  (9) 

3.2 The Product and Quotient Rules 


190  (6) 

3.3 Rates of Change in the Natural and Social Sciences 


196  (12) 

3.4 Derivatives of Trigonometric Functions 


208  (7) 


215  (9) 

3.6 Implicit Differentiation 


224  (9) 


233  (7) 

Applied Project Where Should a Pilot Start Descent? 


240  (1) 

3.8 Derivatives of Logarithmic Functions 


240  (6) 


246  (7) 


253  (6) 

3.11 Linear Approximations and Differentials 


259  (8) 

Laboratory Project Taylor Polynomials 


266  (1) 


267  (4) 


271  (5) 

4 Applications of Differentiation 


276  (90) 

4.1 Maximum and Minimum Values 


277  (11) 

Applied Project The Calculus of Rainbows 


286  (2) 

4.2 The Mean Value Theorem 


288  (6) 

4.3 How Derivatives Affect the Shape of a Graph 


294  (11) 

4.4 Indeterminate Forms and L'Hospital's Rule 


305  (9) 

Writing Project The Origins of L'Hospital's Rule 


313  (1) 

4.5 Summary of Curve Sketching 


314  (8) 

4.6 Graphing with Calculus and Calculators 


322  (7) 

4.7 Optimization Problems 


329  (11) 

Applied Project The Shape of a Can 


339  (1) 

4.8 Applications to Economics 


340  (5) 


345  (6) 


351  (8) 


359  (4) 


363  (3) 


366  (66) 


367  (11) 

5.2 The Definite Integral 


378  (13) 

Discovery Project Area Functions 


390  (1) 

5.3 The Fundamental Theorem of Calculus 


391  (10) 

5.4 Indefinite Integrals and the Total Change Theorem 


401  (9) 

Writing Project Newton, Leibniz, and the Invention of Calculus 


409  (1) 

5.5 The Substitution Rule 


410  (8) 

5.6 The Logarithm Defined as an Integral 


418  (7) 


425  (4) 


429  (3) 

6 Applications of Integration 


432  (36) 


433  (7) 


440  (11) 

6.3 Volumes by Cylindrical Shells 


451  (5) 


456  (4) 

6.5 Average Value of a Function 


460  (3) 

Applied Project Where to Sit at the Movies 


463  (1) 


463  (2) 


465  (3) 

7 Techniques of Integration 


468  (72) 


469  (7) 

7.2 Trigonometric Integrals 


476  (7) 

7.3 Trigonometric Substitution 


483  (7) 

7.4 Integration of Rational Functions by Partial Fractions 


490  (9) 

7.5 Strategy for Integration 


499  (6) 

7.6 Integration Using Tables and Computer Algebra Systems 


505  (7) 

Discovery Project Patterns in Integrals 


511  (1) 

7.7 Approximate Integration 


512  (11) 


523  (11) 


534  (3) 


537  (3) 

8 Further Applications of Integration 


540  (40) 


541  (7) 

8.2 Area of a Surface of Revolution 


548  (7) 

Discovery Project Rotating on a Slant 


554  (1) 

8.3 Applications to Physics and Engineering 


555  (9) 

8.4 Applications to Economics and Biology 


564  (5) 


569  (7) 


576  (2) 


578  (2) 


580  (60) 

9.1 Modeling with Differential Equations 


581  (5) 

9.2 Direction Fields and Euler's Method 


586  (9) 


595  (8) 

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


602  (1) 

9.4 Exponential Growth and Decay 


603  (10) 

Applied Project Calculus and Baseball 


612  (1) 

9.5 The Logistic Equation 


613  (9) 


622  (6) 

9.7 PredatorPrey Systems 


628  (6) 


634  (4) 


638  (2) 

10 Parametric Equations and Polar Coordinates 


640  (52) 

10.1 Curves Defined by Parametric Equations 


641  (7) 

Laboratory Project Families of Hypocycloids 


648  (1) 


648  (7) 

Laboratory Project Bezier Curves 


655  (1) 

10.3 Arc Length and Surface Area 


655  (13) 


668  (102) 

10.5 Areas and Lengths in Polar Coordinates 


770  (5) 


775  (7) 

10.7 Conic Sections in Polar Coordinates 


782  (6) 


788  (2) 


790  

11 Infinite Sequences and Series 


692  


693  

Laboratory Project Logistic Sequences 


704  (1) 


704  (10) 

11.3 The Integral Test and Estimates of Sums 


714  (7) 

11.4 The Comparison Tests 


721  (5) 


726  (5) 

11.6 Absolute Convergence and the Ratio and Root Tests 


731  (7) 

11.7 Strategy for Testing Series 


738  (2) 


740  (5) 

11.9 Representations of Functions as Power Series 


745  (6) 

11.10 Taylor and Maclaurin Series 


751  (11) 

11.11 The Binomial Series 


762  (4) 

Writing Project How Newton Discovered the Binomial Series 


765  (1) 

11.12 Applications of Taylor Polynomials 


766  (9) 

Applied Project Radiation from the Stars 


774  (1) 


775  (3) 


778  
Appendixes 

A1  (106) 
A Intervals, Inequalities, and Absolute Values 

A2  (8) 
B Coordinate Geometry and Lines 

A10  (6) 
C Graphs of SecondDegree Equations 

A16  (8) 
D Trigonometry 

A24  (13) 
E Sigma Notation 

A37  (5) 
F Proofs of Theorems 

A42  (8) 
G Complex Numbers 

A50  (8) 
H Answers to OddNumbered Exercises 

A58  (49) 
Index 

A107  