Class Calculus I

Calculus I provides the fundamental concepts of mathematical analysis and constitutes a core component of Engineering education. The course covers real-valued functions, limits, continuity, differentiation, integration, and the Fundamental Theorem of Calculus, providing the mathematical foundations required for modelling and solving engineering problems. It develops logical reasoning, analytical thinking, and problem-solving skills that are essential for subsequent engineering courses and for the application of mathematics in scientific and technological contexts.

Not applicable

Introduction. Sets. Real numbers. Notion of function. Module function. Polynomials. Equations and inequations. Functions of real variable. Domain, range, graphic and monotony. Algebraic operations with functions. Composition. Rational functions. Inverse function. Exponential, logarithmic, hyperbolic and trigonometric functions. Limits and continuity. Point of accumulation. Definition of limit. Properties. Sandwich theorem. Properties of continuous functions. Bolzano and Weierstrass Theorems. Differentiability. Definition of derivative at one point. Geometrical interpretation of the derivative. Rules of derivation. Chain rule. Derivative of the inverse function. Rolle, Lagrange and Cauchy Theorems. Monotony, extremes and asymptotes. Study of the graph of a function. Antiderivatives. Immediate and almost immediate antiderivatives. Antiderivatives by substitution and by parts. Antiderivatives of rational functions. Definition of integral. Fundamental Theorem of Calculus. Calculation of areas.

Series of exercises will be proposed with the aim of consolidating knowledge and stimulating problem-solving skills. The evaluation of the discipline, expressed on a scale from 0 to 20 points, will be made at different times, including 2 midterms (40% + 50%) and individual work to be developed outside the classroom (10%). If the weighted average of these evaluations is equal to or greater than 9.5, the student will be successful in the subject, otherwise the student will be able to attend a global frequency. In the final exam, the student can improve the grade. The minimum passing grade for these assessments is also 9.5. Assessment criteria are explained at the beginning of the semester.

Santos, J. P., Cálculo numa Variável Real. IST Press, 2012 Sá, A. A. e Louro, B., Cálculo Diferencial e Integral em R. Escolar Editora, 2022