Session 01-01 - Welcome to the Program

Section 01: Mathematical Foundations & Algebra

Dr. Nikolai Heinrichs & Dr. Tobias Vlćek

Introduction

Teaching Team

Dr. Tobias Vlcek

Dr. Nikolai Heinrichs

About me

  • Field: Optimizing and simulating complex systems
  • Languages: of choice: Julia, Python and Rust
  • Interest: Modelling, Simulations, Machine Learning
  • Teaching: OR, Algorithms, Programming & Mathematics
  • Contact: vlcek@beyondsimulations.com

Nikolai and I really appreciate active participation and interaction!

Teaching Styles

  • Interactive Teaching: Presentation, discussion, and exercises
  • Assignments: To be solved here and at home
  • Mock Exams: We will hold mock exams to help you check your progress and understanding
  • Difficulty of the Feststellungsprüfung: Strongly depends on your background and math experience!

No worries, we will help you out if you have any questions!

Participation & Materials

  • Try actively participating in this course
  • You will find it much (!) easier and more interesting
  • Bring a laptop or tablet, a notebook and pen and a calculator
  • Slides are hosted online here
  • Some material will also be uploaded to Moodle due to copyright

If you find any good materials online, please share them with us and the group.

Calculator: Getting to Know Your Calculator

The MENU Screen

MENU screen showing available applications

  • Press MENU to see all applications
  • Use arrow keys to navigate the icon grid, press number or = to select
  • 1: Berechnungen (Calculate) is your main mode

Calculator: Essential Settings

Before we begin, let’s configure the calculator:

  1. Press SHIFT MENU (Setup)
  2. Select 1: Eingabe/Ausgabe → 1: Math –> Math for natural display
  3. Select 2: Winkeleinheit → 1: Gradmaß (D) for degrees

Settings menu

The natural display mode shows fractions and roots as you write them on paper!

Calculator: Menu Navigation

Keyboard layout with key functions

  • Use SHIFT + template keys for calculus (d/dx, Σ, Π)
  • Press MENU then navigate with arrow keys or press A for Gleichung/Funkt
  • CALC button for evaluating expressions with variables

Calculator vs. When to Show Work

Calculator is useful for:

  • Verifying your manual calculations
  • Complex arithmetic
  • Solving equations to check your work

Exams typically require you to show:

  • Differentiation steps (product rule, chain rule, etc.)
  • Setting up equations from word problems and manipulating them
  • Interpretation of results

Calculator: Common Mistakes to Avoid

Try to avoid the following if possible.

  1. Wrong angle mode - Check D/R indicator before trig calculations
  2. Forgetting parentheses - Use them liberally: \(\sin(30)\) not \(\sin 30\)
  3. Not closing brackets - Natural display helps avoid this
  4. Division issues - Use parentheses: \((a+b)/(c+d)\)
  5. Relying only on calculator - Understand why the answer is correct

Our Plan for Today

  • Introductions and course overview
  • Assigments to asses your current level
  • Group formation and contact exchange

Structure

Course Structure

This course consists of 8 sections plus a final review, approximately 60 sessions total:

  • Section 1-2: Foundations, algebra, and equations
  • Section 3-4: Functions and advanced functions
  • Section 5-6: Differential and integral calculus
  • Section 7: Probability and statistics
  • Section 8: Financial mathematics
  • Final Review: Mock-exam practice for the Feststellungsprüfung
  • Each section ends with a mock assessment in FSP style

Module Objectives

Upon completion, you will be able to:

  • Master mathematical language, notation, and logical reasoning
  • Solve algebraic equations and inequalities
  • Analyze and interpret functions and their properties
  • Apply calculus and financial mathematics to business problems
  • Use probability for risk assessment and decision-making
  • Successfully pass the Feststellungsprüfung and begin your Bachelor

Independent self-study is essential to achieve these objectives.

Sections 1-2

Foundations

  • Mathematical language and notation, number systems
  • Fractions, percentages, manipulating terms
  • Exponents, radicals, and logarithms
  • Linear, quadratic, and fractional equations
  • Systems of equations and the methods behind each

Question

Are these topics you are familiar with?

Sections 3-4

Functions

  • The function concept and its main families
  • Linear and quadratic functions
  • Transformations and inverses
  • Polynomial, power, exponential, rational, and trigonometric functions
  • Analysing the behaviour of functions

Question

What about these topics?

Sections 5-6

Calculus

  • Limits and derivatives, differentiation rules
  • Curve sketching and function families (Funktionsscharen)
  • Antiderivatives and definite integrals
  • Area problems and integration by parts, with applications

Question

How do you feel about these topics?

Sections 7-8

Probability & Financial Mathematics

  • Descriptive statistics and combinatorics
  • Conditional probability, Bayes, contingency tables
  • Binomial distribution
  • Interest, annuities, and cost analysis

Question

How do you feel about these topics?

Final Review

Exam Preparation

  • Consolidation across functions, calculus, integrals, and probability
  • Mock-exam and oral-exam practice for the Feststellungsprüfung
  • Final preparation and exam techniques

Each section ends with a mock assessment in FSP style, and from the first full mock onwards we also practice the oral exam format.

Assessment

  • No formal grading by KLU - we’re here to help you succeed
  • 75% attendance required due to KLU rules, tracked as points: 2 on time, 1 late, 0 absent (your points / max points)
  • At least 60% of test points required for FSP admission
  • Tests at the start of selected lectures, announced a week ahead
  • Smartphones stay on the lecturer’s desk during class; laptops and tablets are welcome
  • Feststellungsprüfung will be written individually in English

Please do not hesitate to contact us if you have any questions!

Assignments

  • You won’t have to hand in most assignments
  • We will discuss the results together in class
  • If you don’t do the assignments, you won’t be prepared!
  • It is your responsibility to prepare yourself for the exam
  • We are just here to help and guide you in the right direction

Assignments are not graded, so please do not worry about that! Ask us any question you have, no matter how stupid it may seem. If you are lost, even after we have explained something multiple times, ask us again! We want you to succeed and pass!

Keys to Success

  • Practice regularly: Mathematics improves with daily problem-solving
  • Participate actively: Ask questions and collaborate with peers
  • Form study groups: Learn collectively and support each other
  • Use mock exams: Use simulated exam conditions
  • Monitor progress: Identify weak points early and address them

Artificial Intelligence

How to use AI

  • We base our assessment on the KLU classification:
    • Level 1: Pause: Use of AI defined by the educator
  • Feel free to use AI tools to support understanding
  • A course chatbot is available on the learning website
  • It is designed to guide your problem-solving process

Usage of AI is not allowed in the Feststellungsprüfung.

How to use the Chatbot

  • Just click the chatbot bubble on the website
  • The chat will open and you can ask your questions
  • It is programmed by us and uses Mistral AI as backend
  • Ask your question as specific as possible
  • This ensures enough context for the model
  • We can see aggregated logs, but cannot identify you
  • Please don’t provide personal information

LLMs and Mathematics

  • Large language models (LLMs) are pattern-based
  • They are not good at providing explicit solutions to mathematical tasks
  • They are better suited to explain concepts

Always question the output of LLMs, especially for math!

There are often better options as LLMs, such as Wolfram Alpha, which use symbolic computation rather than pattern matching, to solve math problems.

Assessment

What’s the status quo?

  • Lets find out how prepared you are currently
  • You will have approximately 2 hours to solve some tasks in the style of a mock exam
  • These will cover all important topics
  • If you can’t answer a task, just shortly note why

Instructions

Work individually and quietly

  • Work through the problems in order; pace yourself
  • If you are stuck, briefly note why and move on
  • No AI tools or internet resources during the assignments
  • Calculators: basic arithmetic use is fine; no CAS features
  • You may ask clarifying questions about wording, not solutions
  • This first assessment is diagnostic only; it does not count toward the 60%

Working Together

Learning Groups

Our idea

  • We will form small groups of 3 students
  • Groups will rotate after each section
  • We will just randomly assing you each time
  • Agree on a weekly study slot and communication channel

Question

Is this in your interest or do you prefer a different way?

That’s it for today.

Any Questions?

References

References

  • Interested in further literature to learn more about Mathematics?
  • Take a look at the reference list of this course
  • Here you can find recommended books and resources

Have you found anything worthwhile that might help your classmates or you? Please share it with us and we will update the references.