Calculus I

Code:

M1001

Acronym:

M1001

Level:

100

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2020/2021 - 1S

Active?	Yes
Web Page:	https://moodle.up.pt/course/view.php?id=527
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	95	Plano de estudos a partir de 2014	1	-	6	56	162
L:F	0	Official Study Plan	2	-	6	56	162
			3
L:G	0	study plan from 2017/18	2	-	6	56	162
			3
L:Q	0	study plan from 2016/17	3	-	6	56	162
MI:ERS	129	Plano Oficial desde ano letivo 2014	1	-	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

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

Presentation of the course material by the teacher, discussion of exercises.

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	52,00
Total:	158,00

Eligibility for exams

No requisites.

Calculation formula of final grade

All enroled students are admitted to the final exam. Both the final exam and the makeup exam will be divided in 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.

Each part of the makeup exam may, at the student's request, 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.

Observations

The first part is valued with 12 points and the second with 8 points.