A Preview of Calculus 

2  (8) 


10  (56) 

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) 


56  (3) 

Principles of Problem Solving 


59  (7) 

2 Limits and Rates of Change 


66  (62) 

2.1 The Tangent and Velocity Problems 


67  (5) 

2.2 The Limit of a Function 


72  (12) 

2.3 Calculating Limits Using the Limit Laws 


84  (10) 

2.4 The Precise Definition of a Limit 


94  (10) 


104  (10) 

2.6 Tangents, Velocities, and Other Rates of Change 


114  (10) 


124  (2) 


126  (2) 


128  (94) 


129  (7) 

Writing Project Early Methods for Finding Tangents 


135  (1) 

3.2 The Derivative as a Function 


136  (11) 

3.3 Differentiation Formulas 


147  (11) 

3.4 Rates of Change in the Natural and Social Sciences 


158  (12) 

3.5 Derivatives of Trigonometric Functions 


170  (7) 


177  (8) 

Applied Project Where Should a Pilot Start Descent? 


199  

3.7 Implicit Differentiation 


185  (7) 


192  (7) 


199  (6) 

3.10 Linear Approximations and Differentials 


205  (9) 

Laboratory Project Taylor Polynomials 


213  (1) 


214  (4) 


218  (4) 

4 Applications of Differentiation 


222  (90) 

4.1 Maximum and Minimum Values 


223  (11) 

Applied Project The Calculus of Rainbows 


232  (2) 

4.2 The Mean Value Theorem 


234  (6) 

4.3 How Derivatives Affect the Shape of a Graph 


240  (9) 

4.4 Limits at Infinity; Horizontal Asymptotes 


249  (14) 

4.5 Summary of Curve Sketching 


263  (8) 

4.6 Graphing with Calculus and Calculators 


271  (6) 

4.7 Optimization Problems 


277  (11) 

Applied Project The Shape of a Can 


287  (1) 

4.8 Applications to Economics 


288  (5) 


293  (6) 


299  (7) 


306  (4) 


310  (2) 


312  (58) 


313  (11) 

5.2 The Definite Integral 


324  (13) 

Discovery Project Area Functions 


336  (1) 

5.3 The Fundamental Theorem of Calculus 


337  (9) 

5.4 Indefinite Integrals and the Total Change Theorem 


346  (10) 

Writing Project Newton, Leibniz, and the Invention of Calculus 


355  (1) 

5.5 The Substitution Rule 


356  (7) 


363  (4) 


367  (3) 

6 Applications of Integration 


370  (36) 


371  (7) 


378  (11) 

6.3 Volumes by Cylindrical Shells 


389  (5) 


394  (4) 

6.5 Average Value of a Function 


398  (3) 


401  (2) 


403  (3) 

7 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. 



7.2 Exponential Functions and Their Derivatives 


416  (12) 

7.3 Logarithmic Functions 


428  (76) 

7.4 Derivatives of Logarithmic Functions 


435  (10) 

7.2(*) The Natural Logarithmic Function 


445  (8) 

7.3(*) The Natural Exponential Function 


453  (7) 

7.4(*) General Logarithmic and Exponential Functions 


460  (9) 

7.5 Inverse Trigonometric Functions 


469  (9) 

Applied Project Where To Sit at the Movies 


478  (1) 


478  (7) 

7.7 Indeterminate Forms and L'Hospital's Rule 


485  (11) 

Writing Project The Origins of L'Hospital's Rule 


496  (1) 


496  (4) 


500  (2) 

8 Techniques of Integration 


502  (72) 


503  (7) 

8.2 Trigonometric Integrals 


510  (7) 

8.3 Trigonometric Substitution 


517  (7) 

8.4 Integration of Rational Functions by Partial Fractions 


524  (9) 

8.5 Strategy for Integration 


533  (6) 

8.6 Integration Using Tables and Computer Algebra Systems 


539  (7) 

Discovery Project Patterns in Integrals 


545  (1) 

8.7 Approximate Integration 


546  (11) 


557  (11) 


568  (2) 


571  (3) 

9 Further Applications of Integration 


574  (40) 


575  (7) 

9.2 Area of a Surface of Revolution 


582  (7) 

Discovery Project Rotating on a Slant 


588  (1) 

9.3 Applications to Physics and Engineering 


589  (9) 

9.4 Applications to Economics and Biology 


598  (5) 


603  (7) 


610  (2) 


612  (2) 

10 Differential Equations 


614  (60) 

10.1 Modeling with Differential Equations 


615  (5) 

10.2 Direction Fields and Euler's Method 


620  (9) 


629  (8) 

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


636  (1) 

10.4 Exponential Growth and Decay 


637  (10) 

Applied Project Calculus and Baseball 


646  (1) 

10.5 The Logistic Equation 


647  (9) 


656  (6) 

10.7 PredatorPrey Systems 


662  (6) 


668  (4) 


672  (2) 

11 Parametric Equations and Polar Coordinates 


674  (52) 

11.1 Curves Defined by Parametric Equations 


675  (7) 

Laboratory Project Families of Hypocycloids 


682  (1) 


682  (7) 

Laboratory Project Bezier Curves 


689  (1) 

11.3 Arc Length and Surface Area 


689  (5) 


694  (10) 

11.5 Areas and Lengths in Polar Coordinates 


704  (5) 


709  (7) 

11.7 Conic Sections in Polar Coordinates 


716  (6) 


722  (2) 


724  (2) 

12 Infinite Sequences and Series 


726  


727  (11) 

Laboratory Project Logistic Sequences 


738  (1) 


738  (10) 

12.3 The Integral Test and Estimates of Sums 


748  (7) 

12.4 The Comparison Tests 


755  (5) 


760  (5) 

12.6 Absolute Convergence and the Ratio and Root Tests 


765  (7) 

12.7 Strategy for Testing Series 


772  (2) 


774  (5) 

12.9 Representations of Functions as Power Series 


779  (6) 

12.10 Taylor and Maclaurin Series 


785  (11) 

12.11 The Binomial Series 


796  (4) 

Writing Project How Newton Discovered the Binomial Series 


799  (1) 

12.12 Applications of Taylor Polynomials 


800  (10) 

Applied Project Radiation from the Stars 


808  (2) 


810  (2) 


812  
Appendixes 

A1  (104) 
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  (10) 
E Sigma Notation 

A34  (5) 
F Proofs of Theorems 

A39  (7) 
G Complex Numbers 

A46  (8) 
H Answers to OddNumbered Exercises 

A54  (51) 
Index 

A105  