## Calculus: Understanding Its Concepts and Methods Darel Hardy, Fred Richman, Carol Walker, and Robert Wisner ©2005 Publisher: MacKichan Software, Inc. ISBN: 0-9766806-9-6 Calculus: Understanding Its Concepts and Methods works only with Scientific Notebook 5.5. Read more about Calculus: Understanding Its Concepts and Methods or about other books from MacKichan Software.

#### Review Chapter: Functions and Their Graphs

0.0 Introduction

0.1 Ways to define a function

0.2 Polynomials and rational functions

0.3 Transcendental functions

0.4 Plotting equations

0.5 Parametric curves

0.6 Shifting, scaling, and combining functions

#### Chapter 1: Tangents and Derivatives

1.0 Introduction

1.2 Local linearity

1.3 Limits

1.4 Continuity

1.5 The derivative at a point

1.6 Derivatives as functions

1.7 Asymptotic behavior

#### Chapter 2: Differentiation Rules and Properties

2.0 Introduction

2.1 Product and quotient rules

2.2 Chain rule

2.3 Derivatives of trigonometric functions

2.4 Differentiating implicit functions

2.5 Higher derivatives

2.6 Differentials

2.7 Parametric curves

#### Chapter 3: Applications of Derivatives

3.0 Introduction

3.1 Extreme values

3.2 Mean value theorem

3.3 Shape of a graph

3.4 Optimization

3.5 Newton's method

3.6 Indeterminate forms

3.7 Related rates

#### Chapter 4: Integrals

4.0 Introduction

4.1 Area function

4.2 Antiderivatives

4.3 Definite integrals

4.4 Fundamental theorem of calculus

4.5 Change of variable

#### Chapter 5: Applications of Integration

5.0 Introduction

5.1 Velocity and acceleration

5.2 Area between curves

5.3 Volume by cross sections

5.4 Volume by cylindrical shells

5.5 Average value of a function

#### Chapter 6: Calculus of Transcendental Functions

6.0 Introduction

6.1 Inverse functions

6.2 Natural logarithm

6.3 Exponential functions

6.4 Inverse trigonometric functions

6.5 Hyperbolic functions

#### Chapter 7: Techniques of Integration

7.0 Introduction

7.1 Integration by parts

7.2 Trigonometric functions

7.3 Trigonometric substitution

7.4 Partial fractions

7.5 Tables of integrals and further substitutions

7.6 Improper integrals

#### Chapter 8: Further Applications of Integration

8.0 Introduction

8.1 Polar coordinates

8.2 Arc length

8.3 Surface of revolution

8.4 Exponential growth and decay

8.5 Moments and center of mass

#### Chapter 9: Function Approximations

9.0 Introduction

9.1 Taylor polynomials

9.2 Polynomial interpolation

9.3 Splines

9.4 Bézier curves

9.5 Rational functions

9.6 Trigonometric functions

#### Chapter 10: Infinite Series

10.0 Introduction

10.1 Sequences

10.2 Series

10.3 Convergence tests

10.4 Power series

10.5 Maclaurin and Taylor series

10.6 Complex functions

#### Chapter 11: Numerical Integration

11.0 Introduction

11.1 Riemann sums

11.2 Simpson's rule

11.3 Taylor polynomials

11.4 Other numerical integration methods

11.5 Euler's method

#### Chapter 12: Vectors in Two and Three Dimensions

12.0 Introduction

12.1 Vectors in the plane

12.2 Vectors in space

12.3 Inner products and projections

12.4 Cross product

12.5 Lines and planes

12.6 Cylindrical and spherical coordinate systems

12.7 Surfaces

#### Chapter 13: Partial Derivatives

13.0 Introduction

13.1 Functions of several variables

13.2 Partial derivatives

13.3 Rules for partial derivatives

13.4 Local linearity

13.5 Directional derivatives and the gradient

13.6 Normals and the tangent plane

13.7 Extrema

13.8 Lagrange multipliers

#### Chapter 14: Multiple Integrals

14.0 Introduction

14.1 Double integrals

14.2 Iterated integrals

14.3 Double integrals in polar coordinates

14.4 Surface area

14.5 Triple integrals

14.6 Cylindrical and spherical coordinates

#### Chapter 15: Vector-Valued Functions

15.0 Introduction

15.1 Space curves

15.2 Derivatives and integrals

15.3 Arc length and curvature

15.4 Velocity and acceleration

#### Chapter 16: Vector Calculus

16.0 Introduction

16.1 Vector fields

16.2 Line integrals

16.3 Green's theorem

16.4 Surface integrals

16.6 Divergence theorem

#### Chapter 17: Differential Equations

17.0 Introduction

17.1 Solutions to differential equations

17.2 Differential equations with separable variables

17.3 Homogeneous differential equations

17.4 Exact differential equations

17.5 Exactness from integrating factors

#### Appendix A: Animations

A0: Introduction

A1: 2D Animations

A2: 3D Plots

A3: 3D Animations

B.1 Marginal analysis

B.2 Interest

B.3 Consumer and producer surplus

B.4 Probability

B.5 Expected value

#### Appendix C: Complex Numbers

C.1 Complex numbers

#### Appendix D: Matrices and Determinants

D0: Introduction

D1: Matrices and vectors

D2: Determinants

D3: Geometric transformations in two dimensions

D4: Geometric transformations in three dimensions

#### Appendix E: Engineering Examples

E.1 Work

E.2 Pressure

E.3 Moments of inertia

#### Explorations

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