Site da USP
Graduate Program in Electrical Engineering (PPGEE) Escola Politécnica da Universidade de São Paulo

Fellowship Exam Program (only for Master’s and Direct Entry Doctorate Students)

Calculator icons set with long shadow effect. Button and mathematics, electronic digit, finance and numeral, display set, illustration
1 – Number Sets

1.1 – Natural numbers, integer numbers: divisibility, lowest common multiples, largest common denominators, prime factorization.

1.2 – Rational numbers and elementary notions of real numbers: operations and properties, order relations, absolute value, inequalities, percentages.

1.3 – Complex numbers: representation and operations with complex numbers in algebraic and trigonometric form, modulus of complex numbers, roots of complex numbers.

1.4 – Numerical sequences. Arithmetic and geometric progressions. Sum of an infinite number in terms of an arithmetic progression and a geometric progression. Notion of the limits of a sequence, sum of the infinite terms of a geometric progression with a common ratio modulus of less than 1. Decimal representation of a real number.

1456354400_package_graphics2 – Planar Geometry

2.1 – Geometric planar figures: lines, rays, line segments, angles, polygons, circumferences, circles.

2.2 – Parallel and perpendicular lines in a plane. Parallel lines cut by transversals; Thales’ Theorem.

2.3 – Triangles: sum of the internal and external angles of a triangle, triangle area, triangle congruence, triangle similarity, triangle relationships, specific properties of right triangles, trigonometry of right triangles.

2.4 – Convex polygons: sum of the internal and external angles, congruence and similarity of polygons, regular polygons, the area and specific properties of trapezoids, parallelograms, rhombuses, rectangles and squares.

2.5 – Circumference and the Circle: circumference relationships, length of a circumference, area of a circle and circle sectors.

2.6 – Geometric constructions using a ruler and a compass.

1456354579_draw-063 – Spatial Geometry

3.1 – Geometric spatial figures: lines and planes in space, dihedral and polyhedral angles, convex polyhedrals, regular polyhedrals.

3.2 – Relative positions of lines and planes: parallelism and perpendicularity in space, skew lines.

3.3 – Prisms, pyramids, cylinders, cones and their respective frustums: calculating their areas and volumes.

3.4 – Spheres and spherical surfaces: calculating surface areas and volumes.

3.5 – Similarity between planar and spatial figures: ratio between lengths, areas and volumes.

1456355074_calculator-pencil4 – Functions

4.1 – Notions of functions. Graphs. Even and odd functions. Increasing and decreasing functions. Maximums and minimums.

4.2 – Modular functions, linear functions, affine and quadratic functions. Equations and inequations involving these functions.

4.3 – Composition of functions and inverse functions.

4.4 – Exponential functions and logarithmic functions: fundamental properties, graphs, equations and inequations involving these functions.

1456355270_calculater5 – Polynomials

5.1 – Degrees of polynomials. Addition and multiplication of polynomials. Polynomial identities.

5.2 – Factoring polynomials. Algorithm to divide polynomials. Dividing polynomials by  x – a.

1456355622_Calculator6 – Algebraic Equations

6.1 – Algebraic equations: definition, radicals, multiplying radicals, number of radicals in an equation.

6.2 – Relationships between coefficients and radicals. Algebraic equations with real coefficients: rational radical expressions, rewriting radicals using complex conjugates.

Calculator icons set with long shadow effect. Button and mathematics, electronic digit, finance and numeral, display set, illustration

7 – Combinations and Probability

7.1 – Counting problems.

7.2 – Multisets, permutations and combinations.

7.3 – Newton’s binomial theorem.

7.4 – Probability: notion and distribution of probabilities, conditional probability and independent events.

7.5 – Notions of Statistics: frequency distributions (averages and medians), measurements of dispersion (variances and standard deviations).

1456355074_calculator-pencil8 – Linear Systems and Matrices

8.1 – Linear systems: resolution and discussion.

8.2 – Matrices: addition and multiplication of matrices and matrix inverses. Matrices associated with linear systems.

8.3 – Determinants: properties and applications for linear systems. Cramer’s rule.

1456354400_package_graphics9 – Analytic Geometry

9.1 – Cartesian coordinates: locating points on a line and a plane using Cartesian coordinates, the distance between two points, the use of Cartesian coordinates for solving simple line and plane geometric problems.

9.2 – Study of lines in planar analytic geometry: normal form of the equation of a line, angular coefficient, parallel and perpendicular lines, two variable first degree equations and inequations, distance between a point and a line.

9.3 – Study of circumference in analytic geometry: equation, intersection of circumferences and lines, tangential lines and circumferences, intersections and tangents of circumferences.

9.4 – Analytic representation of geometric locations, definition and representation of conic sections, reduced equation of a conic section, intersection of lines and conic sections.

1456354579_draw-0610 – Trigonometry

10.1 – Arcs and angles: arc measurement (radians), relationship between arcs and angles.

10.2 – Trigonometric functions: definition, periodicity, parity, calculation of notable angles, graphics.

10.3 – Formulas for addition, subtraction, duplication and bisection of arcs. Transforming the sum of trigonometric functions into products.

10.4 – Basic trigonometric identities. Equations and inequations involving trigonometric functions.

10.5 – Sines and Cosines. Solving triangles.

1456356355_handy-icon_0111 – Bibliography:

IEZZI, Gelson. (2004). Fundamentos de Matemática Elementar. São Paulo: Editora Atual. Volumes 1 a 11.