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ALGEBRA 1 Version 2009
Revised and highly enlarged.
ALGEBRA 2
ALGEBRA 3
ALGEBRA 4
GEOMETRY 1
ELECTRICITY 1
ELECTRICITY 2
ELECTRICITY 1&2
OPTICS 1
OPTICS 2
MECHANICS 3
 
ALGEBRA 3
Analysis, vectors, trigonometry, probabilities…
High school 15-18 years


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   1. Identity which factorizes an - bn and its applications.
  •            1. Definition and properties of a homogeneous monomial of degree n, Mn (a, b).
  •            2. To display homogeneous monomial of degree n.
  •            3. Homogeneous polynomials of degree n. Hn(a, b) polynomials.
  •            4. Discovering the factorization of a n - b n .
  •            5. Proof of an - bn factorization.
  •            6. Review sheet.
  •            7. Maximum factorization of the difference of two variables, raised to the same power.
  •            8. Sum of consecutive terms of a geometrical series.
   2. Definition and properties of polynomials: multiplication, division and divisibility by (x-a).
  •            1. Polynomials of degree n.
  •            2. Vector properties of polynomials of degree n.
  •            3. Multiplication of two polynomials. Degree of their product.
  •            4. Practical multiplication of two polynomials.
  •            5. Division and divisibility of a polynomial by x-a.
  •            6. To determine the quotient polynomial and the remainder of the division by x-a.
  •            7. To effect the division of a polynomial by x-a.
  •            8. Product of factors equal to zero.
  •            9. Roots of a polynomial and factorization.
  •            10. Fundamental theorem of algebra and factorization by second degree polynomials.
   3. Examples of a polynomial's factorizations.
  •            1. To factorize, when possible, second degree polynomials.
  •            2. To factorize a third degree polynomial knowing one of its roots.
  •            3. To factorize a class of polynomials P4(x) into a polynomial's product.
  •            4. To factorize a class of polynomials P4(x) into P2(x) polynomials.
  •            5. To factorize a third degree polynomial knowing two of its roots.
  •            6. To factorize a 4th degree polynomial knowing two of its roots.
  •            7. A ladder with a wedge under it.
   4. Permutations and combinations. Pascal's triangle and Newton's binomial.
  •            1. ( A + B )n expansion in a general non-commutative algebra.
  •            2. (A + B) 5 expansion.
  •            3. Permutations' arborescence of n objects, n!
  •            4. Number of combinations formed by p factors A and (n-p) factors B: Cnp.
  •            5. Cnp 'computation table. Pascal's triangle and the associated relation.
  •            6. Proof of (a + b) n expansion.
  •            7. To expand binomials to a power n.
  •            8. Tree and enumeration of permutations. Transition towards combinations. The p-lists.
  •            9. To construct the tree and the list of permutations of 4 objects.
  •            10. Counting possible finishes in horse racing: betting on two, three, four and five horses.
  •            11. Counting occurrences in a game of 32 cards.
   5. Polynomial functions: recalling of the second, third and fourth degree polynomials.
  •            1. Polynomial functions. Recalling of zero and one degree polynomials.
  •            2. Graph and roots of second degree polynomials. The parabola and its symmetry axis.
  •            3. Variations of second degree polynomial function without the derivative tool.
  •            4. Review sheet: Vertex, variation and roots of P2(x) polynomials.
  •            5. To master the parabola properties, the second degree polynomial's graph.
  •            6. Study of third degree monomials.
  •            7. Reduced form of third degree polynomial: its center of symmetry and graph.
  •            8. Study of ""horse's addle"" polynomial: x ® ax2 ( x2 - a )
  •            9. General properties of polynomials of degree n.
   6. The square root function. Inversely proportional and homographic functions.
  •            1. Definition, properties and graphs of Rac(x) and -Rac(x) functions.
  •            2. Square root of a linear function with a constant.
  •            3. To study and draw the graph of a function in the form y = Rac(ax + b)
  •            4. Inversely proportional variables: xy = a.
  •            5. Properties of y = a/x function: the equilateral hyperbola.
  •            6. Asymptotes, symmetries and graph of the homographic function.
  •            7. A solved exercise: the study and drawing of a homographic function.
  •            8. To study and draw the graph of a homographic function.
   7. Derivative at a point. The derivative function and tangent to the graph. Derivation formulas.
  •            1. Average velocity and instantaneous velocity. Law of motion quadratic in function of time.
  •            2. Velocity and acceleration for a law of motion defined by a function cubic in time.
  •            3. Study of the function xn in the neighborhood of x. Derivative of this function.
  •            4. General definition of the derivative of a function.
  •            5. Geometrical meaning of the derivative at a point: slope of the tangent to the graph at this point.
  •            6. Derivative of the function Rac(x) at x = a > 0. Form of this function in the neighborhood of this point.
  •            7. Derivative of the function (Rac(x))3 at x = a. The form of this function in the neighborhood of this point.
  •            8. Linearity of the derivative. Derivation of a polynomial Pn(x).
  •            9. Derivative of a product of functions: "distributive law" of the derivative operation.
  •            10. Derivative of the quotient of functions: First applications.
  •            11. Training to calculate the derivative of the simplest functions.
   8. Functions defined as rational powers and their derivatives.
  •            1. Definition of a rational power of a real number x.
  •            2. Cube's side and sphere's radius as functions of their volume.
  •            3. Multiplication/division of rational powers of a real variable.
  •            4. Rational powers raised to rational power.
  •            5. Derivative of a function raised to power n.
  •            6. Derivative of a rational power of x.
  •            7. Review sheet.
  •            8. Training to compute derivatives of rational powers of x.
   9. Direction of variation of a function. Extremums. Third and fourth degree polynomials.
  •            1. Geometrical property of the tangent to the parabola at one point, enabling its drawing.
  •            2. Geometrical property of the tangent to the hyperbola at one point, enabling its drawing.
  •            3. Locus of points I, from which one can draw two perpendicular tangents to the parabola.
  •            4. Direction of variation of a function and its derivative's sign. Extremums.
  •            5. To use the derivative in studying the second degree polynomial.
  •            6. Taylor's theorem for polynomial functions. Translated function at a point.
  •            7. Applying Taylor's theorem to study the third degree polynomials function f, with f'(x0)³ 0.
  •            8. Applying Taylor's theorem to study the third degree polynomials function f, with f'(x0) < 0.
  •            9. Applying Taylor's theorem to study fourth degree polynomial functions f.
  •            10. Free activity.
   10. Derivation of a compound function. Absolute value, oblique asymptotes.
  •            1. Theorem of derivation of compound functions.
  •            2. Study of the function f: x ® y = Rac(x2 + a2). Function absolute value of x.
  •            3. Study of the function f: x ® y = Rac(x2 + a2) and its derivative. Function sgn(x).
  •            4. Second derivative of f: x ® y = Rac(x2 + a2).
  •            5. Study of the function f: x ® y = Rac(x2 - a2) and its derivative.
  •            6. A function which presents a oblique asymptote: X ® Y = X + A/X.
  •            7. The quotient of a second degree polynomial by a first degree polynomial. Case without extremums.
  •            8. The quotient of a second degree polynomial by a first degree polynomial. Case with extremums.
   11. Vectors on a plane. Properties and applications of Thales's theorem.
  •            1. In Mathematics, the vectors may be translated freely.
  •            2. Definition and properties of the sum of two vectors.
  •            3. The real numbers may be set in correspondence with a 1-dimensional vector space.
  •            4. Multiplication of a vector by a real number. Collinear vectors.
  •            5. The plane as a 2-dimensional vector space. Vector's components.
  •            6. Vector relations in a triangle. A median's properties.
  •            7. Thales theorem in vector form.
  •            8. Lines joining the middles of an ordinary quadrilateral's sides.
  •            9. The intersection of a parallelogram's sides with a line that originates from a vertex.
  •            10. Two successive applications of Thales's theorem.
   12. Definition and properties of the barycenter. Point and vector notations. Applications.
  •            1. Definition and uniqueness of two point's barycenter with reduced coefficients.
  •            2. The center of gravity in Physics. Definition of the barycenter with arbitrary coefficients.
  •            3. The barycenter enables us to calculate with points.
  •            4. Barycenter construction of two points for 3 configurations.
  •            5. Barycenter of 3 points. Barycenter's associativity theorem.
  •            6. Isobarycenter. The center of gravity of a triangle's vertices.
  •            7. Barycenter of three points bearing positive coefficients.
  •            8. Barycenter of three points bearing one negative coefficient.
  •            9. Relation between partial barycenters of 3 points.
  •            10. Barycenter of n points. Associativity theorem.
  •            11. Isobarycenter of the tetrahedron.
   13. Trigonometric functions: definitions and properties.
  •            1. The measure of an angle in radians.
  •            2. Oriented angles. The trigonometric circle.
  •            3. Conversion of an angle's measure from degrees to radians. The principal measure of a given angle.
  •            4. Definitions and periodicity of the cosine and sine functions.
  •            5. Relations between the sine and cosine functions which result from symmetries.
  •            6. Exploiting symmetry relations to simplify expressions involving sine and cosine functions.
  •            7. Definitions and properties of the tangent and cotangent functions.
  •            8. Relations between the tangent and cotangent functions which result from symmetries.
  •            9. Exploiting symmetry relations to simplify expressions involving tan and cotan functions.
  •            10. Review sheet.
   14. Relations between trigonometric functions. Inverse trigonometric functions.
  •            1. Pythagoras' theorem: A proof.
  •            2. The relation sin2 q + cos2 q = 1: the root's signs according to the location of q .
  •            3. Values of the trigonometric functions for special angles.
  •            4. General relations between the cosine or the sine and the tangent.
  •            5. Inverse of the sine function: the equations sin q = a and sin q = sin q'.
  •            6. Inverse of the cosine function: the equations cos q = a and cos q = cos q'.
  •            7. To solve equations where the sine or the cosine have specific values.
  •            8. Trigonometric equations with two families of solutions.
   15. Scalar product and norm. Applications to geometry.
  •            1. Definition and properties of the norm or magnitude of a vector.
  •            2. Definition and properties of the scalar product of two vectors.
  •            3. Review sheet. Scalar product and norm.
  •            4. Identity (u - v)2=... To determine the angles of a triangle knowing its 3 sides.
  •            5. Identity u2 + v2 = ... Median's theorem. Level's lines MB2 + MC2 = Ka2.
  •            6. Identity u2 - v2 = ... Level's lines MB2 - MC2 = Ka2.
  •            7. Triangle with perpendicular medians. Construction of triangles having this property.
  •            8. Orthogonal lines in a square. A proof using the scalar product.
  •            9. To determine the angle by which a chimney is seen.
  •            10. Euler's identity between 4 points of the plane. Application to the triangle's orthocenter.
   16. Scalar product and norm in an orthonormal base. Basic formulas of trigonometry.
  •            1. Orthonormal base. Components of a vector in an orthonormal base.
  •            2. Transformation of vector's components through a change of base.
  •            3. Scalar product and norm, in function of the components in an orthonormal base.
  •            4. Addition formulas: trigonometric lines of the sum and difference of two angles.
  •            5. Relations between the trigonometric lines of angles a and 2a.
  •            6. Relations between the trigonometric lines of x/2 and x. Use of the parameter t=tan(x/2).
  •            7. Factorization of the sum/difference of two trigonometric lines.
  •            8. Geometrical construction of the regular pentagon and the associated trigonometric lines.
  •            9. Trigonometric lines of angle p/12 and its symmetrical angles.
  •            10. Identities which result from the symmetries of the equilateral triangle.
   17. Distance between two points. Metric relations in a triangle. Equation of a circle.
  •            1. Distance between two points. Equation of a circle.
  •            2. Equation of a circle having a diameter [AB].
  •            3. Triangle's area and the resulting metric relations.
  •            4. The area of a triangle in function of its sides: Heron's formula.
  •            5. Metric relations in a triangle and the subscribed circle.
  •            6. Slopes of two orthogonal vectors. Line drawn from a point, orthogonal to a vector.
  •            7. Distance d of a point to a line with equation ax+by=c. Determination of d as a minimum.
  •            8. Sum of the squared distances from a point to a triangle's vertices.
   18. Sets, counting, and probabilities.
  •            1. Introduction to the theory of sets.
  •            2. Boolean operations on sets.
  •            3. The set of outcomes E. Definition of an event. Events and sets.
  •            4. Favorite sports at a summer camp: presentation using sets and a table.
  •            5. Definition and theorem of probabilities.
  •            6. Sum of points in a throw of two dice. Product figures.
  •            7. Tree and probabilities in 4 tosses of a coin.
  •            8. Result of an inspection of vehicles.
  •            9. Galileo's problem of 3 dice.
  •            10. Simultaneous drawing of two balls from a box containing balls of two colors.
   19. Conditional probabilities, total probablities. The use of trees.
  •            1. Conditional probabilities. Total probabilities.
  •            2. Conditional probability in the throw of two dice.
  •            3. Courses for final exams in a high school. Distribution boys/girls.
  •            4. Promotion made by a record company.
  •            5. A television factory.
  •            6. Conditional probability for a smoker who tries to quit.
  •            7. Law of genetic equilibrium by matching the parents' genes. Law of Hardy-Weinberg.
  •            8. Reliability of a test for an illness.
  •            9. A Sudoku amateur. Adapted from a French High School's final exam (June 2007).
  •            10. Recapitulation exercise. Relations between the probabilities of two partitions.
   20. Random variable and function. Mean expectation value and variance. Binomial distribution.
  •            1. Bernoulli's scheme. Binomial distribution with parameters p and n.
  •            2. Vertical line diagram of binomial distribution.
  •            3. Random variable and probability distribution. Mathematical expectation, variance, standard deviation.
  •            4. Calculation of the variance and standard deviation of binomial distribution B(n, p).
  •            5. Probability, expectation value and standard deviation in a casino game. Winning random function.
  •            6. Is a proposed game of balls worthwhile ? For what bid is the game balanced ?