# CALCULUS 1

*Last update 30/06/2021 21:57*

OVERVIEW

This introductory calculus course builds up on the mathematics learnt during the high school. The main topics of the course are differentiation and integration of functions of one variable.

## AIMS AND CONTENT

LEARNING OUTCOMES

The basic objective of Calculus is to relate small-scale (differential) quantities to large-scale (integrated) quantities. This is accomplished by means of the Fundamental Theorem of Calculus. Students should demonstrate an understanding of the integral as a cumulative sum, of the derivative as a rate of change, and of the inverse relationship between integration and differentiation.

AIMS AND LEARNING OUTCOMES

At the end of this course the students are expected:

- to master the mathematical notation;
- to know the properties of the elementary functions and their graph;
- to be able to follow the mathematical arguments;
- and to solve simple exercises, and discuss the results obtained.

PREREQUISITES

Sets, equalities and inequalities, analytic geometry, trigonometry.

Teaching methods

Both theory and exercises are presented by the teachers. Some tutorials will be carried out during the semester.

SYLLABUS/CONTENT

**The real numbers - **The real numbers, maxima, minima, supremum, infimum.

**Functions **- Elementary functions, composite function, inverse function.

**Limits and continuity - **Limits of functions. Continuity. Global properties of continuous functions. The intermediate value theorem and the extreme value theorem.

**Differentiation -** Derivative of a function. Tangent line. Derivative of the composite function and of the inverse function. The theorems of Rolle, Chauchy and Lagrange. De l’Hôpital's rule.

**Integration** - Riemann and Cauchy sums. Indefinite integral. Area of a planar region. Mean value theorem. Integral functions. The fundamental theorem of calculus. Calculating primitives.

RECOMMENDED READING/BIBLIOGRAPHY

Some notes and exercises are available.

**Recommended books**

- M. Oberguggenberger, A. Ostermann,
*Analysis for Computer Scientists: Foundations, Methods, and Algorithms*, Springer-Verlag,

ISBN 978-0-85729-445-6 - M. Baronti, M., F. De Mari, R. van der Putten, I. Venturi,
*Calculus Problems*, Springer-Verlag, ISBN: 978-3-319-15427-5 - G. Crasta- A. Malusa,
*Elementi di Analisi Matematica e Geometria con prerequisiti ed esercizi svolti,*Edizione La Dotta, ISBN: 978-88-986482-5-2

## TEACHERS AND EXAM BOARD

**Ricevimento:**
The teacher is available for explanations one afternoon a week.

Exam Board

FEDERICO BENVENUTO (President)

GIOVANNI ALBERTI

## LESSONS

Teaching methods

Both theory and exercises are presented by the teachers. Some tutorials will be carried out during the semester.

LESSONS START

According to the accademic calendar

ORARI

L'orario di tutti gli insegnamenti è consultabile su EasyAcademy.

## EXAMS

Exam description

The exam consists of two written tests:

- Test with multiple answers. The test consists of 10 questions with multiple answers. For each question, 3 points are given for the exact answer, -1 points for a wrong answer and 0 points if no answer is provided. In order to participate to the second part of the exam the students need to obtain at least 15 points. Duration of the test: 1 hour.
- Written test with exercises. The test consists of 3 exercises articulated in several questions. In order to pass this part the students need to obtain at least 18. Duration of the test: 2 hours and 30 minutes.

The final mark of the exam is given by

(first part)*1/3 + (second part)*2/3

rounded up to the nearest whole number.

Assessment methods

- The first part of the exam allows us to verify the ability of the students to handle the mathematical notation and to make simple deductive reasonings.
- The second part allows us to verify the ability to solve simple calculations and the knowledge of the main tools related to differentiation and integration.