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
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
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 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
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.