Calculus I

Code	16133
Year	1
Semester	S1
ECTS Credits	6
Workload	TP(60H)
Scientific area	Mathematics
Entry requirements	N.A.
Learning outcomes	With this Curricular Unit it is intended that students acquire basic knowledge of Differential and Integral Calculus of real functions of real variable. At the end of this Curricular Unit the student should be able to: 1) Compute limits of real functions of real variable; 2) Study the continuity of real functions of real variable; 3) Compute derivatives of real functions of real variable; 4) Apply the derivatives to the computation of maxima and minima of real functions of real variable; 5) Compute primitives and integrals of real functions of real variable; 6) Use integral calculus to determine areas and volumes of surfaces generated by revolution, as well as length of plain curves. 7) Study number series
Syllabus	1. Real functions of real variable: generalities and examples 1.1 Generalities about functions 1.2 Examples of functions: exponential and logarithmic; trigonometric and respective inverses; hyperbolic 2. Real functions of real variable: limits and continuity 2.1 Limits 2.2 Continuity 3. Differential Calculus 3.1 Definition and examples; derivation rules 3.2 Rolle's, Lagrange's and Cauchy's Theorems 3.3 Higher order derivatives and Taylor's formula 3.4 Monotony and local extremes; concavity and inflection points 3.5 Applications 4. Integral Calculus 4.1 Immediate primitives 4.2 Primitive techniques 4.3 Riemann integral 4.4 Fundamental Theorem of Integral Calculus 4.5 Change of variable and integration by parts 4.6 Applications of integral calculus to the determination of areas and volumes 5. Numerical Series 5.1 Definition and properties 5.2 Series of non-negative terms 5.3 Simple and absolute convergence
Main Bibliography	Bibliografia Principal: – James Stewart, Cálculo, volume I, 7.ª Edição, Cengage Learning, 2013 Bibliografia Secundária: - Apostol, T.M., Cálculo, Vol. 1, Reverté, 1993 - H. Anton, I. Bivens, S. Davis, Cálculo, volume I, 8.ª Edição, Bookman, 2007 – Demidovitch, B., Problemas e exercícios de Análise Matemática, McGrawHill, 1977 - João Paulo Santos, Cálculo numa Variável Real, IST Press, 2012 – Mann, W. R., Taylor, A. E., Advanced Calculus, John Wiley and Sons, 1983 – Sarrico, C., Análise Matemática – Leituras e exercícios, Gradiva, 3.ª Edição, 1999 – Swokowski, E. W., Cálculo com Geometria Analítica, Vol. 1 e 2, McGrawHill, 1983
Teaching Methodologies and Assessment Criteria	Classes are theoretical and practical. The teacher presents the concepts and results and illustrates the theory with examples and applications. The student is motivated to participate in class, interacting with the teacher and colleagues, reflecting on the topics, formulating and solving problems and doing exercises. Autonomous work is also encouraged. Assessment Criteria: 1. The assessment may be made during the period of classes or in a final examination. 2. The evaluation of knowledge throughout the teaching-learning period will be periodic and will consist of two written tests, each lasting two hours and rated at ten (10) points, to be held on 17th November 2023 and 9th January 2024. 3. Students who obtain a mark of 9.5 or higher in the assessment carried out throughout the teaching activities will be exempted from sitting the final examination. 4. Any attempt of fraud will result in failure in the Calculus I course.
Language	Portuguese. Tutorial support is available in English.