Chapter 1. A Library of Functions 1.1 Functions and Change
Material from former 1.1 and 1.2
1.2 Exponential Functions
Material from former 1.3 and 1.7
1.3 New Functions from Old
Material from former 1.5 and 1.8
1.4 Logarithmic Functions
Material from former 1.6 and 1.7
1.5 Trigonometric Functions Former 1.9
1.6 Powers, Polynomials, and Rational Functions
Material from former 1.4 and 1.10
1.7 Introduction to Continuity
Former 1.11 including Intermediate Value Theorem.
The Binomial Theorem is now a section available on the web site.
Chapter 2. Key Concept: The Derivative 2.1 How Do We Measure Speed?
2.2 Limits
NEW section from former Focus on Theory section.
2.3 The Derivative at a Point
2.4 The Derivative Function
2.5 Interpretations of the Derivative
2.6 The Second Derivative
2.7 Continuity and Differentiability
NEW section from former Focus on Theory section.
Chapter 3. Short-Cuts to Differentiation 3.1 Powers and Polynomials
3.2 The Exponential Function
3.3 The Product and Quotient Rules
3.4 The Chain Rule
3.5 The Trigonometric Functions
3.6 Applications of the Chain Rule.
3.7 Implicit Functions
3.8 Parametric Equations
NEW: Material on Motion and Parametric Curves and Differentiation based on Appendix F and G and 16.1
3.9 Linear Approximations and the Derivative
NEW: Material on Estimating the Error in the Approximation and theory on Differentiability and Local Linearity included.
3.10 Using Local Linearity to Find Limits
Includes L'Hopital's rule. NEW.
Chapter 4. Using the Derivative 4.1 Using First and Second Derivatives
4.2 Families of Curves
4.3 Optimization
4.4 Applications to Marginality
4.5 More Optimization: Introduction to Modeling
4.6 Hyperbolic Functions
4. 7 Theorems about Continuous and Differentiable Functions
NEW. Extreme Value Theorem, Local Extrema and Critical Points, Mean Value Theorem, Increasing Function Theorem, Constant Function Theorem, Racetrack Principle.
Chapter 5. Key Concept: The Definite Integral 5.1 How Do We Measure Distance Traveled?
5.2 The Definite Integral
Now includes general Riemann sum.
5.3 Interpretations of the Definite Integral
Material about integrating rates of change is now in this section.
5.4 Theorems About Definite Integrals
Chapter 6. Constructing Antiderivatives 6.1 Antiderivatives Graphically and Numerically
6.2 Constructing Antiderivatives Analytically
6.3 Differential Equations
6.4 Second Fundamental Theorem of Calculus
6.5 The Equations of Motion
Former Focus on Modeling Section.
Chapter 7. Integration 7.1 Integration by Substitution.
7.2 Integration by Parts
7.3 Tables of Integrals
7.4 Algebraic Identities and Trigonometric Substitutions.
NEW section including partial factions and trigonometric substitutions involving completing the square.
7.5 Approximating Definite Integrals
7.6 Approximating Errors and Simpson's Rule
7.7 Improper Integrals
7.8 More on Improper Integrals
Chapter 8. Using the Definite Integral 8.1 Areas and Volumes.
More accessible introduction to setting up integrals focusing on basic concepts.
8.2 Applications to Geometry
Section simplified and made easier to use.
8.3 Density and Center of Mass
Material on Center of Mass expanded.
8.4 Applications to Physics
Section simplified and made easier to use.
8.5 Applications to Economics
8.6 Distribution Functions
8.7 Probability and More on Distributions
NOTE: Chapter 9 and 10 replace Chapter 9 in the 2nd edition. The material has been expanded and extensively reorganized and rewritten. All sections have new problems. Material is clearly divided between series and convergence (Chapter 9) and approximations of functions (Chapter 10) for users who wish to emphasize one or the other.
Chapter 9. Series9.1 Geometric Series Former 9.4.
9.2 Convergence of Sequences and Series
NEW section from former Focus on Theory Section including new material on integral test added.
9.3 Tests for Convergence From Focus on Theory section with substantial new material on integral test, ratio test, and alternating series added.
9.4 Power Series Material from former 9.2 with substantial new material on intervals and radius of convergence added.
Chapter 10. Approximating Functions10.1 Taylor Polynomials First part of former 9.1.
10.2 Taylor Series
Second part of former 9.1 and 9.2.
10.3 Finding and Using Series Former 9.3
10.4 The Error in Taylor Polynomial Approximations
Former Focus on Theory section, substantially rewritten.
10.5 Fourier Series
Chapter 11. Differential Equations11.1 What Is a Differential Equation?
11.2 Slope Fields
11.3 Euler's Method
11.4 Separation of Variables
11.5 Growth and Decay
11.6 Applications and Modeling
11.7 Models of Population Growth
11.8 Systems of Differential Equations
11.9 Analyzing the Phase Plane
11.10 Second-Order Differential Equations: Oscillations
11.11 Linear Second-Order Differential Equations