Calculus and Linear Algebra I

Jacobs University, Fall 2022

Official Class Description from Campusnet

This module is the first in a sequence introducing mathematical methods at university level in a form relevant for study and research in the quantitative natural sciences, engineering, Computer Science, and Mathematics. The emphasis in these modules lies in 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 done in the first-year modules "Analysis I" and "Linear Algebra". The lecture comprises the following topics

• Brief review of number systems, elementary functions, and their graphs
• Brief introduction to complex numbers
• Limits for sequences and functions
• Continuity
• Derivative
• Curve sketching and applications (isoperimetric problems, optimization, error propagation)
• Introduction to Integration and the Fundamental Theorem of Calculus
• Review of elementary analytic geometry
• Vector spaces, linear independence, bases, coordinates
• Matrices and matrix algebra
• Solving linear systems by Gauss elimination, structure of general solution
• Matrix inverse

News

• See here for exams from 2020 for further practice.
• First class is on Mon., Sep. 5. This will be a hybrid introductory session. Please either join online or come to Conrad Naber Lecture Hall.

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 on Alemira.
• Come to the online tutorial (held by the TAs) on Mondays at 14:15.
• Come to one of the example/question sessions on Wednesdays at 14:15 (held by the instructors). One is held online, one is held in person (Conrad Naber Lecture Hall).
• Work on the moodle exercises and submit them before the following tutorial (i.e., Mondays at 14:15).

Textbooks

• Feldman, Rechnitzer, Yeager: CLP-1 Differential Calculus and CLP-2 Integral Calculus (CLP)
• Marsden, Weinstein: Calculus I and Calculus II (MW)
• Edwards, Penney: Calculus (Prentice Hall)
• Riley, Hobson, Bence: Mathematical Methods for Physics and Engineering (3rd edition, Cambridge, 2006) (RHB)
• Treil: Linear Algebra Done Wrong (LADW)

Description of the textbooks from last year (M. Oliver):

CLP is a good modern Calculus textbook with relatively comprehensive coverage of all Calculus topics, except that some of the preliminary concepts (factorization of polynomials, fundamental theorem of algebra, inequalities, graphs of equations) are not or insufficiently covered, see reading recommendations in the syllabus below. The book is open source and freely available for browsing on the web and as a printable PDF download. I am trying to base the class on this book to the extent possible.

MW is an extremely clear and well-written classic and definitely one of my favorites. Its biggest drawback is that it follows a "late transcendentals" concept, i.e., trigonometric functions, exponentials, and logarithms are introduced only after integration, which limits the variety of available examples at the time when the key ideas are first introduced. However, for self-study or review, this may not matter so much. This book is also freely and legally available as a scanned PDF download.

The book by Edwards and Penney is a good mainstream Calculus textbook. It used to be the default reading for many years at Jacobs, so there are still quite a few paper copies in the IRC.

RHB is a comprehensive textbook on mathematical methods that covers a great variety of concepts, far beyond first-year Calculus. The introductory chapters are somewhat terse and assume familiarity with high-school Calculus without presenting a systematic introduction from scratch. Yet, the choice of examples and focus on advanced topics make this book a lasting resource.

For the short Linear Algebra part at the end, the class mostly follows RHB except for the details of the process of Gaussian eliminationand matrix inversion, which are covered by separate class notes. LADW is good background reading; the book is more advanced than what we are doing, but follows the same style and philosophy. Gaussian elimination is explained in the same way as here (except for a differen sign convention).

Table of Contents

Chapter 1: Functions
1.1: Numbers and Polynomials
1.2: Functions and their Graphs
1.3: Limits and Continuity

Chapter 2: Derivatives
2.1: Introduction to Derivatives and their Properties
2.2: Applications of Differentiation

Chapter 3: Integrals

Chapter 4: Differential Equations

Chapter 5: Vectors and Vector Spaces

Chapter 6: Matrices
6.1: Introduction to Matrices and Link to Linear Operators
6.2: Solving Systems of Linear Equations

Grading

The grade is only based on the final exam. Homework submissions can provide up to 10% bonus points according to the following table:

HW Points	Bonus
100 or more	10%
90 - 99	9%
80 - 89	8%
70 - 79	7%
60 - 69	6%
50 - 59	5%
40 - 49	4%
30 - 39	3%
20 - 29	2%
10 - 19	1%
less than 10	0%

Exams

There will be one final exam (centrally scheduled in December) and one make-up final exam (centrally scheduled in January). For practice, here is a mock exam:

Fall 2022 Mock Exam

Here is the final exam with solutions:

Fall Final Exam

And here are two exams from Fall 2020:

Fall 2020 Midterm Exam (solutions)
Fall 2020 Final Exam (solutions)

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. More practice exercises can be found on the course websites of previous years, e.g., Fall 2020 Course Website (see in particular the Homework Sheets and Solutions).

Moodle Exercises

Please go to Jacobs moodle, login, and select the Calculus and Linear Algebra I class to view the exercises and the solutions (after the due date). Each week on Sunday a new exercise sheet is released, and this is due 8 days later on the Monday before the tutorial, i.e., at 14:15.

Project Exercises

There are 3 project exercises: In Week 7, Week 10, and Week 12 (see the class schedule below). Try to solve them first by yourself; for reference, solutions are provided.

Class Schedule

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 (Sep. 5 - 11, 2022)
Session 1	Topic 1.1.A: Numbers and Roots of Polynomials You will learn about the following topics: Natural numbers, integers, rational numbers, real numbers Polynomials and roots Discriminant, irrationality of square root of 2 Literature: CLP1 0.1, additional reading in CLP1 0.2, 0.3
Session 2	Topic 1.1.B: Complex Numbers and the Fundamental Theorem of Algebra You will learn about the following topics: Complex numbers: real and imaginary part, multiplication Complex polynomials Complex roots of polynomials and the Fundamental Theorem of Algebra Literature: RHB 1.1
Example Session	Numbers, Complex Numbers, Roots of Quadratic Equations, Roots of Polynomials of Degree n > 2.
Exercise Sheet	Solutions
Week 2 (Sep. 12 - 18, 2022)
Session 3	Topic 1.1.C: Misc - Polynomial Long Division, Inequalities, Binomial Coefficients You will learn about the following topics: Polynomial long division Drawing polynomials Solving linear and quadratic inequalities Binomial coefficients Literature: CLP2 Example 1.10.2 (look at the polynomial long division only); Wikipedia article on Polynomial Long Division (American notation), see the German version for the notation used in class; MW R2 (for intervals and inequalities); RHB 1.5 (binomial coefficients)
Session 4	Topic: Equations, Functions and their Inverses, Graphs You will learn about the following topics: Definition of function, domain and range Discussion of standard functions (absolute value, parabola, hyperbola, sin, cos, tan, exponential function, logarithm Inverses of functions Literature: CLP1 0.4-0.6, MW R4-R6
Example Session	Polynomial Long Division, Drawing Polynomials, Binomial Coefficients, Exponential Function.
Exercise Sheet	Solutions
Week 3 (Sep. 19 - 25, 2022)
Session 5	Topic 1.3.A: Definition of Limits and Limit Laws You will learn about the following topics: Formal definition of limits Limit laws Left and right limits Literature: CLP1 1.3 and some parts of 1.4
Session 6	Topic 1.3.B: Asymptotes and Limits of the Exponential Function You will learn about the following topics: Horizontal asymptotes Vertical Asymptotes Limits involving the exponential function, the logarithm, and polynomials Proof method of induction Literature: parts of CLP1 1.4 and 1.5
Example Session	Epsilon-Delta-Exercise, Limit Laws, Asymptotes, Induction.
Exercise Sheet	Solutions
Week 4 (Sep. 26 - Oct. 2, 2022)
Session 7	Topic 1.3.C: 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.)
Session 8	Topic 2.1.A: General Definition You will learn about the following topics: know the general mathematical and practical definition of the derivative be able to apply the mathematical definition to get actual derivatives of certain important functions know what a differentiable function is know the derivatives of sine and cosine Literature: CLP1 2.1-2.3
Example Session	Extreme Value Theorem, Bisection Method, Application of Squeeze Law, The Exponential Function Again, Definition of Derivative.
Exercise Sheet	Solutions
Week 5 (Oct. 3 - 9, 2022)
Session 9	Topic 2.1.B: Differentiation Rules You will learn about the following topics: know and be able to apply all fundamental differentiation rules: Addition rule Product rule Multiplication with constant Quotient rule Chain rule Power rule Derivative of the inverse function know the derivatives of: Exponential function e^x ln(x) tan(x) arctan(x) Literature: Most rules: CLP1 2.4, 2.5 for advanced background reading, 2.6 for examples; Chain rule: CLP1 2.9; Derivatives of exp, log and trigonometric functions: parts of CLP1 2.7, 2.8, 2.10.
Session 10	Topic 2.1.C: Implicit Differentiation and Second Derivative You will learn about the following topics: know how to apply implicit differentiation for general equations know how to compute the second (and higher) derivatives and its relevance as the change of the slope know terminologies such as concave (up/down), convex, and point of inflection and their link to the second derivative Literature: implicit differentiation and derivative of inverse functions: CLP1 2.11, also MW 2.4; curvature of a function as an example (cf. RHB 2.1.9; implicit differentiation makes it much easier).
Example Session	Derivative Examples, Some Missing Proofs, Extension of Power Rule to Rational Exponents, Example Implicit Differentiation, Application of Second Derivative.
Exercise Sheet	Solutions
Week 6 (Oct. 10 - 16, 2022)
Session 11	Topic 2.1.D: Theorems of Differentiation You will learn about the following topics: know the various essential theorems of differentiation and understand their applicability as well as the concept of their proofs: Differentiability => continuity Max or min at c => f’(c)=0 Rolle’s theorem Mean value theorem f’(x)>= 0 (<=0) => f increasing (decreasing) be familiar with the term critical point Literature: parts of CLP1 2.13.
Session 12	Topic 2.2.A: Extreme Value Problems You will learn about the following topics: know the conditions for extrema and be able to find them, both local min and max be able to transform a simple text based extreme value problem into respective equations and solve them, potentially via different solution paths (e.g. direct vs implicit differentiation) Literature: CLP1 2.13, 2.14, parts of 3.5.
Example Session	Important Consequence of MVT, Example of MVT, Curve Discussion, Extreme Value Problem (Snell's Law).
Exercise Sheet	Solutions
Week 7 (Oct. 17 - 23, 2022)
Session 13	Topic 2.2.B: Graph Sketching You will learn about the following topics: know the general checklist for curve sketching and be able to apply all necessary techniques from chapters 1 and 2 to sketch a graph Literature: CLP1 3.6.
Session 14	Topic 3.A: 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	Graph Sketching, Integration by Parts, Integration by Substitution.
Exercise Sheet	Solutions
Project 1 Exercise	Try solving these exercises first before looking at the solutions.
Week 8 (Oct. 24 - 30, 2022)
Session 15	Topic 3.B: Integration of Rational Functions You will learn about the following topics: Definition of rational functions General strategy for integrating rational functions Partial fractions Literature: CLP2 1.10
Session 16	Topic 3.C: Definite Integrals and the Fundamental Theorem of Calculus You will learn about the following topics: Definite integrals and their meaning Properties of definite integrals The Fundamental Theorem of Calculus and its proof Literature: CLP2 1.1-1.3
Example Session	Integrals of Rational Functions, Definite Integral by Hand, Integral Mean-value Theorem.
Exercise Sheet	Solutions
Week 9 (Oct. 31 - Nov. 6, 2022)
Session 17	Topic 3.D: Applications of Integration You will learn about the following topics: Area between curves Volume computations Work in physics Rocket Equation Literature: CLP2 1.5, 1.6, 2.1
Session 18	Topic 3.E: Improper Integrals You will learn about the following topics: Definition of different types of improper integrals Examples of improper integrals Application to escape velocity Literature: CLP2 1.12
Example Session	Area between Curves, Gamma Function, Taylor Series.
Exercise Sheet	Solutions
Week 10 (Nov. 7 - 13, 2022)
Session 19	Topic 4.A: Common 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
Session 20	Topic 5.A: Introduction to Vectors and Vector Operations You will learn about the following topics: be familiar with vectors and their basic operations and applications know the polar decomposition of a vector, its respective magnitude and unit vector (i.e. direction) know and be able to apply scalar and cross product and understand their geometrical interpretation and related rules Literature: selected topics from RHB Chapter 7.
Example Session	Newton's Law of Cooling, Another Example of Separation of Variables, Predator-Prey Models/Lotka-Volterra Equations, Examples of Scalar and Cross Product, Vector Application (Centroid of Triangle).
Exercise Sheet	Solutions
Project 2 Exercise	Try solving these exercises first before looking at the solutions.
Week 11 (Nov. 14 - 20, 2022)
Session 21	Topic 5.B: Lines, Planes, and Vector Spaces You will learn about the following topics: be familiar with different line and plane equations and how to transform one into another know and understand the definition and concept of vector spaces be able to name both examples and counter-examples be familiar with related terms such as linear (in-)dependence of vectors, basis and dimension of a vector space, and the span of vectors to construct vector spaces Literature: Equations for lines and planes: RHB 7.7; Vector spaces etc.: RHB 8.1, 8.1.1.
Session 22	Topic 6.1.A: Linear Operations (Transformations) on Vector Spaces You will learn about the following topics: know the rules for a linear operator and understand how a linear operator is related to a matrix via basis representation know what a matrix is, and understand the general application of a matrix to vectors from a vector space, e.g. matrix-vector multiplication Literature: RHB 8.2, 8.3.
Example Session	Vector Spaces (Examples), Basis, Matrices.
Exercise Sheet	Solutions
Week 12 (Nov. 21 - 27, 2022)
Session 23	Topic 6.1.B: Basic Matrix Operations You will learn about the following topics: understand the correspondence between linear operators and matrices with respect to addition, scalar multiplication, and matrix-matrix multiplication know the definition and application of matrix transpose (and Hermitian conjugate) Literature: RHB 8.4, 8.6, 8.7.
Session 24	Topic 6.2.A: Systems of Linear Equations You will learn about the following topics: understand the correspondence of between linear systems of equations and matrix-vector representation Ax=b know the terminology of homogeneous and inhomogeneous linear systems of equations and their relevance for the existence and uniqueness of solutions for Ax=b be able to apply Gaussian elimination to find solution(s) to Ax=b know the definition of the identity matrix Literature: Gaussian elimination: See example here.
Example Session	Matrix Multiplication, Solving Systems of Linear Equations.
Exercise Sheet	Solutions
Project 3 Exercise	Try solving these exercises first before looking at the solutions.
Week 13 (Nov. 28 - Dec. 4, 2022)
Session 25	Topic 6.2.B: Solution Space of Systems of Linear Equations You will learn about the following topics: understand the meaning and relevance of pivots for the amount of solutions to Ax=b know and understand the concepts of under- and overdetermined in the context of Ax=b know and understand the concepts of rank, kernel, range and nullity of a matrix A and their relevance for solutions spaces of Ax=b be familiar with the rank-nullity theorem and its consequences for existence and uniqueness of solutions of Ax=b know the terms inverse as well as invertible and (non-)singular matrix Literature: RHB 8.18.1.
Session 26	Topic 6.2.C: Inverse of a Matrix You will learn about the following topics: know general rules for the inverse A^-1 of A and theorems for its existence be able to compute the inverse of A (if it exists) understand the concept of change of basis and the consequences for linear operators with respect to different bases Literature: RHB 8.15 first half.
Example Session	Solving Systems of Linear Equations, Change of Basis.
Exercise Sheet	Solutions
Week 14 (Dec. 5 - Dec. 7, 2022)
Review Session
Dec. 12, 2022
Final Exam
Jan. 23, 2023
Final Exam (make-up)