Zenon

1
??? ??

• Zenons Paradox ???? ??? ??
• ????
• ??
• ???, ???
• ??? ??, ??
• ?????
• ??
• ????? ????
• ??, ????

2
Zenos Paradox

• 1??? ??? ½???? ????.
• ½??? ??? ¼??? ????.
• ½ ¼ 1/8 1/16 ..
• ??? ???

1? ???? ???.
3
????

• ???? S a ar ar2 ar3 .
• S r S a S a/(1-r)
• S 1 1 1 1
• -S -1 1 1
• S-(-S) 1 2S 1 S ½ !!!!

4
????

• ???? S a ar ar2 ar3 .
• S r S a S a/(1-r)
• S 1 1 1 1
• -S -1 1 1
• S-(-S) 1 2S 1 S ½ !!!! ??? ?????.

5
???? ?

• 11/2 1/3 ¼ 1/5 ..
• ?/4 1 1/3 1/5 1/7 .
• ?2/6 1 1/22 1/32 1/42 .
• ??? ??? ??? ?????
• ??? ? ?? ??? ????.

6
??? ??? ??

• lim an ? A ?? ?? an ? A? ?????? ?????? ???. ?
  A?? ??? n? ???? ?? ??? ???? ??? ???.
• lim an ? A ?? ??? ?gt 0 ? ??? ?? N? ???? n gt
  N?? A an lt ???.
• ??? ?? ? an A? ??? Sn ? A? ????? ???.

7
??

• x(t) 5t2 5t3 1
• y exp(x), y log(x).
• z x2 y2
• exp(x) 1 x x2/2! x3/3!

8
?????

• Newtons solution
• f(x) x2
• f(x) limh? 0 f(xh) f(x)/h dy/dx ? ??
• (xh)2 x2/h 2xh h2/h 2x h
• limh? 0 f(xh) f(x)/h 2x
• http//math.furman.edu/dcs/java/tangent.html
• http//mss.math.vanderbilt.edu/pscrooke/MSS/tange
  ntline.html
• http//www.math.hmc.edu/calculus/tutorials/tangent
  _line/
• http//www.ies.co.jp/math/java/calc/LimSinX/LimSin
  X.html

9
Infinitesimal (fluxion)

• f(x) x2
• (f(x dx) f(x))/ dx (x2 2x dx dx 2
  x2)/dx 2x dx 2x
• ?ab dx b- a
• Berkeley? ??? ?????????
• A. Robinsons non-standard analysis

10
??? ??, ???, ??

• Weierstrass
• ??? ?? limx? a f(x) A
• ??? ?? limh? 0 (f(x h) f(x))/h
• f(x) xn ? ???? f(x) nxn-1.
• sin x ? cos x, cos x ? -sin x tan x ? 1/cos2x
  ex ? ex ln x ? 1/x

11
?? ???

• dP/dt -r P ? ?? ??? ???? ????. ? P(t)
  Mert M? ?? ?? ? ?? ??? ??? ??, ????? ?? ??
• d P/dt r P P(t) Mert. ????
• d P/dt r P(L-P) ?????P(t) ML/(M (L-
  M)eLrt )

12
??

• Eudoxus of Cnidus (circa 408 BC - circa 347 BC)
  method of exhaustion
• Archimedes "Perhaps the best indication of what
  Archimedes truly loved most is his request that
  his tombstone include a cylinder circumscribing a
  sphere, accompanied by the inscription of his
  amazing theorem that the sphere is exactly
  two-thirds of the circumscribing cylinder in both
  surface area and volume!"

13
Method of Exhaustion

• ????? ??? ?? ???? ??? ??? 1/3
• ???? ????.

14
Cavalieri ? ??

• ???? ??? ???? ?????? ??? ???? ??? ??? ???.
• ????? ??

15
????? ????(??)

• If f(x) is continuous on a,b,then the funcion
  has a derivative at every point in a,b, and
• ???? ?? ??? ???? ??.
• http//archives.math.utk.edu/visual.calculus/4/ftc
  .9/
• http//www.ies.co.jp/math/java/calc/perime/perime.
  html

16
??????

• xn ? xn1/n1
• cos x ? sin x
• sin x ? -cos x
• exp x ? exp x
• 1/x ? ln x

17
?? ??

• 1.  All objects have inertia that is, an object
  will remain at rest or in uniform motion unless
  acted upon by some outside force.
• 2.  An object's acceleration is in the same
  direction as the force exerted on it.   The force
  exerted on that object is equal to the product of
  its mass and acceleration, or  F ma
• 3.  For every action there MUST be an equal and
  opposite reaction.
• http//home.sprintmail.com/speshek/physics.html
• http//csep10.phys.utk.edu/astr161/lect/history/ke
  pler.html

18
???? ??

• L T - U ?????
• T m(l )2 U - mglcos ?
• L m(l )2 mglcos ?
• S ?L(q(t),q(t),t )dt

19
?????

• p
• H(p,q) pq - L(q,q )

20
????

• ?????? ???? ??? ?? ????. ??? ????? ??? ??? ???
  ???. ?????? ???
• ?? ??, ???, ?? ????? ????
• ???? ???? ????? ??? ??? ????