Elements of analytic geometry

The Cartesian plane and space: Cartesian product, representation of R2

on the Cartesian plane, subsets of R2 and subsets of the Cartesian plane, R3

and its representation on the Cartesian space.

Lines on the plane: slope of the linear function; equation of the line given

two points; equation of the line given a point and the slope; conditions

for parallelism and for perpendicularity of two lines.

Quadratic functions: equation of a quadratic function with vertical axis of symmetry, vertex, convexity and concavity of a quadratic function.

Equation of a quadratic function with horizontal axis of symmetry.

Elementary Functions

Introductive notions: the concept of function, independent and

dependent variables, composite function, inverse function, restriction

of a function, prolongation of a function.

Real Numbers

Order relation and algebraic structure of R, bounded sets and extremes of a set, metric properties of the real numbers, elements of topology in R.

Functions from R to R

Introductive notions: the concept of function, independent and dependent variables, graph, image of a function, inverse image of a function, increasing (decreasing) function, local and absolute maxima (minima), bounded functions, extremes of a function, even functions, odd functions, inverse function.

Elementary functions, graphs, geometrical properties, analytical properties: sign function, identical function, linear and affine function, absolute value function, power function, root function, power function with real exponent, exponential function, logarithm function, exponential equations and inequalities, logarithmic equations and inequalities.

Limits

Possible cases: Finite limit at a finite point, limit from the right, limit from

the left, bilateral limit, finite limit at infinity, infinite limit at a finite point,

infinite limit at infinity.

Some theorems: existence theorem for monotone functions,

squeeze theorem, limits of elementary functions, theorem of the

operations with limits, indeterminate forms, limit and comparison of

elementary functions.

Continuous functions

Generalities: definition of continuity of a function, discontinuity points of a

function and their classification, continuity of elementary functions, continuity

with respect to algebraic operations.

Continuous function on intervals: Intermediate value theorem, Bolzano

theorem for continuous functions, Weierstrass theorem.

Composite functions: limits and continuity.

Limits for common functions: logarithm, exponential, indices.

Derivatives:

Slope of a non-linear function: difference quotient; derivative of a function; differentiable function; the relationship between differentiability and continuity; derivability of elementary functions; second order derivatives; Ck functions, points of non-derivability of a function.

Algebra of derivatives: the derivative of a constant; the derivative of an indices, sum rule; product rule; quotient rule; the chain rule. Derivative of the exponential function; derivative of the logarithmic function.

Applications

Tangent to a curve: secant, tangent, equation of the tangent line.

Differential: Differential of a function and its geometric interpretation.

Graphs of functions: Fermat theorem; Lagrange theorem; Rolle theorem; monotonicity criteria, convex (concave) function, convexity criteria.

Graph of polynomial functions: asymptotes; the procedure for the study of the graph.

Graph of Rational Functions: domain, asymptotes, procedure for the study of the graph.

Graph of non-elementary Functions: procedure for the study of the graph, De L'Hopital's theorem.

Integration

Indefinite integral: Torricelli - Barrow theorem; primitive for a function; two primitives differ by a constant (proposition); indefinite integral; linearity; of integration.

Methods of integration: some basic antiderivatives; integration by parts (proposition and application); integration by substitution (proposition and application).

Definite integral: geometric aspects, fundamental formula of integral calculus, calculation of the definite integral.