Calculus I

Code	15751
Year	1
Semester	S1
ECTS Credits	6
Workload	TP(60H)
Scientific area	Mathematics
Entry requirements	-
Learning outcomes	This Course Unit is intended for students to acquire and apply basic knowledge of Differential and Integral Calculus of real functions of real variable. At the end of this course students should be able to: - solve inequalities involving rational expressions and absolute values; - determine domains and sketch the graph of functions; - compute limits of functions; - study the continuity of functions; - compute derivatives of functions; - know how to approximate functions by Taylor's polynomials; - apply the derivatives to compute maximums and minimums and sketch the graph of functions; - integrate functions; - apply integrals to compute plane areas, to compute length of curves and to compute areas of surfaces and volumes of solids generated by revolution.
Syllabus	1. GENERALITIES AND EXAMPLES OF FUNCTIONS Real numbers Generalities about functions Inverse and composition of functions Polynomial, rational, absolute value, exponential, logarithmic, trigonometric, trigonometric inverse and hyperbolic functions 2. LIMITS AND CONTINUITY Topological notions Limits Continuity Bolzano and Weierstrass theorems Infinite limits, limits at infinite and asymptotes 3. DIFFERENTIAL CALCULUS Definition, rules and examples Fermat, Rolle, Lagrange and Taylor theorems Cauchy's Rule Applications 4. INTEGRAL CALCULUS Antiderivatives. Definition, properties and technics of integration Definition and properties of Riemann integral Fundamental Theorem of Calculus Applications
Main Bibliography	Main Bibliography: – James Stewart, Daniel Clegg, Saleem Watson – Cálculo, Volume 1, Cengage (2022) Additional Bibliography: – Apostol, T.M., Cálculo, Vol. 1, Reverté, 1993 – H. Anton, I. Bivens, S. Davis, Cálculo, volume I, 8.ª Edição, Bookman, 2007 – Adams, Robert Alexander_ Essex, Christopher - Calculus a complete course, Pearson (2018) – João Paulo Santos, Cálculo numa Variável Real, IST Press, 2012 – Mann, W. R., Taylor, A. E., Advanced Calculus, John Wiley and Sons, 1983 – Swokowski, E. W., Cálculo com Geometria Analítica, Vol. 1 e 2, McGrawHill, 1983
Teaching Methodologies and Assessment Criteria	Classes are both theoretical and practical. The teacher introduces key concepts and results, illustrating the theory with examples and applications. Students are encouraged to participate actively, engaging with the teacher and peers, reflecting on the topics, formulating and solving problems, and completing exercises. Independent work is also promoted. Assessment Criteria: 1. The assessment may take place during the course or in a final examination. 2. Knowledge evaluation during the course will be conducted periodically through two written tests, each lasting two hours and worth ten (10) points. These tests will be held on 4th November 2024 and 7th January 2025. 3. Students who achieve a score of 9.5 or higher in the course assessments will be exempt from the final examination. 4. Any attempt at academic dishonesty will result in failure of the Calculus I course.
Language	Portuguese. Tutorial support is available in English.