0


PRELIMINARIES


0.1

Real Numbers, Logic and Estimation


0.2

Inequalities and Absolute Values


0.3

The Rectangular Coordinate System


0.4

Graphs of Equations

1


FUNCTIONS


1.1

Functions and Their Graphs


1.2

Operations on Functions


1.3

Exponential and Logarithmic Functions


1.4

The Trigonometric Functions & Their Inverses


1.5

Chapter Review

2


LIMITS


2.1

Introduction to Limits


2.2

Rigorous Study of Limits


2.3

Limit Theorems


2.4

Limits Involving Transcendental Functions


2.5

Limits at Infinity, Infinite Limits


2.6

Continuity of Functions


2.7

Chapter Review

3


THE DERIVATIVE


3.1

Two Problems with One Theme


3.2

The Derivative


3.3

Rules for Finding Derivatives


3.4

Derivatives of Trigonometric Functions


3.5

The Chain Rule


3.6

HigherOrder Derivatives


3.7

Implicit Differentiation


3.8

Related Rates


3.9

Differentials and Approximations


3.10

Chapter Review

4


APPLICATIONS OF THE DERIVATIVE


4.1

Maxima and Minima


4.2

Monotonicity and Concavity


4.3

Local Extrema and Extrema on Open Intervals


4.4

Graphing Functions Using Calculus


4.5

The Mean Value Theorem for Derivatives


4.6

Solving Equations Numerically


4.7

Antiderivatives


4.8

Introduction to Differential Equations

5


THE DEFINITE INTEGRAL


5.1

Introduction to Area


5.2

The Definite Integral


5.3

The 1st Fundamental Theorem of Calculus


5.4

The 2nd Fundamental Theorem of Calculus



and the Method of Substitution


5.5

The Mean Value Theorem for Integrals & the Use of Symmetry


5.6

Numerical Integration


5.7

Chapter Review

6


APPLICATIONS OF THE INTEGRAL


6.1

The Area of a Plane Region


6.2

Volumes of Solids: Slabs, Disks, Washers


6.3

Volumes of Solids of Revolution: Shells


6.4

Length of a Plane Curve


6.5

Work and Fluid Pressure


6.6

Moments, Center of Mass


6.8

Probability and Random Variables


6.8

Chapter Review

7


TECHNIQUES OF INTEGRATION &



DIFFERENTIAL EQUATIONS


7.1

Basic Integration Rules


7.2

Integration by Parts


7.3

Some Trigonometric Integrals


7.4

Rationalizing Substitutions


7.5

The Method of Partial Fractions


7.6

Strategies for Integration


7.7

Growth and Decay


7.8

FirstOrder Linear Differential Equations


7.9

Approximations for Differential Equations


7.10

Chapter Review

8


INDETERMINATE FORMS &



IMPROPER INTEGRALS


8.1

Indeterminate Forms of Type 0/0


8.2

Other Indeterminate Forms


8.3

Improper Integrals: Infinite Limits of Integration


8.4

Improper Integrals: Infinite Integrands


8.5

Chapter Review

9


INFINITE SERIES


9.1

Infinite Sequences


9.2

Infinite Series


9.3

Positive Series: The Integral Test


9.4

Positive Series: Other Tests


9.5

Alternating Series, Absolute Convergence,



and Conditional Convergence


9.6

Power Series


9.7

Operations on Power Series


9.8

Taylor and Maclaurin Series


9.9

The Taylor Approximation to a Function


9.10

Chapter Review

10


CONICS AND POLAR COORDINATES


10.1

The Parabola


10.2

Ellipses and Hyperbolas


10.3

Translation and Rotation of Axes


10.4

Parametric Representation of Curves


10.5

The Polar Coordinate System


10.6

Graphs of Polar Equations


10.7

Calculus in Polar Coordinates


10.8

Chapter Review

11


GEOMETRY IN SPACE, VECTORS


11.1

Cartesian Coordinates in ThreeSpace


11.2

Vectors


11.3

The Dot Product


11.4

The Cross Product


11.5

Vector Valued Functions & Curvilinear Motion


11.6

Lines in ThreeSpace


11.7

Curvature and Components of Acceleration


11.8

Surfaces in Three Space


11.9

Cylindrical and Spherical Coordinates


11.10

Chapter Review

12


DERIVATIVES OF FUNCTIONS OF



TWO OR MORE VARIABLES


12.1

Functions of Two or More Variables


12.2

Partial Derivatives


12.3

Limits and Continuity


12.4

Differentiability


12.5

Directional Derivatives and Gradients


12.6

The Chain Rule


12.7

Tangent Planes, Approximations


12.8

Maxima and Minima


12.9

Lagrange Multipliers


12.10

Chapter Review




13


MULTIPLE INTEGRATION


13.1

Double Integrals over Rectangles


13.2

Iterated Integrals


13.3

Double Integrals over Nonrectangular Regions


13.4

Double Integrals in Polar Coordinates


13.5

Applications of Double Integrals


13.6

Surface Area


13.7

Triple Integrals (Cartesian Coordinates)


13.8

Triple Integrals (Cyl & Sph Coordinates)


13.9

Change of Variables in Multiple Integrals


13.1

Chapter Review

14


VECTOR CALCULUS


14.1

Vector Fields


14.2

Line Integrals


14.3

Independence of Path


14.4

Green's Theorem in the Plane


14.5

Surface Integrals


14.6

Gauss's Divergence Theorem


14.7

Stokes's Theorem


14.8

Chapter Review



APPENDIX


A.1

Mathematical Induction


A.2

Proofs of Several Theorems


A.3

A Backward Look
