Calculus: Concepts and Connections Book

0 Preliminaries

0.1 Polynomial and Rational Functions

0.2 Graphing Calculators and Computer Algebra Systems

0.3 Inverse Functions

0.4 Trigonometric and Inverse Trigonometric Functions

0.5 Exponential and Logarithmic Functions

0.6 Transformations of Functions

0.7 Parametric Equations and Polar Coordinates

1 Limits and Continuity

1.1 A Brief Preview of Calculus

1.2 The Concept of Limit

1.3 Computation of Limits

1.4 Continuity and its Consequences

1.5 Limits Involving Infinity

1.6 Limits and Loss-of-Significance Errors

2 Differentiation

2.1 Tangent Lines and Velocity

2.2 The Derivative

2.3 Computation of Derivatives: The Power Rule

2.4 The Product and Quotient Rules

2.5 The Chain Rule

2.6 Derivatives of Trigonometric and Inverse Trigonometric Functions

2.7 Derivatives of Exponential and Logarithmic Functions

2.8 Implicit Differentiation

2.9 The Mean Value Theorem

3 Applications of Differentiation

3.1 Linear Approximations and Newton’s Method

3.2 Indeterminate Forms and L'Hopital's Rule

3.3 Maximum and Minimum Values

3.4 Increasing and Decreasing Functions

3.5 Concavity and Overview of Curve Sketching

3.6 Optimization

3.7 Rates of Change in Economics and the Sciences

3.8 Related Rates and Parametric Equations

4 Integration

4.1 Area Under a Curve

4.2 The Definite Integral

4.3 Antiderivatives

4.4 The Fundamental Theorem of Calculus

4.5 Integration by Substitution

4.6 Integration by Parts

4.7 Other Techniques of Integration

4.8 Integration Tables and Computer Algebra Systems

4.9 Numerical Integration

4.10 Improper Integrals

5 Applications of the Definite Integral

5.1 Area Between Curves

5.2 Volume

5.3 Arc Length and Surface Area

5.4 Projectile Motion

5.5 Applications of Integration to Physics and Engineering

5.6 Probability

6 Differential Equations

6.1 Growth and Decay Problems

6.2 Separable Differential Equations

6.3 Euler's Method

6.4 Second Order Equations with Constant Coefficients

6.5 Nonhomogeneous Equations: Undetermined Coefficients

6.6 Applications of Differential Equations

7 Infinite Series

7.1 Sequences of Real Numbers

7.2 Infinite Series

7.3 The Integral Test and Comparison Tests

7.4 Alternating Series

7.5 Absolute Convergence and the Ratio Test

7.6 Power Series

7.7 Taylor Series

7.8 Applications of Taylor Series

7.9 Fourier Series

7.10 Power Series Solutions of Differential Equations

8 Vectors and the Geometry of Space

8.1 Vectors in the Plane

8.2 Vectors in Space

8.3 The Dot Product

8.4 The Cross Product

8.5 Lines and Planes in Space

8.6 Surfaces in Space

9 Vector-Valued Functions

9.1 Vector-Valued Functions

9.2 Parametric Surfaces

9.3 The Calculus of Vector-Valued Functions

9.4 Motion in Space

9.5 Curvature

9.6 Tangent and Normal Vectors

10 Functions of Several Variables and Differentiation

10.1 Functions of Several Variables

10.2 Limits and Continuity

10.3 Partial Derivatives

10.4 Tangent Planes and Linear Approximations

10.5 The Chain Rule

10.6 The Gradient and Directional Derivatives

10.7 Extrema of Functions of Several Variables

10.8 Constrained Optimization and Lagrange Multipliers

11 Multiple Integrals

11.1 Double Integrals

11.2 Area, Volume and Center of Mass

11.3 Double Integrals in Polar Coordinates

11.4 Surface Area

11.5 Triple Integrals

11.6 Cylindrical Coordinates

11.7 Spherical Coordinates

11.8 Change of Variables in Multiple Integrals

12 Vector Calculus

12.1 Vector Fields

12.2 Curl and Divergence

12.3 Line Integrals

12.4 Independence of Path and Conservative Vector Fields

12.5 Green's Theorem

12.6 Surface Integrals

12.7 The Divergence Theorem

12.8 Stokes' Theorem

12.9 Applications of Vector Calculus

Appendix A Graphs of Additional Polar Equations

Appendix B Formal Definition of Limit

Appendix C Complete Derivation of Derivatives of sin x and cos x

Appendix D Natural Logarithm Defined as an Integral; Exponential Defined as the Inverse of the Natural Logarithm

Appendix E Conic Sections in Polar Coordinates

Appendix F Proofs of Selected Theorems