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Elements of Calculus

Constructor University, Spring 2025

Official Class Description from Campusnet

This module is the second in a sequence introducing mathematical methods at the university level in a form relevant for study and research in the quantitative natural sciences, engineering, Computer Science. The emphasis in these modules is on training operational skills and recognizing mathematical structures in a problem context. Mathematical rigor is used where appropriate. However, a full axiomatic treatment of the subject is provided in the first-year modules "Analysis". The lecture comprises the following topics

• Sets, basic operations, and relations
• Functions, basic operations, compositions of functions, graphs of functions
• Brief introduction to real and complex numbers
• Limits for sequences and functions
• Continuity
• Derivatives of functions and its geometric interpretations
• Computing derivatives: linearity, product rule, chain rule
• Applications of derivatives, optimization for one-variable functions
• Introduction to Integration and the Fundamental Theorem of Calculus
• Differential equations, modeling simple dynamical systems
• Discrete derivative, summations, difference equations
• Functions of several variables, representations using graphs and level curves
• Basic ideas of multivariable calculus
• Partial derivatives and directional derivatives, total derivative
• Optimization in several variables, gradient descent, Lagrange multipliers
• Ordinary differential equations with several variables, simple examples
• Linear constant-coefficient ordinary differential equations
• Fourier series and their applications

News

• Please note that the tutorial on Thu Mar 6 will be in SAC Hall 3.
• In the second week of classes we will offer four sessions: Class sessions on Mon Feb 10 (14:15 CNLH) and Tue Feb 11 (14:15 ICC East Wing), and the tutorial sessions on Wed Feb 12 (SAC Hall 3) and Thu Feb 13 (ICC East Wing).
• The first week of classes has to be canceled due to sickness.

Contact Information

Time and Place

How is this class organized?

In each week, you are supposed to:

• Watch the videos of the two corresponding sessions via MS Teams (links below).
• Come to class (example/question session) on Mondays at 14:15 (held by the instructor).
• Come to the tutorial on Tuesdays or Thursdays at 14:15 (held by the TAs).
• Work on the moodle exercises/quizzes (online multiple choice questions) and submit them before the following tutorial.
• Every other week, work on the homework sheets (linked below as pdfs), and submit your handwritten solution to moodle on the due date before the tutorial.

Textbooks

• Riley, Hobson, Bence (RHB): Mathematical Methods for Physics and Engineering (3rd edition, Cambridge, 2006)
• Feldman, Rechnitzer, Yeager: CLP-1 Differential Calculus and CLP-2 Integral Calculus (CLP)
• Marsden, Weinstein: Calculus I and Calculus II (MW)

Table of Contents

Chapter 1: Basic Calculus Review
1.1: Sets, Numbers and Polynomials
1.2: Functions

Chapter 2: Limits and Continuity
2.1: Sequences and Limits
2.2: Series and Power Series
2.3: Limits of Functions
2.4: Continuity

Chapter 3: Differentiation in One Variable

Chapter 4: Integration in One Variable

Chapter 5: ODEs

Chapter 6: Multivariable Calculus

Chapter 7: ODEs and PDEs

Chapter 8: Fourier Series

Grading

The grade is only based on the final exam. Moodle-quizzes and bi-weekly homework submissions can each provide up to 5% bonus points (i.e., up to 10% bonus points in total can be achieved) according to the following table:

HW percentage solved	Bonus percentage
80 or more	5
60 - 79	4
40 - 59	3
20 - 39	2
5 - 19	1
less than 5	0

Exams

There will be one final exam (centrally scheduled in May) and one make-up final exam (centrally scheduled in August).

Practice, Practice, Practice

An essential component for doing well in this class is to work on practice exercises. Math is about problem solving (as are almost all sciences)! During this course lots of possibilities for solving exercises are provided on moodle, in the example sessions, and in the tutorial, see below.

Moodle Exercises

Please go to moodle, login, and select the Elements of Calculus class to view the exercises, and the solutions (after the due date). Each week on Monday a new quiz is released, and this is due the following week before the tutorial.

Homework Exercises

These are released bi-weekly, and scans of handwritten solutions are to be uploaded to moodle before the due date.

Extra Material

• Learn the Greek alphabet with this wikipedia article.

Class Schedule

Will be updated while class is progressing.

Below, please click on the date to download the lecture notes of this day.

(Note that the book references given below offer only a rough orientation. Sometimes, only parts of a particular chapter are covered in class.)

Date	Topics
Week 1 (Feb. 3 - 9, 2025)
Session 1 Notes Video	Topic: Review of Sets, Numbers, Polynomials, and their Properties You will learn about the following topics: Sets and elementary set operations Review of the definition of natural, integer, rational, real, and complex numbers Polynomials and the Fundamental Theorem of Algebra Inequalities and Intervals Binomial coefficients Literature: any Calculus textbook, RHB 1.1, 1.5, 1.6, CLP Ch. 0.1, 0.2, 0.3, RW R.1, R.2
Session 2 Notes Video	Topic: Basics of Functions and their inverses You will learn about the following topics: Definition of function, domain, range, graph Brief review of standard functions: absolute value, parabola, hyperbola, sin, cos, tan, exponential function Inverse of a function The logarithm Injective, surjective, bijective Literature: CLP 0.4, 0.5, 0.6
Example Session	Polynomial Interpolation, Roots of Polynomilas of Degree bigger than 2, Proof by Induction
pdf of moodle quiz	Please submit on moodle
pdf of homework sheet	Covering Weeks 1 and 2. Please submit on moodle
Week 2 (Feb. 10 - 16, 2025)
Session 3 Notes Video	Topic: Sequences, Limits, Cauchy Sequences You will learn about the following topics: Definition of convergence of a sequence Limit laws Squeeze or Sandwich Theorem Definition of Cauchy sequence Inf, sup, liminf, limsup Literature: any Analysis textbook, see also the Wikipedia entry
Session 4 Notes Video	Topic: Series and Convergence Tests, Power Series and Radius of Convergence You will learn about the following topics: Partial sums Arithmetic series Geometric series Comparison test Ratio test Definitions of power series and their radius of convergence Literature: parts of RHB Ch. 4
Example Session	Examples of Limits, The Exponential Function
pdf of moodle quiz	Please submit on moodle
Week 3 (Feb. 17 - 23, 2025)
Session 5 Notes Video	Topic: Limit of Functions and Asymptotes You will learn about the following topics: Definition of limit of a function Limit laws for functions Limit from the right and left Horizontal and vertical asymptotes Growth of Exponentials and Logarithms vs polynomials Literature: CLP1 1.3 and some parts of 1.4 and 1.5
Session 6 Notes Video	Topic: Continuity and the Intermediate Value Theorem You will learn about the following topics: Definition of continuity Examples of continuous and discontinuous functions Extreme Value and Intermediate Value Theorems Continuity laws Literature: CLP1 1.6. (Take a look at CLP1 3.5.2 for the Extreme Value Theorem.)
Example Session	Limits Involving the Exponential Function and the Logarithm, Limit Laws, Asymptotes, Bisection Method, Application of Squeeze Law
pdf of moodle quiz	Please submit on moodle
pdf of homework sheet	Covering Weeks 3 and 4. Please submit on moodle
Week 4 (Feb. 24 - Mar. 2, 2025)
Session 7 Notes Video	Topic: Definition of Differentiation and Differentiation Rules You will learn about the following topics: Definition of differentiability and derivative Some equivalent definitions of derivative Differentiability and continuity Differentiation rules: product rule, quotient rule, chain rule, inverse function rule Derivative of a power series Literature: CLP1 2.1-2.10
Session 8 Notes Video	Topic: Implicit Differentiation You will learn about the following topics: Graph of an equation Implicit equations and parametrizations Implicit Differentiation Derivative of a parametrization The osculating circle, curvature, and convex and concave Literature: CLP1 2.11, also MW 2.4; also RHB 2.1.9.
Example Session	Must-Know Derivatives, More Examples of Derivatives
pdf of moodle quiz	Please submit on moodle