Thomas' Calculus Part One, Vol. 1 Book

(Practice Exercises, Additional Exercises, and Questions to Guide Your Review appear at the end of each chapter.)

Preliminaries

Real Numbers and the Real Line

Lines, Circles, and Parabolas

Functions and Their Graphs

Identifying Functions; Mathematical Models

Combining Functions; Shifting and Scaling Graphs

Trigonometric Functions

Graphing with Calculators and Computers

2. Limits and Derivatives

Rates of Change and Limits

Calculating Limits Using the Limit Laws

Precise Definition of a Limit

One-Sided Limits and Limits at Infinity

Infinite Limits and Vertical Asymptotes

Continuity

Tangents and Derivatives

3. Differentiation

The Derivative as a Function

Differentiation Rules

The Derivative as a Rate of Change

Derivatives of Trigonometric Functions

The Chain Rule and Parametric Equations

Implicit Differentiation

Related Rates

Linearization and Differentials

4. Applications of Derivatives

Extreme Values of Functions

The Mean Value Theorem

Monotonic Functions and the First Derivative Test

Concavity and Curve Sketching

Applied Optimization Problems

Indeterminate Forms and L'Hopital's Rule

Newton's Method

Antiderivatives

5. Integration

Estimating with Finite Sums

Sigma Notation and Limits of Finite Sums

The Definite Integral

The Fundamental Theorem of Calculus

Indefinite Integrals and the Substitution Rule

Substitution and Area Between Curves

6. Applications of Definite Integrals

Volumes by Slicing and Rotation About an Axis

Volumes by Cylindrical Shells

Lengths of Plane Curves

Moments and Centers of Mass

Areas of Surfaces of Revolution and The Theorems of Pappus

Work

Fluid Pressures and Forces

7. Transcendental Functions

Inverse Functions and their Derivatives

Natural Logarithms

The Exponential Function

ax and loga x

Exponential Growth and Decay

Relative Rates of Growth

Inverse Trigonometric Functions

Hyperbolic Functions

8. Techniques of Integration

Basic Integration Formulas

Integration by Parts

Integration of Rational Functions by Partial Fractions

Trigonometric Integrals

Trigonometric Substitutions

Integral Tables and Computer Algebra Systems

Numerical Integration

Improper Integrals

9. Further Applications of Integration

Slope Fields and Separable Differential Equations

First-Order Linear Differential Equations

Euler's Method

Graphical Solutions of Autonomous Equations

Applications of First-Order Differential Equations

10. Conic Sections and Polar Coordinates

Conic Sections and Quadratic Equations

Classifying Conic Sections by Eccentricity

Quadratic Equations and Rotations

Conics and Parametric Equations; The Cycloid

Polar Coordinates

Graphing in Polar Coordinates

Area and Lengths in Polar Coordinates

Conic Sections in Polar Coordinates

11. Infinite Sequences and Series

Sequences

Infinite Series

The Integral Test

Comparison Tests

The Ratio and Root Tests

Alternating Series, Absolute and Conditional Convergence

Power Series

Taylor and Maclaurin Series

Convergence of Taylor Series; Error Estimates

Applications of Power Series

Fourier Series

Appendices

Mathematical Induction

Proofs of Limit Theorems

Commonly Occurring Limits

Theory of the Real Numbers

Complex Numbers

The Distributive Law for Vector Cross Products

Determinants and Cramer's Rule

The Mixed Derivative Theorem and the Increment Theorem

The Area of a Parallelogram's Projection on a Plane