Calculus Early Transcendentals

1



PRELIMINARIES



1.1

Real Numbers, Estimation, and Logic



1.2

Inequalities and Absolute Values



1.3

The Rectangular Coordinate System



1.4

Graphs of Equations



1.5

Functions and Their Graphs



1.6

Operations on Functions



1.7

Exponential and Logarithmic Functions



1.8

The Trigonometric Functions



1.9

The Inverse Trigonometric Functions



1.10

Chapter Review

2



LIMITS



2.1

Introduction to Limits



2.2

Rigorous Study of Limits



2.3

Limit Theorems



2.4

Limits at Infinity; Infinite Limits



2.5

Limits Involving Trigonometric Functions



2.6

Natural Exponential, Natural Log, and Hyperbolic Functions



2.7

Continuity of Functions



2.8

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

Higher-Order Derivatives



3.7

Implicit Differentiation



3.8

Related Rates



3.9

Derivatives of Exponential and Logarithmic Functions



3.10

Derivatives of Hyperbolic and Inverse Trigonometric Functions



3.11

Differentials and Approximations



3.12

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

Practical Problems



4.5

Graphing Functions Using Calculus



4.6

The Mean Value Theorem for Derivatives



4.7

Solving Equations Numerically



4.8

Antiderivatives



4.9

Introduction to Differential Equations



4.10

Exponential Growth and Decay



4.11

Chapter Review

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 and 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

Integration of Rational Functions Using Partial Fractions



7.6

Strategies for Integration



7.7

First-Order Linear Differential Equations



7.8

Approximations for Differential Equations



7.9

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 in the Plane



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 Three-Space



11.2

Vectors



11.3

The Dot Product



11.4

The Cross Product



11.5

Vector Valued Functions & Curvilinear Motion



11.6

Lines and Tangent Lines in Three-Space



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 FOR 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 and Approximations



12.8

Maxima and Minima



12.9

The Method of 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



13.10

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

15



DIFFERENTIAL EQUATIONS



15.1

Linear Homogeneous Equations



15.2

Nonhomogeneous Equations



15.3

Applications of Second-Order Equations



15.4

Chapter Review





APPENDIX



A.1

Mathematical Induction



A.2

Proofs of Several Theorems