北京 景山 学校 数学系
Elective / Pre-Calculus @ MATHEMATICS @ Senior 1+  。2011-12
jiguanglaoshi@gmail.com

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[2] November
[updated : 2011/11/21]
Date
 Activities / Objectives
  Images / Home work / Assignments
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Monday

Nov. 7

3:15 -4:45
  • Introduction to Numerical Sequences :
      • Fibonacci Sequences
      • the decimals of π

    • 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

    Example of def. by recursion :
    rec_form

    rec_intro

    Pb : what happens if :
    • u0 = 0
    • u0 = 1
    • u0 = 1.5
    • u0 = 2





    MEMO
    Arithmetic
    &
    Geometric
    SEQUENCES





    Assignment # 5
    Monday

    Nov. 14

    3:15 -4:45
    Study of recurrent sequences defined by a decreasing function :

    • Affine function :  suite_affine_non_mon.png
      with :  wn=Un-4|3.png

    • Homographic function : Ass.#5  p.2/2
      • graph of the first terms of the sequence
      • study of the boundaries
      • Use of an auxiliairy geometric sequence to prove the limit.
    		 Un+1=-.5Un+2.png


         		Study of the "Chaos" sequence
    defined by :
    Un+1 = r.Un(a - Un)
    U0 = 1
    Chaos_0.4.png 
    r = 0.4 ; a = 8
    A change of 1/100 for r
    changes the whole picture !
    		Assignment # 6
    Monday

    Nov. 21

    3:15 -4:45
    		 Study of the Hanoï Towers problem
    Mn = Minimum Nb of moves :
    Mn+1 = Mn + 1 + Mn
    Hn+1= Mn+1 +1 = 2 (Mn + 1) = 2Hn
    Hn = 2n
    Limits
    of arithmetic and of geometric sequences :
    1. Un = U0 + n.r :
      1. r > 0 => lim Un = +∞
      2. r < 0 => lim Un = -∞
      3. r =0 => lim Un = U0

    2. Vn = V0. qn:
      1. q > 1  and V0 > 0 => lim Vn = +∞
      2. q > 1  and V0 < 0 => lim Vn = - ∞
      3. q = 1 => lim Vn = V0
      4. |q| < 1=> lim Vn = 0

    3. Examples : Assignment # 7

    4. Series :  Sn = U0 + U1 + U2 +...+Un  
      SumSerieGeom.png
    Hanoi_towers.png
    Mn = 2n - 1
    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. ?
    Paradox of Zenon
    Achilles & the Tortoise
    Achille_et_la_tortue.png
    kub

    Each box side is 3/4 of the previous one,
    what is the total area covered ?



    candy
    Each bar height is 1/2 of the previous one,
    what is the total lenght ?

     candy-sum


    Hanoï Towers
    [game on line]



    Presentation of the
    recursive sequences
    by 吴鹏老师.ppt
    [École alsacienne
    Jan.2009]




    Assignment # 7
    Monday

     Nov. 28

    3:15 -4:45
    		 Formal definitions of LIMITS
    of Sequences

    Applications to the study
    of the sequence defined by :
    Seq_Irrationnal.png

    Graph of the sequence defined by
    Graph_rad(6-x).png

    Graph_rad(6-x).png



    In this last sequence the formal proof of the limit will be given in the next assigment.
    The previous methods using an auxiliairy geometric sequence does not apply here.

    Assignment # 8