北京 景山 学校 数学系
Calculus ++ @ MATHEMATICS @ Senior 2  。2009-2010
jiguanglaoshi@gmail.com

<== previous month                   Home Page                  Next month  ==>
[4] December
[updated : 2009/12/20 ]
Date
 Activities / Objectives
  Images / Home Work
 Remarks / Images
 References / Links
Tuesday

Dec. 1

15:45-16:30

16:40 -17:25
90 minutes TEST
study of two functions :
  • limits
  • asymptotes
  • derivatives
  • tangents
  • singular points
 TEST of Dec.1
Tuesday

Dec. 8

15:45-16:30

16:40 -17:25
  • Return of the Test marked Papers
  • Comments about the answers
  • Definition of the differential of a function in one point. Notation : dy = f '(x)dx
    ∆y = f(x+ h) - f(x) = f '(x).h + h.e(h)
  • if  dx = h is small compared to 1, then the quantity h.e(h) is negligible and we have :
  • ∆y ≈ dy = f '(x)dx 
  •  f(x+ dx) ≈ f(x) + f '(x).dx
  • The notation dy/dx for the derivative of a function in one point is then justified and used very often in Physics to study the variations of a function by measure the small variations dy and dx to find f '(x) and then f(x).
T1.1
T1.2



ANSWERS to TEST of Dec.1
Tuesday

Dec. 15

15:45-16:30

16:40 -17:25
  • Euler's method of construction of a function from it's differential equation :
    f(a+ h) ≈ f(a) + h.f '(a)
  • Example of the Exponential function :
    f '(x) = f(x) and f(0) = 1
    f(nh) = (1 + h)n
    Let e
    then exp
  • Other examples :
    • f '(x) = 1 / (1+x2) and f(0) = 0
    • f '(x) = 2x and f(0) = 0
  • Demo of the fundamental property of the exponential function :
    from the hypothesis f '(x) = f(x) and f(0) = 1
    we proved that for any Real numbers u&v Exp(u + v) = Exp(u) • Exp(v)
Euler 1707-1783
Euler (1707 - 1783)
[Video taken]

Exp(u+v) = Exp(u). Exp(v)

Steps of the Demo :
  1. Hypothesis :
         f '(x) =k. f(x) ; f(0) = 1
  2. Let F(x) = f(a+x). f(-x)
    and prove that F'(x) = 0
    then F(x) = f(a) for any x
  3. f(a+x). f(-x) = f(a) for any x and a
  4. Concluson :
    let a = u+v ; x = -v  then a+x = u
           we get  f(u) . f(v) = f(u+v)
    (last step found by     "Jimmy"- LiuYuTao)

Euler_s_Biography
(wikipedia)

Euler_s_method.ppt
(to download)


Abstract of the above file
(printable .pdf to download)

Tuesday

Dec. 22

15:45-16:30

16:40 -17:25
 Last class !
  • Demo of the reciproqual of the fundamental property of the exponential :
    Any function which has a derivative on R
    and which transform Sums into Products
    f(u+v) = f(u) . f(v)
    ("Isomorphism" of [R;+] over [R*; x])
    satisfies the fundamental relationship :
    f '(x) = k.f(x) and f(0) = 1
Please continue to cultivate your capacities in fast calculations but try to remember that Mathematics
involve a lot of rich concepts that are not to be used in a grocery store ...

Last class !

The last period will be used for an open discussion...