# CALCULUS 1 (UNIT 2)

## LEARNING OUTCOMES OF THE COURSE UNIT

Knowledge and understanding:

At the end of this course the student should know the essential definitions

and results in analysis in one variable, and he should be able to grasp how

these enter in the solution of problems.

Applying knowledge and understanding:

The student should be able to apply the forementioned notionsto solve medium

level problems, and to understand how they will be used in a more applied

context.

Making judgements:

The student should be able to evaluate coherence and correctness of the

results obtained by him or presented him.

Communication skills:

The student should be able to communicate in a clear and precise way, also

in a context broader than mere calculus.

## PREREQUISITES

No

## COURSE CONTENTS SUMMARY

Real analysis, functions of one variable, sequences, series

## RECOMMENDED READINGS

E. Acerbi, G. Buttazzo: Primo corso di Analisi Matematica, Ed. Pitagora, 1997.

E. Acerbi, G. Buttazzo: Analisi matematica ABC, Ed. Pitagora, 2000.

M. Bramanti, C.D. Pagani, S. Salsa: Analisi Matematica 1, Ed. Zanichelli, 2008.

M. Giaquinta, L. Modica, Analisi Matematica 1, vol. 1 & 2, Ed. Pitagora, 1998.

E. Giusti, Analisi matematica vol.1, Ed. Boringhieri, 2002

## ASSESSMENT METHODS AND CRITERIA

The examination is both written and oral.

In the written part, the student will show his basis knowledge and his ability in solving some particular paroblem. In the oral part, the student will show his knowledge of the foundamental theorems of Mathematical Analisys 1. The oral exposition must be done using a proper mathematical formalism.

## TEACHING METHODS

classroom lectures and classroom exercises

## COURSE CONTENTS

Integrals

Partitions of an interval; Riemann sums; Riemann integral; integrability of monotonic functions and of continuous functions;

integral mean; fundamental theorem of integral calculus; primitives; integration by parts; integration by substitution; integration of rational functions.

Improper integrals; convergence of the integral, absolute convergence,comparison tests. Integral test for positive valued series.

Asymptotic expansions

Landau symbols; Taylor's theorem; explicit formula of the remainder; Mac Laurin expansion of elementary functions; Taylor's series

Complements

Bolzano-Weirstrass theorem, compactness in the real line; Cauchy sequences; upper and lower limits;

uniform continuity.

Complex numbers.

Definitions, operations, complex plain, polar form, root extraction.

Differential equations

Nomenclature: order, linear and nonlinear; first examples; solutions of linear first order equations; solution of separable differential equations; constant coefficients linear differential equations.