Math105
Calculus 1
Faculty
Andrea Serio
Data Scientist & Ph.D. in Mathematics
Course length
Duration
Total hours
Credits
Language
Course type
Fee for single course
Fee for degree students
Skills you’ll learn
Overview
Calculus 1 introduces students to the fundamental principles of continuous higher mathematics, with a focus on single-variable functions, limits, differentiation, asymptotic analysis, and integration. Students explore the concept of infinity through limits of sequences and functions, develop fluency in differentiation techniques, and apply derivatives to solve optimisation and curve-sketching problems.
The course introduces asymptotic notation (Big O, little o) and Taylor polynomial approximations, bridging pure mathematics with algorithm analysis and computational science. It also covers primitive functions, integration techniques, and the Fundamental Theorem of Calculus.
Through a blend of theoretical rigour, practical problem-solving, and optional computational tools, students acquire the foundational skills required for advanced coursework in computer science, linear algebra, and multivariable calculus.
Learning highlights
- Master the concepts of limits, continuity, and infinite behaviour in sequences and real-valued functions
- Gain proficiency in differentiation techniques, including implicit differentiation and higher-order derivatives
- Apply derivatives to real-world optimisation problems, function analysis, and rate-of-change models
- Understand asymptotic behaviour (Big O and little o) and approximate functions using Taylor polynomials
- Master integration techniques (such as substitution and integration by parts) and apply the Fundamental Theorem of Calculus
- Use computational tools to visualise calculus concepts, including Riemann sums and function approximations
Course outline
15 classes
Introduction to Calculus & Sequences
- Course overview and review of prerequisites.
- Sequences of real numbers, monotonicity, boundedness, and convergence.
- Definition, properties and intuitive meaning of the limit of a sequence.
Limits of Functions
- Definition of the limit of a real function f(x) as x → a and x → ±∞.
- One-sided limits, squeeze theorem, and fundamental trigonometric limits. E.g.: limx →0 sin(x)/x = 1.
- Limit laws and standard limit evaluation techniques.
Continuity & Asymptotes
- Continuous functions, types of discontinuities (removable, jump, essential).
- Properties of continuous functions: Extreme Value Theorem (EVT) and Intermediate Value Theorem (IVT).
- Vertical, horizontal, and slant asymptotes of real functions.
The Derivative & Tangent Lines
- The rate of change problem and geometric tangent line definition.
- Formal definition of the derivative as a limit of difference quotients.
- Differentiability vs. continuity; physical interpretation (velocity, instantaneous change).
Differentiation Techniques & Rules
- Power rule, constant multiple, sum, product, and quotient rules.
- Derivatives of exponential, logarithmic, and trigonometric functions.
- The Chain Rule for composite functions.
Advanced Differentiation & Higher Order Derivatives
- Implicit differentiation and tangent lines to implicit curves.
- Derivatives of inverse trigonometric functions.
- Logarithmic differentiation.
- Higher-order derivatives f(n)(x).
Mean Value Theorem & Optimisation
- Fermat's Theorem, Rolle's Theorem, and the Mean Value Theorem (MVT).
- Critical points, local and absolute extrema on closed intervals.
- Applied single-variable optimisation problems with examples.
Curve Sketching & L'Hôpital's Rule
- Mid-term evaluation.
- First and second derivative tests: monotonicity, concavity, and inflection points.
- Curve sketching framework.
- Indeterminate limits ( 0/0, ±∞/∞) and L'Hôpital's Rule.
Asymptotic Analysis (Big-O & Little-o notation)
- Concept of local and asymptotic function comparison.
- Definition and properties of Big- O and little-o notations.
- Hierarchy of function growth rates (logarithmic, polynomial, exponential, factorial) and application to algorithm analysis.
Local Approximations & Taylor Polynomials
- Linear (tangent line) and quadratic local approximations.
- Taylor and Maclaurin polynomials of degree n.
- Taylor's Theorem with Lagrange remainder; error estimation for function approximations.
Primitive functions & Indefinite Integration
- The concept of primitive function and basic indefinite integrals.
- Differential equations preview: simple initial value problems ( y' = f(x)).
- Integration technique I: Integration by Substitution.
Integration Techniques
- Integration technique II: Integration by Parts.
- Overview of algebraic integration (partial fraction decomposition for simple rational functions).
- Practical strategy for choosing integration methods.
Definite Integrals & Fundamental Theorem of Calculus
- The area problem, net signed area, and formal definition of Riemann sums.
- Definite integrals and their algebraic properties.
- Fundamental Theorem of Calculus: connecting differentiation and integration.
Applications of Integrals & Course Review
- Applications of the integrals with examples.
- Comprehensive review of the course and preparation for the final exam
Final Examination
- Final Assessment covering topics from the whole course.
Course materials
Books
Media
Prerequisites
High-school algebra, trigonometry, exponential and logarithmic functions.
Recommended prerequisite: Course Intro to Higher Mathematics (or equivalent familiarity with basic mathematical logic, sets, and notation).
Understanding of basic mathematical notation.
Basic Python programming skills for numerical experiments.
Methodology
The course follows a 3-hour daily session structure over 3 weeks (14 teaching sessions + 1 exam session):
* Lectures & Interactive Seminars (60%): Interactive presentation of core theorems, intuitive explanations, and live problem-solving on the whiteboard.
* Hands-on Workshops & Problem Sets (30%): Collaborative in-class practice sessions where students solve problem sheets with immediate instructor feedback.
* Optional code demonstrations are provided for visual and numerical intuition.
* Self-Study & Individual Homework (10%): Daily practice assignments reinforcing theoretical proofs and computational techniques.
Grading
Apply for this course
Calculus 1
by Andrea Serio
Total hours
45 Hours
Dates
Nov 30 - Dec 18, 2026
Fee for single course
€1500
Fee for degree students
€750
How to secure your spot
Complete the form below to kickstart your application
Schedule your Harbour.Space interview
If successful, get ready to join us on campus
FAQ
Will I receive a certificate after completion?
Yes. Upon completion of the course, you will receive a certificate signed by the director of the program your course belonged to.
Do I need a visa?
This depends on your case. Please check with the Spanish or Thai consulate in your country of residence about visa requirements. We will do our part to provide you with the necessary documents, such as the Certificate of Enrollment.
Can I get a discount?
Yes. The easiest way to enroll in a course at a discounted price is to register for multiple courses. Registering for multiple courses will reduce the cost per individual course. Please ask the Admissions Office for more information about the other kinds of discounts we offer and what you can do to receive one.