Precalculus 1: Basic notions
A solid preparation for Calculus, with elements of discrete mathematics: logic, sets, functions, relations, and proofs
What you will 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 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.
Why take this course?
🌟 Precalculus 1: Basic Notions 🎓
Course Overview:
Embark on a journey through the fundamental concepts of Precalculus, which will lay the groundwork for your future studies in Calculus. This course intertwines key elements of discrete mathematics, including logic, sets, functions, relations, and proofs. Precalculus 1 is designed to enhance your understanding and proficiency in mathematical concepts essential for university-level mathematics.
What You'll Learn:
S1. Introduction to the Course 📘
- An overview of the course content and strategies for effective studying.
S2. Magical Letters and Symbols ✨
- Understand Greek and Latin letters and their significance in mathematics.
- Familiarize yourself with essential mathematical symbols that will be used throughout this course.
S3. Numbers and Arithmetic 🧮
- Explore the different kinds of numbers and their arithmetic, including natural, integer, rational, irrational, real numbers, and more.
S4. Absolute Value and Distances 🌍
- Master the Cartesian coordinate system and learn how to calculate distances using absolute values in various metric spaces.
S5. Equations and Inequalities 📈
- Dive into solving linear equations, including those with absolute values, and understand the concept of solution sets.
S6. Functions 📈
- Delve into the world of functions, covering domain, range, graph interpretation, function types, and transformations.
S7. Logic 🧠
- Gain a solid understanding of logical symbols, rules, quantifiers, and proof techniques that are fundamental to mathematical reasoning.
S8. Sets 🔄
- Learn about set theory, operations like union, intersection, and the importance of cardinality and inclusion-exclusion principles.
S9. Relations 🌀
- Explore different types of relations, including RST properties and the concept of equivalence classes and order relations.
S10. Functions as Relations ↔️
- Understand how functions can be viewed as special kinds of relations between sets, exploring injections, surjections, and bijections.
S11. Axioms, Definitions, Theorems, and Proofs 📚
- Decipher the differences between axioms, definitions, theorems, and more, and learn various types of proofs to apply in your problem-solving.
S12. Sequences and Series; AP, GP, HP 📉
- Discover how to work with sequences and series, including arithmetic, geometric, and harmonic progressions.
S13. Extras 🎉
- Get a preview of the additional resources we offer and a glimpse into future course plans.
Additional Resources:
- Detailed Course Content: A comprehensive list of all videos and problems, including titles and texts for each problem, is provided in the resource file "001 List_of_all_Videos_and_Problems_Precalculus_1.pdf". This content is fully explained in the first video, "Introduction to the Course."
Important Notes:
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Exam Preparation: Ensure you align with your professor on the specific parts of the course that will be relevant for your final exam, as requirements can vary greatly.
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Course Content Updates: Keep an eye out for updates and future courses, which we plan to release (with a disclaimer that these are hypothetical and subject to change).
Join us on this mathematical adventure where you'll gain the skills necessary to excel in Precalculus and beyond. With expertly crafted lessons and a wealth of resources at your fingertips, you're set for success! 🌟
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Our review
It's great to see such positive feedback for Hania's precalculus course! It seems that students appreciate her detailed, clear, and thorough approach to teaching mathematics, as well as her patience in answering questions. Many learners find her explanations engaging and enjoy the way she connects the fundamentals to more complex concepts, emphasizing the beauty of mathematics.
Her ability to teach at a pace that is understandable and allow for replaying difficult sections is particularly valued by adult learners or those who prefer a more comprehensive understanding before moving on to new topics. The fact that she answers questions quickly and in a way that is accessible to non-native English speakers also adds to the course's value.
Overall, it's clear that Hania's course not only helps students with their immediate learning goals but also equips them with a deeper understanding of mathematics that will be beneficial as they progress to more advanced topics like calculus and beyond. Her teaching style is commended for its effectiveness in both supplementing existing educational structures and serving as a standalone resource for self-motivated learners.
If you're considering taking this course or any other mathematical courses, it seems like Hania's offerings are highly recommended based on the experiences shared by these learners.