Calculus I
Objectives
At the end of this course the student will have acquired knowledge and skills that will enable him (i) to know and to use the concepts of majorant, minorant, supremum and infimum of a set, (ii) to deal with the absolute value function and to use the triangular inequality to estimate errors, (iii) to know and to work with the inverse of trigonometric functions, in particular, identifying domains and solving equations and inequations involving these functions, (iv) to compute limits and to solve indeterminate forms, (v) to know Taylor’s formula and to use it to obtain polynomial approximations estimating
the errors, (vi) to compute antiderivatives and integrals, (vii) to know the concept of improper integral and to know methods to study convergence, (viii) to solve some first order differential equations, (ix) to conceive and to interpret algorithms to several of the topics considered.
The student will be able to a deeper study of infinitesimal analysis.
General characterization
Code
12900
Credits
6.0
Responsible teacher
Ana Luísa da Graça Batista Custódio
Hours
Weekly - 5
Total - 65
Teaching language
Português
Prerequisites
The student must master the mathematical knowledge lectured until the end of Portuguese high school teaching.
Bibliography
1 - H. Anton, I. Bivens & S. Davis, Calculus, 8th edition, John Wiley & Sons, 2005.
2 - K. E. Hirst, Calculus of One Variable, Springer undergraduate mathematics series, 2006.
3 - A. A. Sá & B. Louro, Cálculo Diferencial e Integral em R, Escolar Editora, 2022.
4 - G. Simmons, Calculus with Analytic Geometry, 2nd edition, McGraw Hill, 1996.
5 - M. Spivak, Calculus, 3rd edition, Publish or Perish Inc., 2006.
6 - G. Strang, Calculus I, OpenStax, 2018.
Teaching method
Classes are theoretical/practical with oral presentation of concepts and results by the professor, together with their proofs, whenever possible, and complemented with examples and applications. A list of exercises and problems is provided to the students to be solved independently. The student is encouraged to use/conceive computational programs related to the topics under study. Specific difficulties could be addressed during classes or in individual sessions scheduled with the professor.
Students need to attend a minimum of two thirds of the theoretical/practical classes lectured in order to be evaluated. Continuous evaluation is based on two tests. If a student does not obtain approval through continuous evaluation, he can try it in a resit exam.
Evaluation method
Calculus I Knowledge Assessment Regulations (1st Semester of 2024/25)
This document regulates the knowledge assessment process of the Calculus I curricular unit, in the first semester of the academic year 2024/25. In any omission, the Knowledge Assessment Regulations of the Faculty of Science and Technology of the Universidade Nova de Lisboa, dated May 29, 2024, apply.
Frequency
Attendance is granted to any student who attends at least 2/3 of the theoretical/practical classes taught.
Students with student-worker status are exempt from obtaining attendance. However, it is recommended that they attend classes with the same periodicity as students who are not exempt from obtaining it.
Only students with attendance will have a final grade in the curricular unit.
Continuous Assessment
The continuous assessment of the curricular unit is carried out using Theoretical-Practical Assessment, including two face-to-face tests, each lasting 1h30.
Let T1 and T2 be the classifications of each of the two tests, expressed on a scale of 0 to 20 values, rounded to the nearest tenth. A student will have a final grade of 0.5 T1 + 0.5 T2, rounded to the nearest integer.
The student will obtain approval in the curricular unit if this classification is greater than or equal to 10 points. Otherwise, the student will have failed the course through continuous assessment.
Resit Examination
Students who fail continuous assessment may present themselves to the resit exam. Let E be the classification obtained in the exam, expressed on a scale of 0 to 20 values, rounded to the nearest integer.
The student will obtain approval in the curricular unit if this classification is greater than or equal to 10 points. Otherwise, the student will have failed the course.
Note Defense
The professor reserves the right to request a supplementary grade defense assessment from all students with a final grade greater than or equal to 18 points. Failure to take this assessment, if requested, implies a final grade of 17 points for the curricular unit.
Grade Improvement
Students approved in the curricular unit may request Grade Improvement, in accordance with the procedure described in Article 23 of the Knowledge Assessment Regulations of the Faculty of Science and Technology of Universidade Nova de Lisboa, of May 29, 2024.
The student’s provisional classification will be equal to E, the classification obtained in the resit exam, rounded to the nearest integer.
The professor reserves the right to request a supplementary grade defense assessment from all students with a provisional classification greater than or equal to 18 points. Failure to take this assessment, if requested, implies a provisional classification of 17 points for the curricular unit.
The final grade improvement classification for the curricular unit will be the maximum between the provisional classification and the student''s current classification.
Subject matter
1. Absolute value; errors; majorant, minorant, supremum, and infimum.
2. Inverse of trigonometric functions; Cauchy’s rule; Taylor’s formula.
3. Antiderivatives.
4. Integration (definite and indefinite integral, improper integral).
5. Differential equations (first order linear equations; equations with separable variables; change of variables).
Programs
Programs where the course is taught: