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

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

compositions

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

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

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

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 http://math.etsu.edu/multicalc/

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

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.