MATH 2412 - Precalculus
(Revised August 2014)
Course Description: Precalculus. Topics include elementary theory of functions
and equations, analytic geometry, vectors, introductory logic, mathematical
induction, sequences and finite series.
4 credits. (4 lecture)
Prerequisites:
Math 1314: Pass with a “C” or better.
Math 1316: Pass with a “C” or better.
Course Intent: This course is intended primarily to prepare students for calculus.
It can also be used for general mathematics credit.
Student Learning Outcomes
1. The student will represent and manipulate algebraic and trigonometric
functions and relations algebraically, graphically, and numerically, including
partial fraction decomposition and finding zeroes of functions.
2. The student will engage in algebraic and trigonometric problem solving and
modeling.
3. The student will synthesize algebraic and trigonometric facts and laws into
proofs.
4. The student will analyze and manipulate equations between various two
dimensional systems such as rectangular, polar, vector representations, conic
systems and axes manipulations, as well as solving equations in these systems.
5. The student will investigate and perform summations and predictions on
geometric, algebraic, and binomial sequences and series.
Course Objectives: Upon completion of this course, a student should be able to:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Develop and use various problem-solving techniques.
Recognize functions as ordered pairs.
Determine the graph of an algebraic equation or function.
Develop partial fraction decomposition.
Find the zeros of real functions
Utilize the six basic trigonometric functions.
Verify various trigonometric identities.
Apply the Law of sines and the Law of cosines for various types of
situations.
Find the powers and roots of complex numbers using DeMoivre’s
Theorem.
Understand basic vectors (2 dimensional).
11.
12.
13.
14.
15.
16.
17.
18.
Convert points in a rectangular coordinate system to polar coordinates.
Recognize algebraic formulas relating to circles, parabolas, ellipses, and
hyperbolas.
Use translation of axes, rotation of axes, and polar equations of conics.
Recognize the use of arithmetic and geometric sequences.
Use summation notation to represent a series.
Understand and use the Binomial theorem.
Understand mathematical induction.
Understand the basic concepts of limits.
Course Outline: Instructors may find it preferable to cover the course topics in
the order listed below. However, the instructor may choose to organize topics in
any order, but all material must be covered.
APPROXIMATE TIME
TEXT REFERENCE
Unit I – Factoring with negative rational exponents Section: Addendum 1
Difference Quotient
Section: Addendum 2
Partial Fractions
Section: 7.3
(3 hours)
Review Topics include the following: Graphs and graphing utilities, lines in the
plane, slope, functions, polynomial functions of higher degree, synthetic division,
real zeros of polynomial functions, and the intermediate value theorem. Required
topics are: Factoring with negative rational exponents, finding the difference
quotient, and partial fraction decomposition.
Unit II – Trigonometry (review)
Sections: {4.2, 4.5 - 4.7}
Analytic Trigonometry (review)
Sections: {5.1 – 5.3}
Analytic Trigonometry
Sections: 5.4, 5.5
{7 hours}
(4 hours)
This unit contains Trigonometric Functions, the unit circle, graphs of the
trigonometric functions, inverse trigonometric functions, verifying identities, sum
and difference formulas, double angle and half-angle formulas, sum-to-product
and product-to-sum formulas, and solving trigonometric equations.
Unit III – Applications of Trigonometry
(12 hours)
Sections:
Chapter 6
This unit includes Law of Sines, Law of Cosines, Polar coordinates, graphs of
Polar equations, DeMoivre’s Theorem, vectors, and the dot product.
Unit IV – Conic Sections and Analytic Geometry
(15 hours)
Sections:
Chapter 9
Topics include the ellipse, the hyperbola, the parabola, rotation of axes,
parametric equations, and conic sections in polar coordinates.
Unit V – Sequences, Induction, and Probability
(14 hours)
Sections:
10.1 – 10.5
This unit contains Sequences and summation notation, arithmetic sequences,
Geometric Sequences and Series, Mathematical Induction, and The Binomial
Theorem.
Unit VI – Introduction to Calculus
(10 hours)
Sections:
11.1 – 11.4
This unit contains an introduction to limits using tables and properties, continuity,
and an introduction to derivatives.
Textbook: Precalculus, Robert Blitzer, 5e, 2013, Pearson Prentice Hall
Suggested Methods: It is helpful to begin each class with questions concerning
the material discussed and the assigned homework problems. In presenting new
material, it is suggested that an explanation be followed by students working
examples in class. Students should be encouraged to work the review exercises
at the end of each chapter. Also, they should be encouraged to visit the
Academic Support Center at their respective colleges
Departmental Policies:
1. Each instructor must cover all course topics by the end of the semester.
The final exam is comprehensive and questions on it can deal with any of
the course objectives.
2. Each student should receive a copy of the instructor’s student syllabus for
the course during the first week of class. The syllabus should also be
available online.
3. A minimum of three in class tests and a comprehensive final examination
must be given. The final examination must be taken by all students.
4. All major tests should be announced at least one week or the equivalent in
advance.
5. The final exam must count between 25 percent and 40 percent of the final
grade.
6. The final course average will be used in the usual manner (A = 90–100; B
= 80-89; C = 70-79; D = 60-69; F = below 60).
7. An open book or a take home major test may be given at the discretion of
the instructor.
8. Any review sheet should be comprehensive and the student should not
feel that classroom notes, homework, and test may be ignored in favor of
the review sheet for any examination.
Resource Materials: Any student enrolled in Math 2412 at HCCS has access to
the Academic Support Center where they may get additional help in
understanding the theory or improving their skill. The Center is staffed with
mathematics faculty and student assistants, and offers tutorial help. A Chapter
Tests preparation video CD comes with the text. A Student’s Solution Manual
and MyMathLab are also available.
Americans With Disabilities Act (ADA): Any student with a documented
disability (e.g. physical, learning, psychiatric, vision, hearing, etc.) who needs to
arrange accommodations must contact the Disability Services Office at their
respective college at the beginning of each semester. Faculty are authorized to
provide only the accommodations requested by the Disability Support Services
Office.
Academic Honesty: At Houston Community College, academic integrity is
expected of all its members and stakeholders. Academic dishonesty includes, but
is not limited to, the willful attempt to misrepresent one’s work, cheat, plagiarize,
or impede other students’ scholastic progress. Scholastic dishonesty is treated
with the utmost seriousness by the instructor and the College. Please refer to the
Student Handbook for specific information related to professional conduct and
scholastic dishonesty.
Addendum 1
Factoring with negative rational exponents
p. 67 Example 13
Exercises:
p. 69 #93 – 101
Suggested supplementary problems
1. Factor: 2  2x  3  4 x  1
1
2. Factor:  3 x  4  2 
3. Factor: 4  x  5 

3
2
1
2
1
 2  4 x  1 2
1
3

 x  5  3x  4  2
2

5
3

 4x  7  x  5  2
2
4. Factor: 3  5 x  3  2x  1
1


1
2
5. Factor: 8  7 x  3  2  2x  5 
1
 6  5 x  3   2 x  1 2
2

1
3
3
2
 4  7 x  3  2  2x  5  3
Addendum 2
Difference Quotient
The difference quotient is basically the slope formula but used with nonlinear
functions. Since a nonlinear function does not have a slope in the traditional
sense, we cannot refer to this as the slope of the curve. The formula for the
difference quotient is:
f  x  h  f  x 
h
Example 1
Let f  x   x 2  4 , find the difference quotient.
Solution:
2
f  x  h   x  h  4
 x 2  2 xh  h 2  4
remember to square the binomial

f  x  h   f  x   x 2  2 xh  h 2  4  x 2  4

 x 2  2 xh  h 2  4  x 2  4
 2 xh  h 2
f  x  h  f  x 
h


2 xh  h 2
h
h  2x  h 
h
 2x  h
Exercises
Find the difference quotient for each function.
1. f  x   x 2  1
2. f  x   x 2  3x  4
3. g  x   x 3
4. f  x   x 3  3x 2
5. g  x   2x 3  4x  8
be careful when subtracting