

1  (72) 

Functions and Their Representations 


1  (9) 

A Catalog of Essential Functions 


10  (14) 


24  (11) 


35  (10) 


45  (11) 

Limits Involving Infinity 


56  (17) 


69  (4) 


73  (69) 

Derivatives and Rates of Change 


73  (10) 

The Derivative as a Function 


83  (11) 

Basic Differentiation Formulas 


94  (12) 

The Product and Quotient Rules 


106  (7) 


113  (8) 


121  (6) 


127  (6) 

Linear Approximations and Differentials 


133  (9) 


138  (4) 

Applications of Differentiation 


142  (52) 

Maximum and Minimum Values 


142  (7) 


149  (6) 

Derivatives and the Shapes of Graphs 


155  (8) 


163  (7) 


170  (9) 


179  (6) 


185  (9) 


191  (3) 


194  (52) 


194  (11) 


205  (12) 

Evaluating Definite Integrals 


217  (10) 

The Fundamental Theorem of Calculus 


227  (9) 


236  (10) 


242  (4) 

Inverse Functions: Exponential, Logarithmic, and Inverse Trigonometric Functions 


246  (60) 


246  (8) 

The Natural Logarithmic Function 


254  (8) 

The Natural Exponential Function 


262  (6) 

General Logarithmic and Exponential Functions 


268  (7) 

Exponential Growth and Decay 


275  (8) 

Inverse Trigonometric Functions 


283  (6) 


289  (6) 

Indeterminate Forms and l'Hospital's Rule 


295  (11) 


303  (3) 

Techniques of Integration 


306  (53) 


306  (6) 

Trigonometric Integrals and Substitutions 


312  (10) 


322  (8) 

Integration with Tables and Computer Algebra Systems 


330  (5) 


335  (12) 


347  (12) 


356  (3) 

Applications of Integration 


359  (53) 


359  (5) 


364  (11) 

Volumes by Cylindrical Shells 


375  (5) 


380  (6) 

Applications to Physics and Engineering 


386  (13) 


399  (13) 


409  (3) 


412  (72) 


412  (10) 


422  (9) 

The Integral and Comparison Tests 


431  (8) 


439  (10) 


449  (5) 

Representing Functions as Power Series 


454  (6) 

Taylor and Maclaurin Series 


460  (13) 

Applications of Taylor Polynomials 


473  (11) 


481  (3) 

Parametric Equations and Polar Coordinates 


484  (35) 


484  (6) 

Calculus with Parametric Curves 


490  (8) 


498  (8) 

Areas and Lengths in Polar Coordinates 


506  (5) 

Conic Sections in Polar Coordinates 


511  (8) 


517  (2) 

Vectors and the Geometry of Space 


519  (74) 

ThreeDimensional Coordinate Systems 


519  (5) 


524  (8) 


532  (7) 


539  (8) 

Equations of Lines and Planes 


547  (8) 

Cylinders and Quadric Surfaces 


555  (6) 

Vector Functions and Space Curves 


561  (11) 


572  (8) 

Motion in Space: Velocity and Acceleration 


580  (13) 


589  (4) 


593  (72) 

Functions of Several Variables 


593  (10) 


603  (8) 


611  (8) 

Tangent Planes and Linear Approximations 


619  (8) 


627  (8) 

Directional Derivatives and the Gradient Vector 


635  (11) 

Maximum and Minimum Values 


646  (8) 


654  (11) 


661  (4) 


665  (62) 

Double Integrals over Rectangles 


665  (11) 

Double Integrals over General Regions 


676  (8) 

Double Integrals in Polar Coordinates 


684  (6) 

Applications of Double Integrals 


690  (5) 


695  (10) 

Triple Integrals in Cylindrical Coordinates 


705  (4) 

Triple Integrals in Spherical Coordinates 


709  (6) 

Change of Variables in Multiple Integrals 


715  (12) 


724  (3) 


727  


727  (6) 


733  (11) 

The Fundamental Theorem for Line Integrals 


744  (9) 


753  (6) 


759  (8) 

Parametric Surfaces and Their Areas 


767  (10) 


777  (11) 


788  (5) 


793  


799  


1  (72) 


1  (9) 


10  (14) 


24  (5) 

Answers to OddNumbered Exercises 


29  (44) 
Index 

73  