A PREVIEW OF CALCULUS 

2  (8) 


10  (84) 

1.1 Four Ways to Represent a Function 


12  (14) 

1.2 New Functions from Old Functions 


26  (15) 

1.3 Graphing Calculators and Computers 


41  (7) 


48  (8) 

Laboratory Project: Families of Hypocycloids 


55  (1) 

1.5 Exponential Functions 


56  (7) 

1.6 Inverse Functions and Logarithms 


63  (12) 

1.7 Models and Curve Fitting 


75  (9) 


84  (3) 

Principles of Problem Solving 


87  (7) 


94  (96) 

2.1 The Tangent and Velocity Problems 


96  (5) 

2.2 The Limit of a Function 


101  (10) 

2.3 Calculating Limits Using the Limit Laws 


111  (9) 


120  (10) 

2.5 Limits Involving Infinity 


130  (12) 

2.6 Tangents, Velocities, and Other Rates of Change 


142  (9) 


151  (7) 

Writing Project: Early Methods for Finding Tangents 


157  (1) 

2.8 The Derivative as a Function 


158  (13) 

2.9 Linear Approximations 


171  (4) 

2.10 What Does f' Say about f? 


175  (7) 


182  (4) 


186  (4) 


190  (76) 

3.1 Derivatives of Polynomials and Exponential Functions 


192  (9) 

3.2 The Product and Quotient Rules 


201  (6) 

3.3 Rates of Change in the Natural and Social Sciences 


207  (13) 

3.4 Derivatives of Trigonometric Functions 


220  (7) 


227  (12) 

Laboratory Project: Bezier Curves 


237  (1) 

Applied Project: Where Should a Pilot Start Descent? 


238  (1) 

3.6 Implicit Differentiation 


239  (8) 

3.7 Derivatives of Logarithmic Functions 


247  (7) 

Discovery Project: Hyperbolic Functions 


253  (1) 

3.8 Linear Approximations and Differentials 


254  (6) 

Laboratory Project: Taylor Polynomials 


259  (1) 


260  (3) 


263  (3) 

4 APPLICATIONS OF DIFFERENTIATION 


266  (82) 


268  (6) 

4.2 Maximum and Minimum Values 


274  (9) 

Applied Project: The Calculus of Rainbows 


282  (1) 

4.3 Derivatives and the Shapes of Curves 


283  (11) 

4.4 Graphing with Calculus and Calculators 


294  (7) 

4.5 Indeterminate Forms and L'Hospital's Rule 


301  (9) 

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


310  (1) 

4.6 Optimization Problems 


310  (12) 

Applied Project: The Shape of a Can 


321  (1) 

4.7 Applications to Economics 


322  (5) 


327  (5) 


332  (8) 


340  (4) 


344  (4) 


348  (98) 


350  (11) 

5.2 The Definite Integral 


361  (11) 

5.3 Evaluating Definite Integrals 


372  (11) 

Discovery Project: Area Functions 


382  (1) 

5.4 The Fundamental Theorem of Calculus 


383  (10) 

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


392  (1) 

5.5 The Substitution Rule 


393  (9) 


402  (6) 

5.7 Integration Using Tables and Computer Algebra Systems 


408  (8) 

Discovery Project: Patterns in Integrals 


414  (2) 

5.8 Approximate Integration 


416  (11) 


427  (10) 


437  (5) 


442  (4) 

6 APPLICATIONS OF INTEGRATION 


446  (56) 


448  (7) 


455  (9) 


464  (5) 

6.4 Average Value of a Function 


469  (4) 

Applied Project: Where to Sit at the Movies 


472  (1) 

6.5 Applications to Physics and Engineering 


473  (11) 

6.6 Applications to Economics and Biology 


484  (5) 


489  (7) 


496  (3) 


499  (3) 


502  (56) 

7.1 Modeling with Differential Equations 


504  (5) 


509  (5) 


514  (4) 


518  (9) 

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


526  (1) 

7.5 Exponential Growth and Decay 


527  (10) 

Applied Project: Calculus and Baseball 


536  (1) 

7.6 The Logistic Equation 


537  (9) 

7.7 PredatorPrey Systems 


546  (7) 


553  (3) 


556  (2) 

8 INFINITE SEQUENCES AND SERIES 


558  


560  (10) 

Laboratory Project: Logistic Sequences 


569  (1) 


570  (9) 

8.3 The Integral and Comparison Tests; Estimating Sums 


579  (10) 

8.4 Other Convergence Tests 


589  (8) 


597  (6) 

8.6 Representations of Functions as Power Series 


603  (5) 

8.7 Taylor and Maclaurin Series 


608  (11) 


619  (5) 

Writing Project: How Newton Discovered the Binomial Series 


623  (1) 

8.9 Applications of Taylor Polynomials 


624  (9) 

Applied Project: Radiation from the Stars 


632  (1) 

8.10 Using Series to Solve Differential Equations 


633  (4) 


637  (3) 


640  
APPENDIXES 

A1  (112) 
A Intervals, Inequalities, and Absolute Values 

A2  (5) 
B Coordinate Geometry 

A7  (12) 
C Trigonometry 

A19  (13) 
D Precise Definitions of Limits 

A32  (8) 
E A Few Proofs 

A40  (2) 
F Integration of Rational Functions by Partial Fractions 

A42  (9) 
G Polar Coordinates 

A51  (16) 
H Complex Numbers 

A67  (9) 
I Answers to OddNumbered Exercises 

A76  (37) 
INDEX 

A113  