About

This is a pre-University course and is one part of a standard mathematics curriculum which also includes elementary algebra, combinatorial probability, statistics and co-ordinate geometry. The current offering is tailored to the Indian School Certificate Mathematics Curriculum for students of Rishi Valley School. However, depending on time and interests there may be occasional deeper dives or tangents to related topics which are indicated by the asterisk (*) in the syllabus we have covered, sometimes using video content. Some topics may have been covered via exercises or homework and are indicated by (**). Topics which we have not covered but I consider essential are also listed italicised for interested students.

Syllabus

Naive Set Theory Introduction to Sets Defining Sets: Roster and Set-Builder Cardinality of Finite Sets Equal and Equivalent Sets Proper and Improper Subsets The Power Set Operations on Sets Unions and Intersections of Sets Set Difference and Symmetric Difference *Algebraic Properties of Sets Cartesian Product of Sets Relations Relations on Sets Equivalence Relations Additional Topics Equivalence Classes*	Logic and Proofs Propositional Logic Propositions and Truth Tables Negations, Disjunctions and Conjunctions Conditionals and Implications Logic and Mathematics Principle of Explosion and Vacuous Truth Existential and Universal Quantifiers Informal Introduction to Axiomatic Systems Standard Proofs Direct Proof and Proof by Contradiction Proof by Contrapositive Proof by Induction Consistency and Completeness Additional Topics Boolean Algebra and Universality* Gödel's Incompleteness Theorems
Functions Introduction to Functions Functions as Relations Domain and Co-domain Algebraic Functions Comment on Singularities Various Types of Functions Logarithms and Exponents Trigonometric Functions Step Functions, Square Root and Modulus Invertible Functions Image and Preimage Injectivity and Surjectivity Invertible Functions Composition of Functions Additional Topics Infinite Sets and Countability Binary Operations as Functions	Sequences and Series Introduction and Terminology Sequences as Functions Summation and Product Notation The Telescoping Property Arithmetic and Geometric Progressions Arithmetic Progressions (AP) Partial Sum of an AP Sum of the first N Naturals Geometric Progressions (GP) Partial Sum of a GP and Geometric Series Some Special Sequences and Sums Sum of the first N squared/cubed Naturals Inequality of Arithmetic and Geometric Means Arithmetico-Geometric Progressions and Series Additional Topics Informal and visual notion of convergence Rigorous definition of convergent sequences*
Limits and Differentiability Introduction to Limits The meaning of tends to Limits from graphs of functions Left-Hand and Right-Hand limits Evaluating and Proving Limits Rigorous definition of a Limit Algebraic properties of limits Sandwich Theorem Continuity and Differentiability Continuity of functions Differentiable functions Standard derivatives Properties of derivatives *L'Hospital's Rule for limits

Resources

Stitz & Zeager — Precalculus

Open

MIT OCW — Calculus

Open

Recommended external literature for self-study and deeper dives.

Helpful playlists from The Bright Side of Mathematics by Dr. Julian Grossmann, MIT OpenCourseWare featuring Professor Gilbert Strang and also 3 Blue 1 Brown's Essence of Calculus courtesy of Grant Sanderson.

Exercises and Tests

All previous assignments, soft homework and test papers will be available here shortly. If there are any errors, please do not hesitate to contact me. If any students wish to contribute to official solutions to the problems, they may contact me and I will host their solutions on the website if satisfactory.

General Updates

All students are required to join the google classroom for real-time updates and announcements.
All students are required to share a google document or overleaf file for their projects (by 25.10).
The above document is to discuss, review, finalise and clarify issues regarding the project and its draft.
Failure to do the above without proper reason and its communication will result in a loss of 10 project points.
All students are required to have begun their project draft by 30.11 in the file shared earlier.

Projects

The list of projects for this year will soon be available to see here. Upto four selected projects may get featured on my blog :)

The projects will be primarily evaluated on the following criteria:

Naive Set Theory Introduction to Sets Defining Sets: Roster and Set-Builder Cardinality of Finite Sets Equal and Equivalent Sets Proper and Improper Subsets The Power Set Operations on Sets Unions and Intersections of Sets Set Difference and Symmetric Difference *Algebraic Properties of Sets Cartesian Product of Sets Relations Relations on Sets Equivalence Relations Additional Topics Equivalence Classes*	Logic and Proofs Propositional Logic Propositions and Truth Tables Negations, Disjunctions and Conjunctions Conditionals and Implications Logic and Mathematics Principle of Explosion and Vacuous Truth Existential and Universal Quantifiers Informal Introduction to Axiomatic Systems Standard Proofs Direct Proof and Proof by Contradiction Proof by Contrapositive Proof by Induction Consistency and Completeness Additional Topics Boolean Algebra and Universality* Gödel's Incompleteness Theorems
Functions Introduction to Functions Functions as Relations Domain and Co-domain Algebraic Functions Comment on Singularities Various Types of Functions Logarithms and Exponents Trigonometric Functions Step Functions, Square Root and Modulus Invertible Functions Image and Preimage Injectivity and Surjectivity Invertible Functions Composition of Functions Additional Topics Infinite Sets and Countability Binary Operations as Functions	Sequences and Series Introduction and Terminology Sequences as Functions Summation and Product Notation The Telescoping Property Arithmetic and Geometric Progressions Arithmetic Progressions (AP) Partial Sum of an AP Sum of the first N Naturals Geometric Progressions (GP) Partial Sum of a GP and Geometric Series Some Special Sequences and Sums Sum of the first N squared/cubed Naturals Inequality of Arithmetic and Geometric Means Arithmetico-Geometric Progressions and Series Additional Topics Informal and visual notion of convergence Rigorous definition of convergent sequences*
Limits and Differentiability Introduction to Limits The meaning of tends to Limits from graphs of functions Left-Hand and Right-Hand limits Evaluating and Proving Limits Rigorous definition of a Limit Algebraic properties of limits Sandwich Theorem Continuity and Differentiability Continuity of functions Differentiable functions Standard derivatives Properties of derivatives *L'Hospital's Rule for limits