Math 3018 (3171-72) Precalculus


Precalculus Mate 3018 (or its two semester variant 3171+3172) is a “pre-mathematics” course reviewing topics which should have been —but which frequently are not— covered in the high school. Warning: The one semester variant. Math 3018, is intended for students whose college board score in mathematics is greater than 720. A grade of A in the remedial course Math 3001 is NOT , on its own, sufficient preparation for Math 3018. Indeed, the two semester sequence Math 3171-72 is more suitable for most students.

2022-2023– Semester I

LESSON            TOPICs
1Set theory, propositional and predicate logic
2Binary relations and functions
3Mathematical proof. The natural numbers
4Mathematical Induction
5Sequences and summation notation
EXAM ITBA
6The real numbers. Order and absolute value
7The Euclidean plane. Relations in the plane
8Functions of a real variable
9The Algebra of functions
10Exponential and logarithmic functions
EXAM IITBA
11Polynomials. I
12Polynomials II
13Circular functions
EXAM IIITBA
14Trigonometric equations and identities
15Applications of Trigonometry
EXAM IVTBA

MATH 3171 covers Topics 1-8; MATH 3172 covers 8 -16 + systems of linear equations.

  1. Course Philosophy
  2. Syllabus MATH 3018
  3. Generic Course Objectives
  4. Classroom Behaviour
  5. On the use of electronic aparati
  6. Recomendaciones para Estudiantes Nuevos
  7. In Praise of Lectures

Topic 1. Set theory, propositional and predicate logic

  1. On Definition (Definition according to Aristotle)
  2. On Truth
  3. On implication
  4. Validity
  5. Review of Elementary Set Theory and Logic
  6. The Square of Opposition
  7. Truth tables
  8. A list of tautologies

Topic 2: Binary Relations and Functions

  1. Binary Relations and Functions
  2. Image of a function

Topic 3: Mathematical proof. The natural numbers

  1. Read:  Elementary version of fractions.
  2. Read: The division theorem.
  3. Read: The division algorithm  (The example in Egyptian is optional but might help understanding). 
  4. Read Section 1.2 of Santos and do  the exercises 1.2.1-1.2.6
  5. Read items 1 up to 19 inclusive of the notes on the real line and do exercises 1,2,3,4

Topic 4: Mathematical Induction

  1. Mathematical Induction

Topic 5: Sequences and summation notation

  1. Sequences
  2. Optional Calculus  Reading – The Limit of a Sequence pages 1-4.

Topic 6: The real numbers. Order and absolute value

  1. Properties of the real numbers
  2. Order and Inequalities

Topic 7: The Euclidean plane I.

  1. Read notes on the Cartesian plane and do the exercises therein.
  2. Read notes on lines and do the exercises therein. 
  3. Straight Lines
  4. The Plane
  5. Some Standard Forms of the Straight Line
  6. Read notes on graphing and do the exercises therein.
  7. Graphing examples
  8. The Circle
  9. The Parabola

Topic 8: The Euclidean plane II.

  1. Review again the basic material on functions assigned earlier in the course.
  2.  Read notes on the parabola and do all exercises therein.

Topic 9: Exponential and logarithmic functions

  1. Review again, the  inverse functions studied earlier and complete the corresponding exercises on page  13.
  2. Read class note on  the exponential and logaritmic functions and do the exercises therein.
  3. Exponential functions–review

Topic 10: Polynomials. I

  1. Read proof of the synthetic division algorithm and do exercises   1 – 8 inclusive.

Topic 11: Polynomials II

  1. Complex Numbers
  2. Read notes on the rational root theorem and do exercises therein.
  3. Complete all the exercises.

Topic 12: Circular functions.

  1. Do exercises 1-27  on the trigonometric functions.

Topic 13: Trigonometric equations and identities

  1. The Four Trigonometrical Addition Formulae
  2. The Solution of Triangles

Old Exams

Textbooks

  1. (Free) Precalculus Book by David Santos
  2. (Free) Ossifrage-And-Algebra-2007 by David Santos
  3. (Free) Pre-y Cálculo Criollos (Spanish Version) by David Santos
  4. (Free) Andragogic Propaedeutic Mathematics by David Santos
  5. (Free) Differential Calculus by David Santos

Additional Notes

  1. Some Standard Forms of the Straight Line
  2. Unix File Permissions
  3. Logic Set Correspondence
  4. Venn Diagrams
  5. The Limit of a Sequence
  6. Greek Alphabet
  7. LateX Examples-pdf
  8. Download LaTex Source Code

Notes on the Bibliography

A Bibliography is included in the course syllabus above. Please read the following remarks before deciding on which books to purchase.

The great price (and weight) of precalculus textbooks bears no relation to their academic quality.   The very excellent Elementary Algebra and Precalculus texts by David Santos can be legally downloaded (See syllabus for a link).   For math majors and students aiming to achieve a good scientific level, these are preferable to the current official text.

The following books contain material relevant to the course.   The translation of Mathematics, Japanese grade 9 by Kunihiko Kodaira is an excellent review of material from high school algebra and geometry.   It also includes a section on probability and statistics. Mathematics, Japanese grade 10 contains most, though not all, of the material in MATH 3171 as well as some trigonometry from Math 3172.   It manages to achieve in 200 pages what the recommended text struggles to do in 1000.   The text Algebra and Geometry, Japanese Grade 11 covers plane and solid geometry, vectors and matrices.

The text Introductory Analysis by Dolciani et al. has been used by the Department in previous years so it can be obtained at a modest price in the street or from Amazon (for less than $10 used).   It is more carefully written than the official text and at a higher mathematical level.

As its title suggests, the book Foundations of Mathematics by Stewart and Tall gives a more profound view of the foundations of Mathematics and the construction of the numbers than the other texts.   Any student who is serious about achieving a good level in Mathematics should know the material in this book.  

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