Infinitesimal Calculus I

Code:

M111

Acronym:

M111

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2011/2012 - 1S

Active?	Yes
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Bachelor in Physics

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L:AST	40	Plano de Estudos a partir de 2008	1	-	7,5	-
L:F	45	Plano de estudos a partir de 2008	1	-	7,5	-
L:M	100	Plano de estudos a partir de 2009	1	-	7,5	-
MI:EF	50	Plano de Estudos a partir de 2007	1	-	7,5	-

Teaching language

Portuguese

Objectives

Acquiring knowledge of the basic concepts, results and techniques of the differential and integral calculus on one variable, including competences on the effective computation of derivatives, primitives and integrals, and approximations of functions by their Taylor polynomials and series.

Program

0. INTRODUCTION

The set of real numbers: algebraic structure, ordering and completeness. Sequences: definition, uniqueness of its limit, monotone and bounded sequences, subsequences.

1. DERIVATIVES AND ANTIDERIVATIVES

Geometric motivation and physical significance of the notion of derivative of a real function at a point. Definition and uniqueness of the limit of a function at a point. Lateral limits and the arithmetic of limits. Definition of derivative and lateral derivatives at a point. Antiderivatives. Derivatives and antiderivatives of elementary functions. Continuous functions. Relationship between continuity and derivability. Squeeze theorem. Derivatives and antiderivatives of sums and products by a scalar. Derivatives of products and quotients. Chain rule and associated integration rule. Derivative of the inverse function. Inverses of trigonometric functions and corresponding derivatives. Leibniz’s differential notation. Integration by substitution. Integration by parts. Integration of rational functions.

2. INTEGRALS

Concept of area: area of a rectangle, approximation of the area of a plane region by sums of areas of rectangles, properties of area, existence of area. Integral of a bounded function over an interval. Integrable functions. Basic properties of integrals. Average value of a function. The area function. The Fundamental Theorem of Calculus and its consequences. Computation of integrals. Integration by substitution and integration limits. Improper integrals: the case of continuous functions defined on unbounded intervals and the case of continuous unbounded functions defined on an interval.

3. THE FUNDAMENTAL THEOREMS OF CALCULUS AND APPLICATIONS

Theorems of continuity: the permanence of sign in the neighbourhood of a continuity point, Theorem of Intermediate Values, Theorem of Weierstrass. Theorems on differentiable functions: the derivative is zero in local extreme points (for functions with open domains), Mean Value Theorems (Rolle’s, Lagrange’s, Cauchy’s). Applications: determination of extremes, proof that a function defined on an interval with nil derivative is constant, determination of monotony intervals and concavity, classification of critical points. Limits at infinity and horizontal and oblique asymptotes. Infinite limits and vertical asymptotes. Indeterminate forms. L’Hôpital’s Rule. Heine’s characterization of limit. Functions of class $c^k$.

4. POLYNOMIAL APPROXIMATION AND SERIES

Polynomial approximation of functions:

Taylor polynomials, tangent of degree $n$ of a function and its Taylor polynomial of degree $n$ at a given point, Lagrange’s formula for the remainder. Application: irrationality of Neper’s number.

Numerical series:

Series of real numbers, sequence of partial sums, convergence. The general term converging to zero is a necessary condition for convergence of a series. Geometric series and the harmonic series. Leibniz’s criterion for alternate series. Arithmetic of series. Non-associativity of the terms in a series, associativity in the convergent case. Non-comutativity of the terms in a series, comutativity in the absolutely convergent case. Relationship between absolute convergence and convergence. The comparison criterion. Ratio and integral criteria for series of positive terms.

Power series:

Power series centred at a point, convergence domain. Convergence at a point. Interval and radius of convergence, and their determination. Differentiation and integration term by term of power series (proof omitted). Taylor series. Determination of the Taylor series of various elementary functions and application to the computation of the sum of numerical series.

Mandatory literature

000097905. ISBN: 978-0-495-38273-7
000098594. ISBN: 85-221-0479-4 (Vol. I)

Complementary Bibliography

Spivak, Michael; Calculus, Houston : Publish or Perish, 1994

Teaching methods and learning activities

Exposition of the theory by the teachers. Slides for study and support of the theoretical classes are available. Exercise sheets with previous indication of the exercises to be actually discussed in the theoretical-practical classes in each week are available, to stimulate previous work by the students. The webpage of the course contains other materials, e.g. tests and resolutions from previous years. Regular tutorial time to provide individual support to the students. The students have access to the evaluation tests and exams, and are entitled to receive all the explanations and corrections they require.

Evaluation Type

Distributed evaluation with final exam

Assessment Components

Description	Type	Time (hours)	Weight (%)	End date
Attendance (estimated)	Participação presencial	75,00
	Total:	-	0,00

Eligibility for exams

The students are not required to attend the classes. Missing more than a mini-test implies that the student must be approved in the final exam.

Calculation formula of final grade

1. The students have the opportunity of doing a total of 4 mini-tests during the semester.

2. The mini-tests are done in the practical classes (except possibly for the last one) during 1 hour.

3. Each mini-test will be marked with a maximum of 5 values.

4. If the sum of the markings of a student in the 4 mini-tests is at least 9.5 values, having at least 1.5 values in three of them, (s)he is not required to do the final exam at the regular period. In that case, the final classification, corresponds to the sum of values obtained in the mini-tests, after rounding, except possibly if that sum is equal or greater than 16,5 (see 6).

5. If the student completes and delivers the exam of the regular period, the final classification will be entirely determined by the result obtained in this exam. In particular, the student may fail to pass, independently of the markings obtained in the mini-tests. The same does not apply however to the exam of the appeal period, which will always be considered as the lawful attempt to get a better classification.

6. Those students who obtain a minimum of 16.5 points in the four mini-tests or in the exam, have the right to do a (written) exam of excellence, in a date to be announced after the publication of the regular exam results. If a student chooses not to do this excellence exam, his/her final classification will be 16. Neither (s)he will risk getting a final mark below these values, whatever the result of the excellence exam will be.

7. A student obtaining a mark superior to 8,5 and inferior to 9,5 in na exam has the right to require, in the two days following the publication of the results in the webpage, na oral exam.

Special assessment (TE, DA, ...)

The exams required under the special cases previewed in the law will be written, but may be preceded by na oral exam to establish if the student should be admitted or not to the written exam.

Classification improvement

The student has the right to make a (single) attempt to improve his final classification by doing the exam in one of the two exam periods following the one when he was approved. The final marking is the highest among the original marking and the marking of the new exam.

Observations

Article 13th of the General Regulation for Students’ Evaluation in the University of Porto, approved the 19th May 2010:

``Any student who commits fraud in an exam or test fails that exam and will face disciplinary charges by the University.''