Subject: Calculus I
Scientific area:
Mathematics
Contact hours:
96 Hours
Number of ECTS:
7,5 ECTS
Language:
Portuguese
General Objectives:
1 - Master concepts and techniques of linear algebra, differential and integral calculus of single-variable functions, and the method of mathematical induction.
2 - Recognize the need for and the importance of math in solving problems related to the main areas of your course.
3 - Acquire the mathematical tools needed to do well in the more advanced course units.
2 - Recognize the need for and the importance of math in solving problems related to the main areas of your course.
3 - Acquire the mathematical tools needed to do well in the more advanced course units.
Contents / Syllabus:
1 - Matrices: Basics of matrices. Matrices operations. Classification Matrices.
2 - The method of Mathematical Induction.
3 - Determinants of square matrices: Calculation of determinants. Rule Sarrus. Laplace theorem and its applications. Applications of determinants to the linear systems: systems of linear equations. Gauss. Cramer's rule. Calculation of the inverse of an invertible matrix.
4 - Eigenvalues and eigenvectors: Basic concepts of eigenvalues and eigenvectors. Eigen subspace associated with an eigenvalue. Algebraic and geometric multiplicity. Characteristic polynomial. Cayley-Hamilton theorem. Diagonalization.
5 - Sequences of real numbers: bounded, monotone, and convergent sequences. Fundamental theorems. Study and calculation of limits.
6 - Real functions of a real variable: Domain, codomain, composite function, monotonicity of functions, inverse. Exponential, logarithmic, trigonometric functions and their inverses, piecewise functions. Bolzano's and Weierstrass' theorems. Limits and continuity.
7 - Differentiability. Fundamental theorems of differential calculus: Rolle, Lagrange, and Darboux. Cauchy's and l'Hôpital's rules. Calculation of indeterminate forms. Polynomial approximation of functions. Taylor's formula. Application of these concepts to a complete study of a function.
8 - Integration of functions of one variable. Indefinite integral. Immediate integration by parts and by substitution. Definite integral and its application. Fundamental theorem of integral calculus. Derivative of an integral
9 - Improper integrals. Definition and classification. Concept of improper integral convergent and divergent. Properties of improper integrals of 1st and 2nd kind. Discussion of the nature of improper integrals by comparison. Criteria for convergence of improper integrals. Study of the nature of the integral of 3rd kind.
2 - The method of Mathematical Induction.
3 - Determinants of square matrices: Calculation of determinants. Rule Sarrus. Laplace theorem and its applications. Applications of determinants to the linear systems: systems of linear equations. Gauss. Cramer's rule. Calculation of the inverse of an invertible matrix.
4 - Eigenvalues and eigenvectors: Basic concepts of eigenvalues and eigenvectors. Eigen subspace associated with an eigenvalue. Algebraic and geometric multiplicity. Characteristic polynomial. Cayley-Hamilton theorem. Diagonalization.
5 - Sequences of real numbers: bounded, monotone, and convergent sequences. Fundamental theorems. Study and calculation of limits.
6 - Real functions of a real variable: Domain, codomain, composite function, monotonicity of functions, inverse. Exponential, logarithmic, trigonometric functions and their inverses, piecewise functions. Bolzano's and Weierstrass' theorems. Limits and continuity.
7 - Differentiability. Fundamental theorems of differential calculus: Rolle, Lagrange, and Darboux. Cauchy's and l'Hôpital's rules. Calculation of indeterminate forms. Polynomial approximation of functions. Taylor's formula. Application of these concepts to a complete study of a function.
8 - Integration of functions of one variable. Indefinite integral. Immediate integration by parts and by substitution. Definite integral and its application. Fundamental theorem of integral calculus. Derivative of an integral
9 - Improper integrals. Definition and classification. Concept of improper integral convergent and divergent. Properties of improper integrals of 1st and 2nd kind. Discussion of the nature of improper integrals by comparison. Criteria for convergence of improper integrals. Study of the nature of the integral of 3rd kind.
Bibliography / Information Sources:
T. Apostol , 1998 , Cálculo , Ed. Reverté, Lda.
N. Piskounov , 1977 , Cálculo Diferencial e Integral , Lopes da Silva, Editora
James Stewart; Daniel K. Clegg; Saleem Watson , 2021 , Calculus , Cengage
R.Courant, F. John , 1999 , Introduction to Calculus and Analysis , Springer-Verlag
Luís T. Magalhães , 1998 , Álgebra Linear como Introdução a Matemática Aplicada , Texto Editora
N. Piskounov , 1977 , Cálculo Diferencial e Integral , Lopes da Silva, Editora
James Stewart; Daniel K. Clegg; Saleem Watson , 2021 , Calculus , Cengage
R.Courant, F. John , 1999 , Introduction to Calculus and Analysis , Springer-Verlag
Luís T. Magalhães , 1998 , Álgebra Linear como Introdução a Matemática Aplicada , Texto Editora
Assessment Methods and Criteria:
Classification Type: Quantitativa (0-20)
Evaluation Methodology:
The teaching of the curricular unit is done through 96 hours of theoretical-practical classes. During the classes, students will be encouraged to participate in solving exercises and problems. Exercise sheets will include work to be done outside the classroom, and students can clarify any doubts about them during classes or during consultation hours in the lecturers' offices, at times scheduled at the beginning of the semester. Some electronic tools may be used to help and motivate students to deduce certain properties or answer icebreaker-type questions. Assessment is done through 2 exams, to be completed individually and without the use of digital or electronic tools. Each test contributes 50% (10 points) to the final grade, and passing the course unit requires a minimum score of 3.0 points (on a scale from 0 to 10) in each test. During the regular examination period, the final grade is calculated as the sum of the grades obtained in the tests, rounded to the nearest whole number (without rounding the individual test grades). The course unit is considered successfully completed when the final grade (after rounding to the nearest whole number) is 10 points or higher. Students who do not pass the course unit during the regular examination period are automatically enrolled in the resit examination period. During this period, they can, by registering with the teachers, take only one assessment, while maintaining the minimum grade requirements for each partial evaluation and the same criteria for calculating the final grade. If they do not inform the teachers that they want to be assessed only on one part, they will be registered for a comprehensive exam during the resit period. The special exam takes place through a test that covers all the material of the course unit and requires prior registration with the academic office. Detecting any form of cheating during an assessment results in the cancellation of that test. Whenever a teacher has a well-founded suspicion of cheating, the students involved will be called for an oral exam covering the part(s) of the assessment where doubts remain. Nevertheless, each teacher responsible for teaching and assessment may set an alternative assessment method, as long as it fully complies with the University's Learning Assessment Regulations currently in force at the University of Madeira.
Evaluation Methodology:
The teaching of the curricular unit is done through 96 hours of theoretical-practical classes. During the classes, students will be encouraged to participate in solving exercises and problems. Exercise sheets will include work to be done outside the classroom, and students can clarify any doubts about them during classes or during consultation hours in the lecturers' offices, at times scheduled at the beginning of the semester. Some electronic tools may be used to help and motivate students to deduce certain properties or answer icebreaker-type questions. Assessment is done through 2 exams, to be completed individually and without the use of digital or electronic tools. Each test contributes 50% (10 points) to the final grade, and passing the course unit requires a minimum score of 3.0 points (on a scale from 0 to 10) in each test. During the regular examination period, the final grade is calculated as the sum of the grades obtained in the tests, rounded to the nearest whole number (without rounding the individual test grades). The course unit is considered successfully completed when the final grade (after rounding to the nearest whole number) is 10 points or higher. Students who do not pass the course unit during the regular examination period are automatically enrolled in the resit examination period. During this period, they can, by registering with the teachers, take only one assessment, while maintaining the minimum grade requirements for each partial evaluation and the same criteria for calculating the final grade. If they do not inform the teachers that they want to be assessed only on one part, they will be registered for a comprehensive exam during the resit period. The special exam takes place through a test that covers all the material of the course unit and requires prior registration with the academic office. Detecting any form of cheating during an assessment results in the cancellation of that test. Whenever a teacher has a well-founded suspicion of cheating, the students involved will be called for an oral exam covering the part(s) of the assessment where doubts remain. Nevertheless, each teacher responsible for teaching and assessment may set an alternative assessment method, as long as it fully complies with the University's Learning Assessment Regulations currently in force at the University of Madeira.