Student Outlines Part One for University Calculus

Front Cover
Title Page
Copyright Page
Table of Contents
Preface to the Student
Study Notes
1 Functions
2 Limits and Continuity
3 Differentiation
4 Applications of Derivatives
5 Integration
6 Applications of Definite Integrals
7 Techniques of Integration
8 Infinite Sequences and Series
9 Polar Coordinates and Conics
Appendix A Brief Table of Integrals
1 Functions
1.1 Functions and Their Graphs
1.2 Combining Functions; Shifting and Scaling Graphs
1.3 Trigonometric Functions
1.4 Exponential Functions
1.5 Inverse Functions and Logarithms
1.6 Graphing with Calculators and Computers
Chapter 1 Self-Test
Solutions to Chapter 1 Self-Test
2 Limits and Continuity
2.1 Rates of Change and Tangents to Curves
2.2 Limit of a Function and Limit Laws
2.3 The Precise Definition of a Limit
2.4 One-Sided Limits and Limits at Infinity
2.5 Infinite Limits and Vertical Asymptotes
2.6 Continuity
2.7 Tangents and Derivatives at a Point
Chapter 2 Self-Test
Solutions to Chapter 2 Self-Test
3 Differentiation
3.1 The Derivative as a Function
3.2 Differentiation Rules for Polynomials, Exponentials, Products, and Quotients
3.3 The Derivative as a Rate of Change
3.4 Derivatives of Trigonometric Functions
3.5 The Chain Rule and Parametric Equations
3.6 Implicit Differentiation
3.7 Derivatives of Inverse Functions and Logarithms
3.8 Inverse Trigonometric Functions
3.9 Related Rates
3.10 Linearization and Differentials
3.11 Hyperbolic Functions
Chapter 3 Self-Test
Solutions to Chapter 3 Self-Test
4 Applications of Derivatives
4.1 Extreme Values of Functions
4.2 The Mean Value Theorem
4.3 Monotonic Functions and the First Derivative Test
4.4 Concavity and Curve Sketching
4.5 Applied Optimization
4.6 Indeterminate Forms and L'Hôpital's Rule
4.7 Newton's Method
4.8 Antiderivatives
Chapter 4 Self-Test
Solutions to Chapter 4 Self-Test
5 Integration
5.1 Estimating with Finite Sums
5.2 Sigma Notation and Limits of Finite Sums
5.3 The Definite Integral
5.4 The Fundamental Theorem of Calculus
5.5 Indefinite Integrals and the Substitution Rule
5.6 Substitution and Area Between Curves
5.7 The Logarithm Defined as an Integral
Chapter 5 Self-Test
Solutions to Chapter 5 Self-Test
6 Applications of Definite Integrals
6.1 Volumes by Slicing and Rotation About an Axis
6.2 Volumes by Cylindrical Shells
6.3 Lengths of Plane Curves
6.4 Areas of Surfaces of Revolution and the Theorems of Pappus
6.5 Exponential Change and Separable Differential Equations
6.6 Work
6.7 Moments and Centers of Mass
Chapter 6 Self-Test
Solutions to Chapter 6 Self-Test
7 Techniques of Integration
7.1 Integration by Parts
7.2 Trigonometric Integrals
7.3 Trigonometric Substitutions
7.4 Integration of Rational Functions by Partial Fractions
7.5 Integral Tables and Computer Algebra Systems
7.6 Numerical Integration
7.7 Improper Integrals
Chapter 7 Self-Test
Solutions to Chapter 7 Self-Test
8 Infinite Sequences and Series
8.1 Sequences
8.2 Infinite Series
8.3 The Integral Test
8.4 Comparison Tests
8.5 The Ratio and Root Tests
8.6 Alternating Series, Absolute and Conditional Convergence
8.7 Power Series
8.8 Taylor and Maclaurin Series
8.9 Convergence of Taylor Series
8.10 The Binomial Series
Chapter 8 Self-Test
Solutions to Chapter 8 Self-Test
9 Polar Coordinates and Conics
9.1 Polar Coordinates
9.2 Graphing in Polar Coordinates
9.3 Areas and Lengths in Polar Coordinates
9.4 Conic Sections
9.5 Conics in Polar Coordinates
9.6 Conics and Parametric Equations; The Cycloid
Chapter 9 Self-Test
Solutions to Chapter 9 Self-Test
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