Course details; Programme of mathematics

Can you correct this detailed list of the mathematics contents that I took in the three final years of my secondary education? It is really important because it is part of the application procedure.

Here it is what they ask: You must provide a detailed list of the mathematics subjects and contents taken in the three final years of your secondary education. The list (an English translation is required) needs to be signed by your math teacher (including contact details: name and email address) and stamped by your school.

And here is the list:

PROGRAMME OF MATHEMATICS
Class B Bilingue
Third year
ALGEBRA
 First-degree and second-degree inequalities (integer, fractional, digit and literal inequalities)
 Irrational, integer and fractional equations and inequalities

PLANE ANALYTICAL GEOMETRY
 System of abscissas on an oriented straight line
 Absolute and relative distance between two points
 Abscissa of the midpoint of a segment
 Cartesian coordinates in two dimensions
 Distance between two points in the Cartesian coordinate plane
 Cartesian coordinates of the midpoint of a segment
 Segments in a given proportion
 Centroid of a triangle
 The locus of an equation in the Cartesian coordinate plane
 Segment axis
 Condition for a point to belong to a curve or to a line
 Points of intersection of two loci
 Implicit and explicit form
 X-axis symmetry, y-axis symmetry and origin symmetry
 Analytical method and theorems

The straight line
 Equation of parallel and perpendicular lines
 Equation of a straight line passing through the origin
 Slope of a straight line
 Translation of the coordinate system
 General position of a straight line
 Condition of parallelism and perpendicularity for straight lines
 General form of a straight line
 Reciprocal position of two straight lines
 Locus: y - y0 = m (x - x0)
 Locus: y = m1x + q , with m1 as a given number
 Point-slope form of a straight line
 Slope of a line passing through two points
 Line through two points
 Axis of a segment as a straight line passing through its midpoint
 Distance from a point to a line
 Angle bisector

The circle
 Definition of a circle as a locus
 Equation in terms of center and radius
 The expanded form of the equation of the circle
 Circles in particular positions
 Reciprocal position of a straight line and a circle
 Conditions for determining if an equation is a circle
 Circle through three points: using the circumcenter or three conditions
 Tangent line to a circle: Î" = 0 method, d = r, using the fact that the tangent is perpendicular to the radius from the point it meets the circle, formulae of splitting

 Reciprocal position of two circles and radical axis
 Locus: λ(x2 + y2 + a1x + b1y + c1) + ľ(x2 + y2 + a2x + b2y + c2)
 Particular cases of the equation λ(x2 + y2 + a1x + b1y + c1) + ľ(x2 + y2 + a2x + b2y + c2): passing through two given points, tangent to a given straight line in a point

 Amenable curves to the equation of a circle

The parabola
 Construction of a parabola with ruler and compasses
 The parabola as a locus. Equation of the parabola with axis of symmetry coinciding with the y-axis and with the vertex in the origin

 Concavity and width
 Equation of the parabola with axis of symmetry parallel to the y-axis, vertex, focus, equation of the axis of symmetry and directrix
 Parabolas in particular positions
 Parabola passing through three points
 Conditions in order to write the equation of a parabola
 Reciprocal position of a straight line and a parabola
 Tangents to a parabola
 y = x as an axis of symmetry
 Equation of the parabola with axis of symmetry parallel to the x-axis
 Locus: y - ax2 - bx - c + k(y - a2x2 - b2x - c2)
 Deducible curves from the parabola

The ellipse
 The ellipse as a locus
 Equation of an ellipse whose major and minor axes coincide with the Cartesian axes
 Symmetries, vertices and foci
 Eccentricity of the ellipse
 Translated ellipse, completing-the-square method
 Deducible curves from the parabola

The hyperbola
 The hyperbola as a locus
 Equation of a hyperbola whose foci lying on the Cartesian axes
 Eccentricity of the hyperbola
 Translated hyperbola
 Deducible curves from the hyperbola

Fourth Year
REAL FUNCTIONS
 Definition of function, classification of functions
 Definition of domain, image of an element, range, graph
 X-intercept and y-intercept, sign of a function
 Preimage, inverse function
 Properties of some functions: even, odd, injective, surjective, bijective, increasing and decreasing functions in an interval, monotonic, bounded, unlimited, periodic functions
 Graphs and characteristics of elementary functions: constant, linear, quadratic, direct and inverse proportionality, homographic, root, nowhere continuous, absolute value, piecewise functions

GONIOMETRY
 Oriented angles and how to measure them: sexagesimal system and decimal form, the radian
 How to pass form a system of measurement to another
 Definition of sine, cosine and tangent of an oriented angle
 The unit circle
 Trigonometric functions defined using the unit circle
 Variations of sine, cosine and tangents and respective graphs
 Trigonometric standard identities and relations between trigonometric functions
 Secant, cosecant and cotangent; variations of the graph of the cotangent
 Trigonometric functions of particular angles: 45°, 30°, 60° and 18°
 Periodic functions and period of the trigonometric functions
 Inverse trigonometric functions and their graphs
 Related angles
 Opposite angles, complementary angles
 First quadrant and first octant reduction
 Trigonometric formulae: sum and difference formulae, double-angle formulae, sum-to-product formulae, product-to-sum formulae, half-angle formulae, parametric formulae
 Trigonometric identities
 Elementary trigonometric equations with sine, cosine and tangent and equations attributable to them
 Second degree trigonometric equations with only one trigonometric function
 Solvable equations with appropriate divisions
 Deducible equations to elementary equations using trigonometric formulae
 Linear equations with sine, cosine and tangent: graphic method, normalizing method and using parametric functions
 Homogeneous and non-homogeneous second degree equations with sine, cosine and tangent
 Elementary trigonometric inequalities with sine, cosine and tangent
 Integer and fractional trigonometric inequalities

TRIGONOMETRY
 Theorems for right angles
 How to solve problems with right angles: the four cases
 Area of a triangle if two sides and the included angle are known
 Chord theorem
 Law of sines
 Law of cosines
 How to solve problems with any triangle
 Plane geometry problems
 Gradient of a straight line
 Angle formed by two straight lines
 Cartesian and Polar coordinates

EXPONENTIAL FUNCTIONS AND LOGARITHMS
 Powers with rational and irrational exponents
 Real-exponent power properties
 The exponential function, its properties and graphs
 Exponential equations: elementary equations or equations that can be transformed into elementary equations, solvable equations through the substitution of the variable
 Exponential inequalities: in canonical form or in forms that can be conducted to the canonical form, solvable inequalities through the substitution of the variable

 Definition of logarithm and its fundamental properties
 The logarithmic function, its properties and graphs
 Comparison between the exponential and the logarithmic curves
 Common logarithm (log10(x) ) and natural logarithm ( loge(x) )
 Fundamental theorems and identities of logarithms
 Change of base
 Exponential equations and inequalities that are solvable using logarithms
 Logarithmic equations and conditions of acceptability of the answers
 Logarithmic inequalities

GEOMETRICAL TRANSFORMATIONS
 X-axis symmetry
 Y-axis symmetry
 Symmetry with respect to the bisector y = x
 Origin symmetry
 Horizontal and vertical translation
 Horizontal and vertical stretching and compressing
 Graph of the functions |f(x)|, f(|x|), f(-x) and -f(x)
 Application of the geometrical transformations to the exponential, the logarithmic and the trigonometric functions

Fifth year

FUNCTIONS
Definition of function. Classification of functions. Domain and range, intercepts with the Cartesian axes, sign of a function. Properties of some functions: even, odd, injective, surjective, bijective, increasing and decreasing functions in an interval, monotonic, bounded, unlimited, periodic functions. Graphs and characteristics of elementary functions: constant, linear, quadratic, direct and inverse proportionality, homographic, root, exponential, logarithmic, trigonometric functions. Absolute value function. Inverse function. Piecewise and composite functions. Neighbourhood. Isolated point and limit point.

ALGEBRA OF LIMITS AND CONTINUOUS FUNCTIONS
Definition of limit: finite limit of a function as x approaches a finite value, infinite limit of a function as x approaches a finite value, finite limit of a function as x approaches infinity, infinite limit of a function as x approaches infinity. How to prove the limits using their definition. Left-hand limit and right-hand limit. Definition of a continuous function at a point. Continuity of elementary functions. Fundamental theorems on limits: theorem of the uniqueness of limits (with demonstration), theorem of the permanence of the sign (with demonstration), the squeeze theorem (with demonstration) and its consequences. Calculation of limits using the squeeze theorem.

Limits for sum, difference, product and ratio. Indeterminate forms. Limit of rational integer and fractional algebraic functions as x approaches a finite value or infinity in the indeterminate forms. Limit of composite functions. Substitution of the variable when calculating the limit. Limit of exponential functions with variable basis. Notable special limits (with demonstration). Calculation of limits using notable limits.

CONTINUOUS FUNCTIONS
Discontinuity of functions. Classification of discontinuities of a function. Properties of continuous functions (without demonstration): Bolzano's theorem, Weierstrass's theorem, Darboux's theorem. Application of the Bolzano's theorem to equations and bisection method for the approximation of the solutions.

DERIVATIVE OF A FUNCTION
Differential quotient and its geometrical meaning. Derivative of a function and its geometrical meaning. Right-hand derivative and left-hand derivative. Fundamental derivatives (derivatives of elementary functions). Equation of the straight line that is tangent to the graph of a function. Continuity of derivable functions (with demonstration). Points of continuity, but of non derivability: cusp, angular point, vertical inflection point. Theorems to calculate derivatives: sum, product and quotient rules. Derivative of the exponential function with variable base. Rule of derivation of the inverse function (geometrically demonstrated) and derivative of the inverse of trigonometric functions. Higher-order derivatives. Stationary point of inflection. Tangents to an angular point and angle that they form.

THEOREMS ON DERIVABLE FUNCTIONS
Rolle's theorem (with demonstration) and its geometrical meaning. Lagrange's theorem (with demonstration) and its geometrical meaning. Cauchy's theorem (without demonstration). Applications of Rolle's theorem and Lagrange's theorem. Consequences of Lagrange's theorem: functions with

f|(X) = 0 in an interval (with demonstration), functions with the same derivatives in an interval (with demonstration), sufficient condition to determine on which intervals the function is constant, ascending or descending and "inverse" theorem (both with demonstration). De L'HĂ'pital's theorem (without demonstration) and its applications to solve indeterminate forms. Sufficient condition of derivability (without demonstration).

MAXIMA, MINIMA AND POINT OF INFLECTION AND STUDY OF FUNCTIONS
Definition of local maximum and minimum, definition of inflection. Theorems on local maxima and minima: necessary condition for the existence of a local maximum and a local minimum for derivable functions (with demonstration), sufficient condition for the determination of the points of maximum and minimum and of stationary points of inflection. Research of absolute and local maxima and minima. Definition of point of inflection, ascending and descending point of inflection. Determination of the intervals and points on which the function is convex or concave. Criterion for the determination of the points of inflection and the type of concavity of a function using the second derivative. Asymptotes: vertical, horizontal and oblique asymptotes. Research of the oblique asymptote of a particular rational fractional function. Study of the point of non derivability using the limit of the first derivative. Study of functions that depend on a parameter. Problems of maximum and minimum.

INDEFINITE INTEGRALS
Antiderivative of a function. Definition of indefinite integral and its properties. Immediate formulae of integration. Integral of a function that has an antiderivative that is a composite function. Integral of the functions that have as antidederivatives the inverse trigonometric functions. Integration of rational algebraic fractional functions with first or second degree denominator. Integration by parts. Integration by substitution.

DEFINITE INTEGRALS
Superior and inferior integral sums. Definition of definite integral of a continuous function and its properties. Geometrical interpretation for a function that is not negative. Mean value theorem. The integral function. Fundamental theorem of calculus. Fundamental formula of calculus. Calculation of areas. Volume of a rotating solid.

Prof. XXX
E-mail: mail@mail.com

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