北京 景山 学校 数学系
Elective / Pre-Calculus @ MATHEMATICS @ Senior 1+  。2010
jiguanglaoshi@gmail.com

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[3] November
[updated : 2010/11/21]
Date
 Activities / Objectives
  Images / Home work / Assignments
 Remarks / Images
 References  / Links


Monday

Nov. 01

14:25 -15:05

15:15 -15:55
Review and complements on
Associated functions
Associated functions / geometric transformations
    Examples  - Equations - Graphs - Exercises
    Use of Mathematical software
Given a function f of which we know the graph we can define the following ones and graph them without having to do anymore calculations.


a)     f1(x) = - f(x) / Ox Symmetry

b)    f2(x) = f(-x) / Oy Symmetry

c)    f3(x) = - f(-x) / Central Symmetry

d)    f4(x) = f(x - L) + H  / Translation

e)   f5(x) = | f(x) |  partial symmetry / Ox

f)  f6(x) = f( |x| ) partial symmetry / Oy

e) f7(x) = | f(|x|) |
Exercises / Assignment #5 p.2
Examples

hyp

All graphs must be
very carefully drawn :
  • No scratches
  • Use a ruler for the asymptotes and the axes of symmetry
  • Use different colors for each curve
  • Mark precisely which is which.

Test # 1
Monday Nov. 15th
  • Elementary functions :
    • First degree (st. lines)
    • Second degree (Parabola)
    • Homographic (Hyperbola)
    • Radical
    • Absolute value
  • Associated functions (see list in blue column)
  • Symmetries :
    • even functions
    • odd functions
    • vertical axes other than Oy
    • center other than O


Assignment #5


Assignment #6
p.1


Assignment #6
p.2
Monday

Nov. 08

14:25 -15:05

15:15 -15:55
  • Check Attendance
  • Correction of exercises in Assignment #5
  • Exercises on Assignment #6 p.1
  • Check understanding of Vocabulary
General problems of symmetry
  • even functions :     f(-x)  =  f(x)
  • odd functions :      f(-x) = - f(x)
  • vertical axis x = a
                    f(a+x) = f(a-x)
  • center I(a;b) :
          f(a + x) + f(a - x) = 2b
  • Use of a change of variable associated to a change of Axes :
  • F(X) = f(a + X)  & F even
  • F(X) = f(a + X) - b  & F odd
Exercises : study the symmetry of the graph of the given functions :
- review of the Parabola :
    f(x) = ax2 + bx + c = a (x  - L)2 + H
     F(X) = f(X + L) = aX2+ H : Even

- review of the Hyperbola :
      Hyp F(X)   

- Other examples : see next column   

Exercises in class :
f1
Axis : x = 2
gr1



f2
Center : I(2;-1)
odd2

 Exercises on Assignment #6 p.1

raducaux

niao


ois


Assignment #6
p.2













Monday

Nov. 15

14:25 -15:05

15:15 -15:55
  • Check attendance
  • Return Assignment # 6 p.1
  • Complete Assignment # 6 p.2

Review TEST # 1
(60 minutes)
  1. Linear programming


  2. Parabola and Hyperbola

  3. Associated functions
 TEST # 1 Answers

1.1   1.2

1.3
 Assignment #6
p.2

has not been prepared by any student !

Here are the answers checked in class
by calculations and computer graphics :


symm





Review TEST # 1




TEST # 1 Answers
Monday

Nov. 22

14:25 -15:05

15:15 -15:55
  • Introduction to Sequences :
      • Fibonacci Sequences
      • the decimals of Pi
    • Examples of Arithmetic sequences :
      • the Natural numbers
      • The Odd numbers
      • The Even numbers
      • General formula :
        U        n = U0 + n.r
    • Examples of Geometric sequences :
      • The powers of 2
      • the powers of 1/2
      • General formula :
        Vn = V0 . qn

  • Exercise on non linear sequences :
    Sequences defined by an elementary function :
    • Un = (4n -2) / (n+1)
      Graph of the first terms to examine :
    • Is the sequence monotonous ?
    • Is the sequence bounded ?
    • Is there a limit ?



f(n)

un=f(n)
  • for any n : -2 ≤ un ≤ 4
  • for any n : un ≤ un+1
  • lim un = 4
  • Test #1 + Answers
    has been returned
    to the students the next
    Tuesday Nov.16
    by 吴鹏老师



Memo of Introduction to Sequences
(to download)


Assignment # 7(sequences)
(to download)
Monday

Nov. 29

14:25 -15:05

15:15 -15:55
  • Check attendance
  • Check and Complete p.1 of Assignment # 7
Review of the properties of the geometric and the arithmetic sequences :
  • Is the sequence monotonous ?
  • Is the sequence bounded ?
  • Is there an adequate formula ?
  • Is there a limit ?
Discussion for the arithmetic and the geometric sequences  case by case :
  1. Un = U0 + n.r :
    1. r > 0
    2. r < 0
    3. r =0
  2. Vn = V0. qn:
    1. q > 1
      1. V0 > 0
      2. V0 < 0
    2. 0 < q < 1
    3. q = 1
    4. -1 < q <0
    5. q =-1
    6. q < -1
  • New Example of sequences :
    • the laughing cow earings :
    the concept of limit.

    Formal defintion of the limit of a sequence :
    • Limit Un = 0 (example)
    • Limit Un = 1 (example)
    Examples sequences defined recursively :
    Construction of the graphic representation of sequences defined by recursion.
    un+1 = f(un) and  u0 = a
    Examples with :
    frec

    un+1 =  - 0.5 un + 2  ;  u0 = 0

    vacheronde
    The original box has a diameter of 10cm
    Each earing is 1/10 of the box.
    How many boxes included inside one another
    can be seen until the diameter of the box
    is less than .5 mm. ?


    Home work :
    Assignment # 7 p.2
    Complete the questions

         fr


     v0 = 0 ;   frec   ; v0 = 4
    Construction of a sequence defined by recursion whith an elementary function :

    f(x) = 0.5 x + 2 ; un+1 = f(un)
    un+1 =  - 0.5 un + 2  ;  u0 = 0
    Un+1=-0,5Un+2.png
    vn =  un -  4/3
    (    vn) is a geometric sequence
    (to be checked)