# Infinitesimal Calculus I (MAY111)

### General

School School of Science Department of Mathematics Undergraduate MAY111 1 Infinitesimal Calculus I Lectures (Weekly Teaching Hours: 5, Credits: 7.5) General Background - Language of Instruction (lectures): Greek.Language of Instruction (activities other than lectures): Greek and EnglishLanguage of Examinations: Greek and English Yes (in English) See eCourse, the Learning Management System maintained by the University of Ioannina.

### Learning Outcomes

Learning outcomes Here, the acronym RFooV stands for Real Function of one Variable. Remembering: Introduction to the sets of Natural Numbers, Integer Numbers, Rational Numbers, Irrational Numbers and Real Numbers, viewed from the aspect of Mathematical Analysis. Bounded and not bounded subsets of such sets. Basic concepts of Trigonometry. The concept of RFooV. Some basic properties of such functions. Elementary RFooVs. The concept of real valued sequences. Study of such sequences, including existence and calculation of limits. Limits and continuity of RFooV, using the (ε-δ) definition and the sequential definition. Basic properties of convergent RFooVs. Basic properties of continuous RFooVs. Classes of non-continuous RFooVs. Derivative of RFooV using the (ε-δ) definition and the sequential definition. Derivatives of elementary RFooVs. Calculation of derivatives. Comprehension: Methods of establishing a mathematical concept based on axioms and based of construction. Calculation and finding properties of sets of real numbers. Minimum upper and maximum lower boundaries. Graphing RFooVs, monotone RFooVs, bounded RFooVs, periodic RFooVs. Subsequences, the Bolzano-Weierstass Theorem, Cauchy sequences. Local behaviour of continuous RFooVs. The Bolzano Theorem and the Intermediate Values Theorem. Properties of continuous RFooVs defined in closed intervals, continuity of reverse continuous RFooVs. Uniform continuity of RFooVs defined in closed intervals. Methods of derivation, higher order derivatives. The Rolle Theorem, the Mean Value Theorem, the Darboux Theorem. The connection between derivative and monotonicity, extrema of RFooVs, convex and concave RFooVs, inflections points. Theorems for the derivation of inverse RFooVs. Generalized Mean Value Theorem, the De L’ Hospital Rule. Studying RFooVs using derivatives. Applying: Existence and uniqueness of solutions of non-linear equations. Finding maximum and minimum values of quantities, which emerge in problems in Natural Sciences. Plotting RFooVs. Evaluating: Teaching undergraduate courses. Creative, analytical and inductive thinking. Required for the creation of new scientific ideas. Working independently. Working in groups. Decision making.

### Syllabus

• Real numbers, axiomatic foundation of the set of real numbers (emphasis in the notion of supremum and infimim), natural numbers, induction, classical inequalities.
• Functions, graph of a function, monotone functions, bounded functions, periodic functions. Injective and surjective functions, inverse of a function. Trigonometric functions, inverse trigonometric functions, exponential and logarithmic functions, hyperbolic and inverse hyperbolic functions.
• Sequences of real numbers, convergent sequences, monotone sequences, sequences defined by recursion, limits of monotone sequences, nested intervals. The notion of subsequence, Bolzano Weierstass’ Theorem, Cauchy sequences. Accumulation points of sequences, upper and lower limit of a sequence (limsup, liminf).
• Continuity of functions, accumulation points and isolated points, limits of functions, one sided limits, limits on plus infinity and minus infinity. Continuity of several basic functions, local behaviour of a continuous function. Bolzano Theorem and intermediate value theorem. Characterization of continuity via sequences, properties of continuous functions defined on closed intervals, continuity of inverse functions.
• Derivative of a function, definition and geometric interpretation, examples and applications in sciences. The derivatives of elementary functions, derivation rules, higher order derivation. Rolle’s Theorem, Mean Value Theorem, Darboux’s theorem. Derivative and the monotonicity of a function, extrema of functions, convex and concave functions, inflection points. Derivation of inverse functions. Generalized Mean Value Theorem, De L’ Hospital rule. Study of functions using derivatives.

### Teaching and Learning Methods - Evaluation

Delivery
1. Lectures in class.
2. Learning Management System (e.g.: Moodle).
Use of Information and Communications Technology
1. Use of Learning Management System, combined with File Sharing Platform as well as Blog Management System for distributing teaching material, submission of assignments, course announcements, gradebook keeping for all students evaluation procedures, and communicating with students.
2. Use of Appointment Scheduling System for organising appointments between students and the teacher.
3. Use of Survey Web Application for submitting anonymous evaluations regarding the teacher.
4. Use of Wiki Engine for publishing manuals regarding the regulations applied at the exams processes, the way teaching is organized, the grading methods, as well as the use of the online tools used within the course.
Teaching Methods
Lectures 65
Study and analysis of bibliography 100
Preparation of assignments and interactive teaching 22.5
Course total 187.5
Student Performance Evaluation

Language of evaluation: Greek and English.
Methods of evaluation:

1. Weekly presentations - oral exams, combined with weekly written assignments.
2. In any case, all students can participate in written exams at the end of the semester.

The aforementioned information along with all the required details are available through the course's website. The information is explained in detail at the beginning of the semester, as well as, throughout the semester, during the lectures. Reminders are also posted at the beginning of the semester and throughout the semester, through the course’s website. Upon request, all the information is provided using email or social networks.

### Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

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