Calculus I

Code:

M1001

Acronym:

M1001

Level:

100

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2021/2022 - 1S

Active?	Yes
Web Page:	https://moodle.up.pt/course/view.php?id=3584
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Bachelor in Computer Science

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L:B	0	Official Study Plan	3	-	6	56	162
L:CC	89	study plan from 2021/22	1	-	6	56	162
			2
L:F	0	Official Study Plan	2	-	6	56	162
			3
L:G	2	study plan from 2017/18	2	-	6	56	162
			3
L:IACD	63	study plan from 2021/22	1	-	6	56	162
L:Q	0	study plan from 2016/17	3	-	6	56	162

Teaching language

Portuguese

Objectives

To become acquainted with the basic concepts and techniques of calculus at the level of real-valued functions of a single real variable, as well as sequences and series.

Learning outcomes and competences

The capacity of solving calculus problems. Autonomy on the solution of exercises.

Working method

Presencial

Program

0. Generalities on functions: 

Polynomial functions. Trigonometric functions. Exponential functions.

1. Limits and continuity:

Sequences of real numbers. Basic results on sequences. Real-valued functions of a real variable. Limits. Continuity. Intermediate Value Theorem and Weierstrass Extreme Value Theorem.

2. Derivatives and antiderivatives:

Derivatives. Geometric and physical interpretation of derivatives. Differentiation rules. Derivative of the inverse. Inverse trigonometric functions and their derivatives. Theorems of Rolle, Lagrange and Cauchy, L ́Hôpital’s Rule. Applications to the study of the behaviour of a function and computation of minima and maxima. Antiderivatives and antiderivatives of elementary functions. Computing antiderivatives by substitution and by parts. Antiderivatives of rational functions.

3. Integration:

Riemann’s integral. Fundamental Theorem of Calculus. Integration by substitution and integration by parts. Computation of areas. Improper integrals.

4. Polynomial approximation and series:

Taylor polynomials. Numerical series. Basic properties. Convergence tests: Leibniz, ratio and integral.

Mandatory literature

James Stewart; Calculus. ISBN: 978-1-305-27237-8
James Stewart; Cálculo. ISBN: 978-85-221-2584-5 2o v.

Complementary Bibliography

James Stewart; Precalculus. ISBN: 978-0-8400-6886-6
Gregory V. Bard; Sage for Undergraduates, American Mathematical Society, 2015. ISBN: 978-1470411114
Spivak Michael; Calculus. ISBN: 0-914098-77-2
Joseph W. Kitchen Jr.; Calculus
Chaves Gabriela; Cálculo Infinitesimal, Universidade do Porto

Teaching methods and learning activities

The teacher presents the course material and discusses the exercises with the students.

Software

sagemath
wolframalpha

Evaluation Type

Evaluation with final exam

Assessment Components

designation	Weight (%)
Exame	100,00
Total:	100,00

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	106,00
Frequência das aulas	56,00
Total:	162,00

Eligibility for exams

No requisites.

Calculation formula of final grade

All enrolled students are admitted to the final exam. Both the final exam and the makeup exam will be divided into two parts, equally valued. The first part covers the first two topics of the program (1. Limits and continuity; 2. Derivatives and antiderivatives), and the second part covers the remaining topics taught.

At the student's request, each part of the makeup exam may be classified with the grade previously obtained in the final exam on that part. Not attending a part is understood as requesting to consider the grade obtained earlier in the corresponding part.

Special assessment (TE, DA, ...)

The exams required under the special legal cases will be written but can be preceded by an oral exam to establish if the student should be admitted to the written 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 during an exam or test fails that exam and will face disciplinary charges by the University."

Jury: Manuel Delgado and Fernando Jorge Moreira.