Precalculus 1: Basic notions

What you'll learn in this Udemy Course

• ✓ 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 R\Q 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.