Table of Contents


Preliminary Section 1: Functions

           Definition of a function, properties, operations, domain and range

Preliminary Section 2: Graphs of Functions

           Definition of the graph of a function, translation, symmetry



              Looking Ahead: A Review of inequalities and piece-wise defined functions


Chapter 1: Limits, Tangents, and Rates of Change

1.      Tangent Lines

              The tangent line to a polynomial is introduced as a means of exploring what calculus

                  is all about.

2.      The Limit Concept

              Limits of polynomials and rational functions are explored graphically, numerically,

                  and analytically.

3.      Definition of the Limit

                  The “zooming” process is used to motivate the definition of the limit

4.      Limits and Infinity

                  Horizontal asymptotes of rational functions

5.      Continuity

              Continuity, limits from the right and left, Intermediate Value Theorem.

6.      Differentiability

                  The tangent line concept is used to motivate the definition of the derivative and the

                  concept of differentiability of a function.

7.      Rates of Change

              The concept of instantaneous rates of change is developed and applied using the derivative.


            Concept Review: Limits and Tangents

                    Self-test and a Next Step entitled “Why Tangent Lines”


            Looking Ahead: A Review of the laws of exponents and trigonometry



Chapter 2: The Derivative

1.      The Derivative Function

              Derivatives, second derivatives, and higher derivatives of polynomials

2.      The Product and Quotient Rules

                  Derivatives of products and quotients

3.      The Chain Rule

                        Composition of functions and a rigorous derivation of the rule for differentiating


4.      Implicit Differentiation

                  Derivatives of implicitly-defined functions, including functions defined by curves of

                  the form x=f(y)

5.       Rates of Change

                  The derivative as a measure of how fast an output variable is changing with respect

                                        to a given input variable

6.      The Exponential Function

                                        The exponential function is introduced with the laws of exponents, the derivative, the

                                        chain rule and some simple applications

7.      The Natural Logarithm

                                        The natural logarithm is introduced as an inverse function

8.      The Sine and Cosine Function

                  The derivative of the sine and cosine, chain rule forms, and rates of change of

                                        harmonic oscillators

9.      Additional Functions

                  The rest of the trig functions, exponentials, and logarithms with arbitrary bases


          Concept Review: The Derivative

                    Self-test, and a Next Step entitled “Why Calculus”



Chapter 3: Applications of the Derivative

1.      The Mean Value Theorem

                  The Mean Value Theorem as the theoretical basis of calculus, primarily through its

                  justification for the use of differentials.

2.      L’Hopital’s Rule

                                        Indeterminate forms and methods for evaluating them, if they exist

3.      Absolute and Relative Extrema

                                        Critical Points and the Extreme Value Theorem

4.      Monotonicity

                                        The first derivative used for simple curve sketching

5.      Concavity

                                        The second derivative is used to determine concavity and extrema

6.      Optimization

                                        Optimization word problems

7.      Least Squares

                  Minimization of Total squared error functions


          Concept Review: Applications of the Derivative

                    Self-test, and a Next Step entitled “Seasonality”



Chapter 4: Integration

1.      Simple Function Approximation

                  Graphs of Functions approximated by rectangles

2.      Riemann Sum Approximations

                                        Simple Function approximation is used to estimate areas and displacements

3.      The Definite Integral

                                        Simple function approximation transformed into the definition of the definite integral

4.      Derivatives of Indefinite Integrals

                  The derivative form of the fundamental theorem and its applications

5.      Antiderivatives

                                        The derivative form of the fundamental theorem motivates the study of the

                                        antiderivative of a function.

6.      The Fundamental Theorem

                                        The evaluation form of the fundamental theorem

7.      Substitution

                  The method of substitution is developed for antiderivatives

8.      Substitution in Definite Integrals

                  Substitution in definite integrals and its applications

9.      Integration by Parts

                  Integration by parts and Tabular integration


          Concept Review: The Definite Integral

                    Self-test, and a Next Step entitled “Fractal Interpolation”



Chapter 5: Applications of the Integral

1.      Area Between Two Curves

                  Type I and Type II regions, and applications

2.      Centroids

                                        Finding the “center” of a region using lamina

3.      Volumes of Solids of Revolution

                                        Volume formulas, Theorem of Pappus, Applications

4.      Arclength

                  The length of the graph of a function over an interval

5.      Additional Applications

                                        Mean Value Theorem for Integrals, Volumes by Slicing, Surface area

6.      Improper Integrals

                                        Discontinuous integrand, infinite intervals

7.      Geometric Probability

                  Densities, distributions, expected value, gaussian distrbutions


          Concept Review: The Definite Integral

                    Self-test, and a Next Step entitled “Conditional Probability ”




Chapter 6: More Functions and Integration

1.      The Hyperbolic Functions

                  Derivatives, antiderivatives, Arclength of catenaries

2.      Inverse Functions

                                        1-1, derivatives of inverses, graphs of inverses

3.      Inverse Trigonometric Functions

                        Inverse sine and inverse tangent, derivatives, antiderivatives

4.      Trigonometric Substitutions

                                        Substitutions in definite and indefinite integrals, elliptic integrals

5.      Additional Antiderivatives

                  Inverse hyperbolic formulas, trigonometric integrals


          Concept Review: More Functions and Integrals

                    Self-test, and a Next Step entitled “Area of an Ellipse ”




Chapter 7: Differential Equations

1.      Separable Differential Equations

                  Separation of Variables, Applications

2.      Growth and Decay

                                        Exponential growth and decay word problems

3.      Partial Fractions

                                        Integration using Partial Fractions, the logistic equation and chemical kinetics

4.      Mathematical Modeling

                                        Parameter estimation, Curve-fitting, Newton’s law of cooling

5.      Second Order Systems

                                        Phase portrait, method of quadratures, applications

6.      Harmonic Oscillation

                                        The second derivative test is applied to the study of harmonic oscillation

7.      Slope fields and Equilibria

                              Slope fields, stability of equilibrium points


                        Concept Review: Series

                                Self-test, and a Next Step entitled “Why Calculus?”





Chapter 8: Sequences and Series

1.      Sequences

                  Sequences, Limits of Sequences, Monotone Convergence

2.      Linear Recursion

                                        Geometric Sequences and their applications

3.      Discrete Dynamical Systems

                                        Web Diagrams and the Fixed Point Theorem

4.      Euler’s and Newton’s methods

                                        Sequences of numerical estimates, Fixed point theorem and Newton’s method

5.      Infinite Series

                                        Partial Sums, geometric series, divergence test

6.      The Integral Test

                                        Integral test, p-series

7.      The Comparison and Other Tests

                              Comparison of Positive term series, absolute convergence, the alternating series test


                        Concept Review: Series

                                Summary, Self-test, and a Next Step entitled “The Riemann Zeta Function”



Chapter 9: Taylor’s Series

1.      Taylor Polynomials

                  Taylor polynomials at a point, Taylor polynomials as extension of linearization

2.      Families of Taylor Polynomials

                                        Generating general formulas for Taylor polynomials of a function at a point

3.      Taylor’s Theorem

                                        Remainder estimation, Convergence of special Maclaurin series

4.      Algebra and Counting

                  Arithmetic with MacLaurin series, Application to combinatorics


            Next Step entitled “How does a Calculator Work? ”


5.      Power Series

                                        Ratio test, intervals of convergence

6.      Taylor’s Series

                                        Calculation of general form of Taylor coefficients for a given function

7.      Calculus with Power Series

                                        Differentiation and Integration of Series, Binomial Theorem

8.      Differential Equations

                  Solutions of differential equations via power series expansions


          Concept Review: Taylor Polynomials

                    Summary, self-test, and a Next Step entitled “Hypergeometric Series ”




Chapter 10:  Fourier Series

1.      Fourier Coefficients

                             Fourier Coefficients for Fourier Sine and Cosine Series

2.      Fourier Series Convergence

                              Limits of Sequences of Partial Sums

3.      Fourier Series on other Intervals

                              Even and odd extensions, general coefficient formulas


          Concept Review: Fourier Series

                                    Summary, self-test, and a Next Step entitled “Signal Processing ”





Chapters 11 – 15 constitute multivariable calculus.  This is an online text for a technology-intensive course using Maple and Java applets.  For details, go to



Chapter 11:  Vectors and Vector-Valued Functions

1.       Vectors

2.       Dot Products

3.       Cross Products

4.       Planes

5.       Vector-valued functions

6.       Velocity and Acceleration

7.       Arclength and the unit tangent

8.      Components of Acceleration


Chapter 12: Functions of 2 Variables

1.       Functions of 2 variables

2.       Limits and Continuity

3.       Partial Derivatives

4.       Partial Differential Equations

5.       Differentiability and Approximation

6.       The Chain Rule

7.       Gradients and Level curves

8.       Optimization

9.       Lagrange Multipliers


Chapter 13: Surfaces and Transformations

1.       Surfaces and Tangent Planes

2.       Parametric Surfaces

3.       Coordinate Transformations

4.       Polar Coordinates

5.       The Jacobian

6.       Cylindrical and Spherical Coordinates

7.       Arclength and Geodesics

8.       The Fundamental Form

9.      Curvature of a Surface


Chapter 14: Integration

1.       Iterated Integrals

2.       Double Integrals

3.       Applications of the Double Integral

4.       Triple Integrals

5.       Change of Variable in Double Integrals

6.       Integration in Polar Coordinates

7.      Integration in Spherical Coordinates


Chapter 15: Generalizing the Fundamental Theorem

1.       Vector Fields and Operations

2.       Line Integrals in Vector Fields

3.       Potentials of Conservative Fields

4.       Green’s Theorem

5.       Surface Integrals

6.       Stoke’s Theorem

7.      The Divergence Theorem


Chapter 16 is a “capstone chapter” which applies the calculus of the first 15 chapters to the investigation of the inverse square problem.