Mathematics 1
Calculus
Complete Course Summary
Foundations & Functions · Limits & Sequences
Dierentiation & its Applications · Integration Techniques & Applications
Lecture Notes, Step-by-Step Solution Guides & Worked Exam Exercises
,Mathematics 1 Calculus: Complete Course Summary 2
Contents
I Lecture Notes & Worked Exercises 4
1 Foundations: Logic, Sets, Inequalities, Induction 4
1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Solving Inequalities: the Sign-Table Method . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Mathematical Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Functions 5
2.1 De
nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Injective, Surjective, Invertible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Polynomials and Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.4 Exponentials, Logarithms and their Laws . . . . . . . . . . . . . . . . . . . . . . . 7
2.5 Trigonometry and Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . 7
2.6 Completeness and Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3 Sequences and Limits 8
3.1 Formal De
nitions (know them by heart) . . . . . . . . . . . . . . . . . . . . . . . 8
3.2 The εδ (εN ) Proof Template . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.3 Limit Laws and Standard Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.4 Convergence of Recursive Sequences (classic exam problem) . . . . . . . . . . . . . 9
3.5 Continuity and the Intermediate Value Theorem . . . . . . . . . . . . . . . . . . . 10
4 Dierentiation 10
4.1 De
nition and Basic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.2 Dierentiation Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.3 Implicit Dierentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.4 Logarithmic Dierentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.5 Linear and Quadratic Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.6 Higher-Order Derivatives and the Shape of the Graph . . . . . . . . . . . . . . . . 12
4.7 Dierentiability at Tricky Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5 Applications of the Derivative 12
5.1 Extrema: De
nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5.2 Rolle's Theorem and the Mean Value Theorem . . . . . . . . . . . . . . . . . . . . 13
5.3 L'Hôpital's Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5.4 The Complete Curve-Sketching Algorithm . . . . . . . . . . . . . . . . . . . . . . . 15
5.5 Bounding Arguments with the MVT . . . . . . . . . . . . . . . . . . . . . . . . . . 16
6 Integration 16
, Mathematics 1 Calculus: Complete Course Summary 3
6.1 Antiderivatives and the Inde
nite Integral . . . . . . . . . . . . . . . . . . . . . . . 16
6.2 The De
nite Integral and the Riemann Sum . . . . . . . . . . . . . . . . . . . . . . 16
6.3 The Fundamental Theorem of Calculus . . . . . . . . . . . . . . . . . . . . . . . . . 17
6.4 Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
6.5 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
6.6 Integrating Rational Functions (Partial Fractions) . . . . . . . . . . . . . . . . . . . 19
6.7 Area and Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
6.8 Average Value and the Mean Value Theorem for Integrals . . . . . . . . . . . . . . 20
6.9 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
II Quick-Reference Step-by-Step Guides 22
7 Guide: Solving an Inequality 22
8 Guide: Finding the Domain, Range and Inverse of f 22
9 Guide: εδ / εN Proof of a Limit 22
10 Guide: Proving an Equation has Exactly One / Exactly n Solutions 23
11 Guide: Sketching a Function 23
12 Guide: Evaluating a Limit that Looks Indeterminate 23
13 Guide: Choosing an Integration Technique 23
14 Guide: Area, Volume and Improper Integrals 24
III Additional Worked Exam Exercises 25
15 Limits, Sequences & Continuity 25
16 Functions, Inverses & Curve Sketching 26
17 Dierentiation & L'Hôpital 27
18 Integration & its Applications 27
Calculus
Complete Course Summary
Foundations & Functions · Limits & Sequences
Dierentiation & its Applications · Integration Techniques & Applications
Lecture Notes, Step-by-Step Solution Guides & Worked Exam Exercises
,Mathematics 1 Calculus: Complete Course Summary 2
Contents
I Lecture Notes & Worked Exercises 4
1 Foundations: Logic, Sets, Inequalities, Induction 4
1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Solving Inequalities: the Sign-Table Method . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Mathematical Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Functions 5
2.1 De
nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Injective, Surjective, Invertible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Polynomials and Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.4 Exponentials, Logarithms and their Laws . . . . . . . . . . . . . . . . . . . . . . . 7
2.5 Trigonometry and Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . 7
2.6 Completeness and Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3 Sequences and Limits 8
3.1 Formal De
nitions (know them by heart) . . . . . . . . . . . . . . . . . . . . . . . 8
3.2 The εδ (εN ) Proof Template . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.3 Limit Laws and Standard Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.4 Convergence of Recursive Sequences (classic exam problem) . . . . . . . . . . . . . 9
3.5 Continuity and the Intermediate Value Theorem . . . . . . . . . . . . . . . . . . . 10
4 Dierentiation 10
4.1 De
nition and Basic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.2 Dierentiation Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.3 Implicit Dierentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.4 Logarithmic Dierentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.5 Linear and Quadratic Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.6 Higher-Order Derivatives and the Shape of the Graph . . . . . . . . . . . . . . . . 12
4.7 Dierentiability at Tricky Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5 Applications of the Derivative 12
5.1 Extrema: De
nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5.2 Rolle's Theorem and the Mean Value Theorem . . . . . . . . . . . . . . . . . . . . 13
5.3 L'Hôpital's Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5.4 The Complete Curve-Sketching Algorithm . . . . . . . . . . . . . . . . . . . . . . . 15
5.5 Bounding Arguments with the MVT . . . . . . . . . . . . . . . . . . . . . . . . . . 16
6 Integration 16
, Mathematics 1 Calculus: Complete Course Summary 3
6.1 Antiderivatives and the Inde
nite Integral . . . . . . . . . . . . . . . . . . . . . . . 16
6.2 The De
nite Integral and the Riemann Sum . . . . . . . . . . . . . . . . . . . . . . 16
6.3 The Fundamental Theorem of Calculus . . . . . . . . . . . . . . . . . . . . . . . . . 17
6.4 Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
6.5 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
6.6 Integrating Rational Functions (Partial Fractions) . . . . . . . . . . . . . . . . . . . 19
6.7 Area and Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
6.8 Average Value and the Mean Value Theorem for Integrals . . . . . . . . . . . . . . 20
6.9 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
II Quick-Reference Step-by-Step Guides 22
7 Guide: Solving an Inequality 22
8 Guide: Finding the Domain, Range and Inverse of f 22
9 Guide: εδ / εN Proof of a Limit 22
10 Guide: Proving an Equation has Exactly One / Exactly n Solutions 23
11 Guide: Sketching a Function 23
12 Guide: Evaluating a Limit that Looks Indeterminate 23
13 Guide: Choosing an Integration Technique 23
14 Guide: Area, Volume and Improper Integrals 24
III Additional Worked Exam Exercises 25
15 Limits, Sequences & Continuity 25
16 Functions, Inverses & Curve Sketching 26
17 Dierentiation & L'Hôpital 27
18 Integration & its Applications 27
