Precalculus 2: Polynomials and rational functions

A solid preparation for Calculus, with elements of discrete mathematics: logic, sets, functions, relations, and proofs

Created by: Hania Uscka-Wehlou
University teacher in mathematics, PhD

Created by: Martin Wehlou
Editor at MITM AB

What Will I Learn?

• How to solve problems in chosen Precalculus topics (illustrated with 236 solved problems) and why these methods work.
• Repetition of chosen aspects of high school mathematics (basic notions as numbers, functions, sets, equations, and inequalities).
• The basic notions will be defined in a more abstract way, and you will get an insight into their place in Calculus and some other branches of university maths.
• By the end of this course you will be able to understand all the elements in the epsilon-delta definition of limit, both symbols and the content.
• Arithmetic with basic rules (associativity and commutativity of addition and multiplication, distributivity) illustrated for positive integers.
• Basic terminology for equations and inequalities, with examples of linear equations and inequalities, with and without absolute value.
• Basic concepts related to functions (domain, range, graph, surjection, injection, bijection, inverse, composition) and how to work with them.
• Functions: monotone (increasing, decreasing), bounded, even, odd; extremum: maximum, minimum, both local and global.
• Geometrical operations on graphs of functions: translations, reflections, shrinking, dilating. Piecewise functions.
• Absolute value and its role in computing distances, with geometrical illustrations and functional approach.
• Equations of straight lines in the plane: slope and intercept; first glimpse into derivative as the slope of the tangent line, and link to monotonicity.
• Introduction to sequences and series; short notation for sum (Sigma) and product (Pi); arithmetic, geometric, and harmonic progressions.
• Logic: symbols and rules (tautologies) used for creating mathematical statements, definitions, and proofs.
• Set theory: union, intersection, complement, set difference; some important rules and notation.
• Relations: RST (equivalence) relations, order relations, functions as relations.
• Necessary and sufficient conditions: definition and examples.
• Various types of proofs: direct, indirect, by contradiction, induction proof, with some examples.
• Building blocks of mathematical theories: axioms, definitions, theorems, propositions, lemmas, corollaries, etc.
• Decimal expansion of rational and irrational numbers. Density of Q and RQ in R.
• Cardinality of sets: finite sets, N, Z, Q, R; countable and uncountable sets.
• You will also get a solid background for any future studies in Real Analysis and other courses in university mathematics.

Requirements

• Basic high school mathematics (mainly arithmetic)
• You are always welcome with your questions. If something in the lectures is unclear, please, ask. It is best to use QA, so that all the other students can see my additional explanations about the unclear topics. Remember: you are never alone with your doubts, and it is to everybody's advantage if you ask your questions on the forum.

Target audience

• Students who plan to study Calculus, Real Analysis, Discrete Mathematics or Abstract Algebra
• High school students curious about university mathematics; the course is intended for purchase by adults for these students
• Everybody who wants to brush up their high school maths and gain a deeper understanding of the subject
• College and university students studying advanced courses, who want to understand all the details they might have missed in their earlier education