1) Definition of function. Numerical sequences. Domain and image of a function. Injective, surjective and bijective functions.
2) Invertible functions. Economic example: inverse demand function of a consumer.
3) Composite functions. Economic example: reduction of the number of variables in a mathematical model.
3) Maximum and minimum of a function. Economic example: rofit function and cost function of a firm.
4) Elementary functions. Linear functions; power functions; exponential functions; logarithms. Economic phenomena represented by elementary functions.
LIMITS AND CONTINUITY OF ONE-VARIABLE FUNCTIONS
5) Definition of limit of a function. Limit of a sequence.
6) Definition of continuous function in a point and in a whole interval.
7) Theorems on continuous functions (Zeros Theorem, intermediate values Theorem, Weierstrass Theorem).
8) Applications to the economic models of theorems on continuous functions: existence of solutions to problems of maximization and minimization, existence of equilibrium prices in a competitive market.
DERIVATIVES OF ONE-VARIABLE FUNCTIONS
9) Average rate of change of a function. Economic examples: average increase in the costs of a company, average increase in consumer utility.
10) Definition of derivative of one-variable functions. Economic examples: marginal cost, marginal utility.
11) Rules of derivation.
12) Theorems on derivable functions: Fermat's Theorem and Lagrange's Theorem. Economic applications: first-order conditions relating to the problems of maximization and minimization in economic models. Economic examples: maximization of consumer's utility, maximization of firm's profits.
13) Taylor's formula of the second order. Necessary and sufficient second order conditions related to maximization and minimization problems in economic models.
14) Concave and convex functions. Procedure for identifying solutions in economic models in which concave or convex functions are involved.
VECTORS AND n-VARIABLE FUNCTIONS
15) Vectors. Scalar product between vectors, norm of a vector, distance between two vectors. Internal points of a set, border points. Open sets and closed sets. Economic application: analysis of the structure of the constraint set in the consumer's problem of utility maximization.
16) Functions of n variables. Economic applications: utility function, production function.
17) Lines of level of a function. Economic example: indifference curves.
18) Continuous functions. Weierstrass's Theorem for functions of n variables. Economic application: existence of the solution in maximization and minimization economic problems.
19) Partial derivatives, gradient.
20) Methods for solving problems of maximization and minimization of functions of n variables with and without constraints. Lagrange multiplier method for maximization and minimization problems with equality constraints. Maximization and minimization problems with inequality constraints. Economic applications: minimizing the costs of a firm, maximizing the utility of a consumer.