Calculus I

High school mathematics knowledge.

Presentation and analysis of the curricular unit’s themes, using the expository and interrogative method, in lectures, theoretical-practical classes and practical classes, where the ability to calculate and the understanding of concepts are encouraged. Interpretation of concepts and solving exercises, in groups or individually, using GeoGebra software, WolframAlpha software and MATLAB programming language.

Differential and integral calculus of real functions of a real variable and its applications.

Topics are presented and discussed by solving exercises and using the following software: GeoGebra, WolframAlpha and MATLAB. Autonomous learning and rigor in the interpretation, use and description of mathematical concepts.

1. Real function of a real variable
Properties of real functions of a real variable, limit and continuity, trigonometric functions and inverse trigonometric functions, exponential function, logarithmic function and hyperbolic functions.

2. Differential calculus
Derivative, properties, derivative of the composition function and the inverse function, theorems of Rolle and Lagrange, undetermined forms and Cauchy’s rule, polynomial approximation: differentials and Taylor polynomial.

3. Primitive of real functions of real variable
Techniques for calculating the primitive function.

4. Integral calculus
Definite Integral, properties, fundamental theorem of calculus, integration by parts and by substitution, applications of the definite integral: area of a plane region, volume of a solid of revolution and length of the arc of a curve; improper integrals: Integrals at unlimited intervals and integrals of unlimited functions.

5. Introduction to the study of ordinary differential equations
First order ordinary differential equations, first order linear equation, Bernoulli equation and equation of separate variables.

6. Component of numerical analysis
Approximation and error in numerical calculation, numerical methods for solving non-linear equations: bisection method and Newton-Raphson method; polynomial interpolation, numerical integration methods: trapezoid rule and Simpson rule; numerical methods for differential equations: Euler’s method.

NAO

Departamento de Física e Matemática, (2005). Quinzena de Uniformização de Conhecimentos. ISEC.

Finney, R. L., Weir, M. D., Giordano, F. R. (2003). Cálculo (de George B. Thomas). Addison Wesley.

Larson, R., Hostetler, R. P., & Edwards, B. H. (2006). Cálculo (Vol. 1). McGraw-Hill.

MathWorks (2022). Getting Started with MATLAB.

Rodrigues, R. (2022). Notas teóricas e exercícios de Análise Matemática. ISEC.