The exam topics
Mathematical Analysis 1
Limits, derivatives, integrals, series and differential equations: the theory and the exercises for the exam.
Sets, induction and elementary functions
The language of sets and the number sets from N to R, the principle of induction and the binomial theorem, then the elementary functions with their graphs, the starting toolkit for all of analysis.
Complex numbers
What complex numbers are, how they are represented in algebraic, trigonometric and exponential form, how powers and roots are computed, and how equations are solved in the complex field.
Limits and continuity
What it means for a function to tend to a value, how limits are computed, which theorems govern them, and what continuity adds.
Differential calculus and Taylor
The derivative as the limit of the difference quotient, the differentiation rules, the theorems of Fermat, Rolle and Lagrange with their consequences, L'Hôpital's rule and Taylor's formula.
Graphing functions
The complete scheme of the study, the criteria for maxima, minima and inflection points, convexity and the construction of the qualitative graph.
Integral calculus
Antiderivatives and integration techniques, the Riemann definite integral with the fundamental theorem of calculus, and the computation of areas.
Improper integrals
How the integral extends to unbounded intervals and to unbounded functions, which benchmark integrals serve as reference, and how the comparison tests establish convergence without computing an antiderivative.
Numerical series
What it means to add infinitely many terms, which tests establish the behavior of a series, and how a power series is studied.
Differential equations
What differential equations and the Cauchy problem are, and the solution methods for first-order equations and for second-order equations with constant coefficients.