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PRECALCULUS:
AN INTERACTIVE APPROACH
Travel and
Explorations
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PRECALCULUS: AN INTERACTIVE APPROACH (in progress)
This is an experiential, problems-oriented precalculus text for
students who are competent with intermediate algebra. No
trigonometry is assumed; this subject is covered in Chapters 5 and
6. The aim of the text is to invite and encourage the student to be
an active co-partner in learning, thinking, and doing mathematics.
The text’s activities, collaborations, problems, and explorations
are intended to nurture and guide what the student does, learns, and
perceives.
Chapter 1 Introductory Concepts
Chapter 2 Functions
Chapter 3 Polynomial and Rational Functions
Chapter 4 Exponential and Logarithmic Functions
Chapter 5 Trigonometric Functions
Chapter 6 Laws of Sines and Cosines; Vectors; Polar Equations
Chapter 7 Conic Sections
Chapter 8 Solving Linear Systems Using Matrices
Appendix A Algebra Appendix B
Geometry
Answers to all Problems
Syllabus and related content In addition to treating
the standard material covered in most precalculus books, this text
introduces and uses sequences, data and regression, parametric
equations, inequalities, local linearity and tangent lines. The
following themes are used throughout the text in order to help
students appreciate the coherence and fundamental processes of doing
mathematics:
Interconnections among different
representations of a problem, concept, solution
Conjecture and verification, extension or
generalization, of a discovered result
Discrete analogs of functions with
continuous domains
The bidirectional linkage between modifying
functions and transforming their graphs
Inverse functions and inverse processes
Locally linear approximations and tangent
lines
Problems and explorations The heart of any text is its
problems. While this work provides a reasonable number of practice
problems, it avoids the more common repetitious type exercises in
most textbooks. An important feature of the problems and
explorations is their thematic multipart investigations. Some theme
problems guide and teach the student how to proceed from a simple
problem to a more general related problem or challenge. Other
problems lead the student to appreciate how solving one problem can
open up new conjectures for investigation. And other multipart
problems encourage the student to progress from problem solver to
problem proposer. Also sprinkled throughout are many “Huh?” problems
that clamor for the student to explain and correct “proofs” of
startling results (e.g., “1 = 0,” “every real number is even,” and
“the square of a real number is negative.”)
The text attempts to convey the pervasiveness and
utility of mathematics throughout nature and across many
disciplines. Thus, in addition to historical perspectives, many of
the explorations and multipart theme problems introduce, develop,
and apply mathematical concepts to other disciplines.
Course options This work can be used for different types of
study and investigation. Here are some course options.
Core course (Families of Functions and their Distinguishing
Properties)
Chapter 1
Sections 2.1 - 2.5 (Section 2.2
can be delayed and considered before Section 6.5)
Sections 3.1 - 3.3
Sections 4.1, 4.2, 4.3
Sections 5.1 - 5.6
Chapter 6
* Core course + Regression theme (Sections 3.4, 4.3, 5.8)
* Core course + Difference Equations theme (Sections
2.6, 4.1)
* Core course + Analytic Geometry theme (Chapter 7)
* Core course + Matrices theme (Chapter 8)
Targeted audiences The material in this
volume would be well suited for a variety of uses.
(1) An Honors Precalculus course, taught in many high
schools and two-year colleges
(2) A rigorous course, in place of Advanced Placement
Calculus, that broadens students’
perspectives
and deepens their understandings of mathematics concepts and ways of
doing mathematics
(3) A source for independent study by mathematically
talented students, as may be found in
high schools
for the “gifted” and universities’ programs for the “gifted” (e.g.,
Stanford,
Northwestern,
Johns Hopkins, Duke, University Denver)
(4) A text for the training and development of
secondary school teachers
(5) A text for students who are studying
for the Mathematics HL Diploma Programme of
the
International Baccalaureate Organization (IBO). This huge and
rapidly growing
program
of accredited study for well prepared students covers:
Algebra, Functions and
Equations, Circular Functions and Trigonometry, Matrices, Vectors,
Statistics and
Probability, Differential and Integral Calculus.
“The majority of these students will be
expecting to include mathematics as a major component of their
university studies, either as a subject on its
own right or within courses such as physics, engineering
and technology. Others may take this subject because they have
strong interest in mathematics and enjoy meeting its challenges
and engaging with its problems. The nature of the subject is
such that it focuses on developing important mathematical
concepts in a comprehensible, coherent, and rigorous way.
Students are encouraged to apply their mathematical knowledge to
solving problems set in a variety of meaningful contexts.”
International Baccalaureate Organization