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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, CLP-2 Integral Calculus, CLP-3 Multivariable Calculus (CLP 1-3)
• 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
3.1: Definition, Properties, and Examples
3.2 Theorems and Applications

Chapter 4: Integration in One Variable

Chapter 5: ODEs

Chapter 6: Multivariable Calculus
6.1: Total and Partial Derivatives
6.2: Optimization in Many Variables

Chapter 7: ODEs and PDEs
7.1: Linear ODEs
7.2: 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
Week 5 (Mar. 3 - Mar. 9, 2025)
Session 9 Notes Video	Topic: Theorems of Differentiation You will learn about the following topics: Maxima and Minima Rolle's theorem Lagrange/Mean Value Theorem (MVT) Cauchy's Mean Value Theorem L'Hopital's rule Literature: parts of CLP1 2.13, RHB 2.1.10
Session 10 Notes Video	Topic: Critical Points, Inflection Points, and Graph Sketching You will learn about the following topics: Critical Points and Extreme Values Sufficient conditions for extreme values Points of inflection Sketching qualitative features of the graph of a function Literature: CLP1 2.13, 2.14, parts of 3.5, also 3.6. RHB 2.1.8, 2.1.9
Example Session	L'Hopital, Graph Sketching
pdf of moodle quiz	Please submit on moodle
pdf of homework sheet	Covering Weeks 5 and 6. Please submit on moodle
Week 6 (Mar. 10 - Mar. 16, 2025)
Session 11 Notes Video	Topic: Taylor Series You will learn about the following topics: Taylor expansion Examples of Taylor expansions Literature: RHB 4.6, 27.1.4; CLP2 3.6
Session 12 Notes Video	Topic: Indefinite Integrals You will learn about the following topics: Antiderivatives Indefinite integral Examples of antiderivatives Integration by parts Integration by substitution Literature: CLP1 4.1; see also CLP2 1.4, 1.7 only the indefinite integral versions.
Example Session	Newton's Method
pdf of moodle quiz	Please submit on moodle
Week 7 (Mar. 17 - Mar. 23, 2025)
Session 13 Notes Video	Topic: Definite Integrals and the Fundamental Theorem of Calculus You will learn about the following topics: The definite integral as the area under a function Definition of the definite integral for continuous functions Examples of definite integrals Properties of definite integrals The mean value theorem of integration The Fundamental Theorem of Calculus and its proof Literature: CLP2 1.1-1.3
Session 14 Notes Video	Topic: Applications of Integration You will learn about the following topics: Area between curves Volume computation Taylor series revisited Integration of power series Literature: CLP2 1.5, 1.6, 3.6; RHB 4.6
Example Session	Integral Mean-value theorem, Area between curves, Rational functions
pdf of moodle quiz	Please submit on moodle
pdf of homework sheet	Covering Weeks 7 and 8. Please submit on moodle
Week 8 (Mar. 24 - Mar. 30, 2025)
Session 15 Notes Video	Topic: Improper Integrals You will learn about the following topics: Definition of improper integrals Examples of improper integrals Applications: Integral test and gamma function Example of arrival times (queueing theory) Literature: CLP2 1.12; RHB Ch. 2.2.10
Session 16 Notes Video	Topic: Ordinary Differential Equations You will learn about the following topics: Definition of first-order ordinary differential equations Autonomous and separable equations Example: Exponential growth Example: Limited growth Equilibrium points and long time behavior of solutions Literature: CLP2 2.4; MW 8.5; RHB 14.2.1
Example Session	Queueing Theory (M/M/1), Another Example of Separation of Variables, Predator-Prey Models / Lotka-Volterra Equations, Blowup of Solutions, Non-uniqueness of Solutions
pdf of moodle quiz	Please submit on moodle
Week 9 (Mar. 31 - Apr. 6, 2025)
Session 17 Notes Video	Topic: Finite Difference Methods You will learn about the following topics: Forward/Backward/Central difference quotients Forward/Backward Euler methods Stability discussion of explicit and implicit schemes Literature: parts of RHB Ch. 27.6
Session 18 Notes Video	Topic: Definitions of Total/Partial and Directional Derivatives You will learn about the following topics: Vector valued functions of many variables Definition of (total) differentiability and the total derivative Uniqueness of the derivative Directional derivatives Partial derivatives Literature: RHB 5.1 and 5.2
Example Session	Finite Differences and Stability for the Damped Harmonic Oscillator, Total/Partial/Directional Derivatives
pdf of moodle quiz	Please submit on moodle
pdf of homework sheet	Covering Weeks 9 and 10. Please submit on moodle
Week 10 (Apr. 7 - Apr. 13, 2025)
Session 19 Notes Video	Topic: Connections between Total/Directional/Partial Derivatives You will learn about the following topics: Examples of total/directional/partial derivatives Jacobian matrix Total differentiability implies partial difffentiability Continuous total differentiability implies continuous partial difffentiability Literature: RHB 5.1 and 5.2
Session 20 Notes Video	Topic: Chain Rule, Gradient, Higher-order Derivatives, Taylor Expansion You will learn about the following topics: Chain Rule Gradient and a necessary conditions for extrema Higher-order derivatives and Schwarz' theorem Hessian matrix Second order Taylor expansion Literature: RHB 5.5 and parts of 5.8
Example Session	Jacobian, Hessian, Taylor; Leibniz integral rule
pdf of moodle quiz	Please submit on moodle
Week 11 (Apr. 21 - Apr. 27, 2025)
Session 21 Notes Video	Topic: Change of Variables, Differentials, Differential Operators You will learn about the following topics: Polar coordinates The gradient in polar coordinates Exact and inexact differentials Gradient, divergence, curl Literature: RHB Ch. 5.6 Change of variables, Ch. 5.3 Exact and inexact differentials, Ch. 10.7 Vector operators (more involved than what we discussed in class)
Session 22 Notes Video	Topic: Critical Points, Maxima and Minima You will learn about the following topics: Critical points, maxima, minima, saddle points, degenerate critical points Using the Hessian to conclude existence of maxima or minima Using the eigenvalues of the Hessian to conclude existence of maxima or minima Literature: RHB Ch. 5.8 Stationary values of many-variable functions. See also the wikipedia article Second partial derivative test
Example Session	Exact and Inexact Differentials, Differential Operator Identities, Optimization
pdf of moodle quiz	Please submit on moodle
pdf of homework sheet	Covering Weeks 11 and 12. Please submit on moodle
Week 12 (Apr. 28 - May 4, 2025)
Session 23 Notes Video	Topic: Lagrange Multipliers You will learn about the following topics: Isoperimetric problem Level sets and the gradient Method of Lagrange multipliers Literature: RHB Ch. 5.9 Stationary values under constraints. CLP 3 Ch. 2.10 Lagrange Multipliers
Session 24 Notes Video	Topic: Least Squares and Gradient Descent You will learn about the following topics: Solving the least square problem with Calculus Gradient descent Potential problems of gradient descent Stochastic gradient descent Literature: Strang - Introduction to Linear Algebra, Ch. 4.3 Least Squares Approximations. See also the wikipedia article Gradient descent
Example Session	Lagrange Multipliers
pdf of moodle quiz	Please submit on moodle
Week 13 (May 5 - May 11, 2025)
Session 25 Notes Video	Topic: Higher Order Linear ODEs You will learn about the following topics: Example of an RLC circuit Definition of higher order linear ODEs General solution to the homogeneous equation Particular solution to the inhomogeneous equation Find solutions to the homogeneous equation using the characteristic polynomial Some strategies for finding particular solutions to the inhomogeneous equation Literature: RHB Ch. 15.1.1, 15.1.2, 15.1.3. See the rest of Ch. 15 for a more general treatment.
Session 26 Notes Video	Topic: Partial Differential Equations You will learn about the following topics: Definition of a PDE Wave equation Heat equation Laplace and Poisson equations Schrödinger equation Solving the one dimensional heat equation with separation of variables Using Fourier series to solve PDEs Literature: A very good reference is the book "Boyce, DiPrima - Elementary Differential Equations and Boundary Value Problems" Chapter 10.5 (or even the whole Chapter 10 for a more comprehensive treatment of PDEs). RHB also discusses PDEs in detail in Ch.s 20 and 21; see in particular the overview from Ch. 20.1 and Ch. 20.5 for a different way of solving the heat equation. Ch. 21.2 is a bit more similar to the method we discuss in class.
Example Session	Solving a Higher Order Linear ODE
pdf of moodle quiz	Please submit on moodle
pdf of homework sheet	Covering Weeks 13 and 14. This sheet cannot be submitted, but solutions will be provided on moodle.
Week 14 (May 12 - May 18, 2025)
Session 27 Notes Video	Topic: Definition of Fourier Series and Examples You will learn about the following topics: Idea behind Fourier series Definition of the Fourier series and Fourier coefficients Example of a step function (square puls) Example of a (periodic) parabola Literature: RHB Ch. 12. Another good reference is "Boyce, DiPrima - Elementary Differential Equations and Boundary Value Problems", Ch.s 10.2, 10.3, 10.4.
Session 28 Notes Video	Topic: Convergence and Applications of Fourier Series You will learn about the following topics: Bessel's inequality Parseval identity Mean-square convergence Behavior at discontinuities and Gibbs phenomenon Applying Fourier series to solve PDEs Definition of Fourier transform Literature: RHB Ch. 12. Another good reference is "Boyce, DiPrima - Elementary Differential Equations and Boundary Value Problems", Ch.s 10.2, 10.3, 10.4.
May 24, 2025
Final Exam
Aug 26, 2025
Final Exam (make-up)