# CALCULUS 1 (UNIT 2)

## Learning outcomes of the course unit

At the end of the course, the student must have acquired the basic knowledge and skills of integro-differential calculus for functions of one real variable; he must also be able to understand how these can be applied to solve concrete problems and to handle them easily also in relation to other areas of mathematics. In particular, the student should:

1) be familiar with the concept of convex function, also in relation to the level of regularity of the function itself; have a solid knowledge of the Riemann integration theory (possibly over unbounded intervals and for unbounded functions) and understand how to apply it to the calculation of integrals. Recognize the various convergence criteria for numeric series and include the generalized integral relation. Know the concept of uniform continuity and its consequences (only MAT). Be able to handle linear differential equations of the first and second order and the associated Cauchy problems (only PHYS) (Knowledge and understanding)

2) apply the theoretical knowledge acquired to the resolution of even complex concrete problems. Understand its applicability to other areas of mathematics (Applying knowledge and understanding)

3) evaluate the consistency and accuracy of the results obtained; analyze the appropriate remedial strategies for solving exercises, relying on the collection of tools they believe to possess (Making judgments)

4) use a formally correct language allowing to communicate clearly, accurately and concisely the content of the program. Frontal lessons and direct comparisons with the teacher will encourage the student to acquire a specific and appropriate scientific vocabulary (Communication skills)

5) deepen their knowledge, starting from the basics provided in the course, in order to be able to appropriately and effectively manage the use of further analytical tools and concepts, both in subsequent courses and in individual study (Learning skills)

At the end of the course, the student must have acquired the basic knowledge and skills of integro-differential calculus for functions of one real variable; he must also be able to understand how these can be applied to solve concrete problems and to handle them easily also in relation to other areas of mathematics. In particular, the student should:

1) be familiar with the concept of convex function, also in relation to the level of regularity of the function itself; have a solid knowledge of the Riemann integration theory (possibly over unbounded intervals and for unbounded functions) and understand how to apply it to the calculation of integrals. Recognize the various convergence criteria for numeric series and include the generalized integral relation. Know the concept of uniform continuity and its consequences (only MAT). Be able to handle linear differential equations of the first and second order and the associated Cauchy problems (only PHYS) (Knowledge and understanding)

2) apply the theoretical knowledge acquired to the resolution of even complex concrete problems. Understand its applicability to other areas of mathematics (Applying knowledge and understanding)

3) evaluate the consistency and accuracy of the results obtained; analyze the appropriate remedial strategies for solving exercises, relying on the collection of tools they believe to possess (Making judgments)

4) use a formally correct language allowing to communicate clearly, accurately and concisely the content of the program. Frontal lessons and direct comparisons with the teacher will encourage the student to acquire a specific and appropriate scientific vocabulary (Communication skills)

5) deepen their knowledge, starting from the basics provided in the course, in order to be able to appropriately and effectively manage the use of further analytical tools and concepts, both in subsequent courses and in individual study (Learning skills)

## Prerequisites

Calculus (Unit 1)

Calculus (Unit 1)

## Course contents summary

Convexity and integral calculus for functions of one variable, series.

Convexity and integral calculus for functions of one variable, series.

## Course contents

Convexity: convex functions, monotonicity of incremental ratios, relation between convexity, first derivative and sign of the second derivative.

Integrals: partitions of an interval; Riemann sums; Riemann integral; integrability of monotone functions and of continuous functions (w/o proof for PHYS); integral mean; fundamental theorem of integral calculus; primitives; integration by parts; integration by substitution; integration of rational functions.

Series: Convergent, divergent and undetermined series; series with positive terms, comparison, ratio and root tests; absolute convergence, Leibnitz criterion; esamples: geometric series, telescopic series, generalized harmonic series, alternating harmonic series.

Improper integrals: definition for bounded and unbounded intervals, convergence of the integral, absolute convergence, comparison tests. Integral test for positive valued series.

Complements (both for MATH and PHYS): countable and noncountable sets, uncountability of real numbers.

Complements (only for MATH): uniform continuity, Heine-Borel theorem, Riemann-integrability of continuous functions (proof).

Complements (only for PHYS): Ordinary differential equations: order of a differential equation, Cauchy problem, separation of variables, explicit solutions to homogeneous first order linear equations and second order linear equations with constant coefficients, variation of constants method for inhomogeneous equations.

Convexity: convex functions, monotonicity of incremental ratios, relation between convexity, first derivative and sign of the second derivative.

Integrals: partitions of an interval; Riemann sums; Riemann integral; integrability of monotone functions and of continuous functions (w/o proof for PHYS); integral mean; fundamental theorem of integral calculus; primitives; integration by parts; integration by substitution; integration of rational functions.

Series: Convergent, divergent and undetermined series; series with positive terms, comparison, ratio and root tests; absolute convergence, Leibnitz criterion; esamples: geometric series, telescopic series, generalized harmonic series, alternating harmonic series.

Improper integrals: definition for bounded and unbounded intervals, convergence of the integral, absolute convergence, comparison tests. Integral test for positive valued series.

Complements (both for MATH and PHYS): countable and noncountable sets, uncountability of real numbers.

Complements (only for MATH): uniform continuity, Heine-Borel theorem, Riemann-integrability of continuous functions (proof).

Complements (only for PHYS): Ordinary differential equations: order of a differential equation, Cauchy problem, separation of variables, explicit solutions to homogeneous first order linear equations and second order linear equations with constant coefficients, variation of constants method for inhomogeneous equations.

## 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.

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.

## Teaching methods

The course includes 5 hours of frontal teaching per week. Lesson will be traditionally held at the blackboard and the topics will be presented rigorously. A direct interaction with students will be encouraged, also in order to highlight any previous gap on the topics dealt with and to promptly attempt to retrieve them. Weekly, a few hours will be devoted to exercises. This will allow to show to the students the applications of the theoretical results and to help them to understand more deeply the theory. Weekly, as part of the individual study, a file will be uploaded to the elly portal. The file will include various exercises to be solved and simple demonstrations, omitted in class, to be completed. Exercise hours will be devoted to the exhaustive resolution of exercises from the previous weeks, considered significant or requested by the students.

The course includes 5 hours of frontal teaching per week. Lesson will be traditionally held at the blackboard and the topics will be presented rigorously. A direct interaction with students will be encouraged, also in order to highlight any previous gap on the topics dealt with and to promptly attempt to retrieve them. Weekly, a few hours will be devoted to exercises. This will allow to show to the students the applications of the theoretical results and to help them to understand more deeply the theory. Weekly, as part of the individual study, a file will be uploaded to the elly portal. The file will include various exercises to be solved and simple demonstrations, omitted in class, to be completed. Exercise hours will be devoted to the exhaustive resolution of exercises from the previous weeks, considered significant or requested by the students.

## Assessment methods and criteria

The evaluation is done in two phases. The first consists in evaluating a written test where the student must solve exercises without the help of books and notes. The first part is successful if the student reaches a score non inferior to 15; the maximum score is 30. The solution of the test will be uploaded to the elly portal as soon as the test is completed; the test will be corrected the same day and the votes will be announced as soon as possible. For students who have passed the two partial written test scheduled in the first Unit, the written test will just focus on the program of the second Unit and the vote of this will average with those of the first two partial tests. For all the other students, it will cover the whole program of the two Units. Exceeding the written test allows the student to take the oral exam, which will concern the whole program of the course and will mainly focus on theoretical aspects (definitions, theorems, proofs). The student will have to show knowledge and appropriate understanding of the program using a correct mathematical formalism and owing a proper language. The final vote will be given by a weighted average of the two votes. The purpose of this type of assessment is to try to reliably evaluate the level of achievement of learning outcomes expected above, in particular points 1) to 4).

The evaluation is done in two phases. The first consists in evaluating a written test where the student must solve exercises without the help of books and notes. The first part is successful if the student reaches a score non inferior to 15; the maximum score is 30. The solution of the test will be uploaded to the elly portal as soon as the test is completed; the test will be corrected the same day and the votes will be announced as soon as possible. For students who have passed the two partial written test scheduled in the first Unit, the written test will just focus on the program of the second Unit and the vote of this will average with those of the first two partial tests. For all the other students, it will cover the whole program of the two Units. Exceeding the written test allows the student to take the oral exam, which will concern the whole program of the course and will mainly focus on theoretical aspects (definitions, theorems, proofs). The student will have to show knowledge and appropriate understanding of the program using a correct mathematical formalism and owing a proper language. The final vote will be given by a weighted average of the two votes. The purpose of this type of assessment is to try to reliably evaluate the level of achievement of learning outcomes expected above, in particular points 1) to 4).