pi pi 2

, 29-30 2014

.

pi

. pi

30 2014

( :

: ( )

:

. pi

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:

(pi, :, Fermat,Descartes, Huygens,Newton.)

pi .

. pi

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- - -pi .

: Isaac Newton,Grantham, Lincolnshire, 1665

. pi

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:

, , , pi,.

:

R.

:

.

. pi

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NEWTON

I R, f , g : I R, a, a + h I .hf (a)hg(a)

:=f (a + h) f (a)g(a + h) g(a) .

pi f pi g a f (a) - g(a).

hf (a)/h(a). . pi

30 2014

: , pi, pi , .a pi,V (a) S(a) pi,

V (a) = a3, S(a) = 6a2.

. pi

30 2014

pi

hV (a)hS(a)

=(a + h)3 a36(a + h)2 6a2 =

3a2 + 3ah + h2

6(2a + h).

pi pi (h = a),

hV (a) =7a18

hS(a).

pi pi (h = a(2 1),)

hV (a) =3a2 + 3ah + h2

6(2a + h)hS(a) u 1.8V (a)

pi V (a + h) u 2.8V (a). . pi

30 2014

pi, pi ,(h u 0.26a),

hS(a) = hV (a)/hV (a)hS(a)

=a3

3a2+3ah+h26(2a+h)

u 3.52a2

pi .

. pi

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: pi USBstick PowerBook G4, pi, :

: pi pi ;

. pi

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t0 , T pi pi , + 1 .

1 = 1, 1V () = 0, 6572 0, 4347 = 0, 2225 GB .pi, pi pi (pipi) - , () .

V () V (t0) t0 =

V V ()t0 + T =

1V ()1

= 0, 2225GBmin

.

. pi

30 2014

pi, pipi

t0 = V (t1)0, 2225 =0, 43470, 2225

= 1.95 u 2 pi

pipi

t0+T = V V ()0, 2225 =3, 32530, 2225

= 14, 9 u 15 pi

, :

. pi

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pi pi, pi , . [ .]

pi pi pi pi .

[] Cava-lieri (1598-1647), - (indivisibles). Geometria indivisibilibus (1635).

= . . pi

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pi :

pipi pi pi pi . pi pi - -, .

. pi

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pi- .

. pi

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Descartes (1596-1650): pi.

Fermat (1601-1665):

pi y = f (x) a,

f (a + h) u f (a),

h. pi pipi a, , pipi, h = 0.

. pi

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:

f (x) = 3x3 2x2 + 1. f (a + h) u f (a)

9a2 + 9ah + 3h2 4a 2h u 0. h = 0, 9a2 4a = 0. a = 0 a = 4/9 f . , , ,, pi.

. pi

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Fermat pipi pi pipi h = 0.

, pi pi pipi, pi pi , h 0, pi:

: pif (x) = x + sin x . f (a + h) u f (a)

h + sin(x + h) u sin x ,

. pi

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pi pi

cos(x +h2

) u h/2sin(h/2)

.

pi -1 h 0.pi pipicos x = 1, pi + 2kpi, k Z, .

. pi

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Newton (1642-1727): pi pi pi .

Barrow (1630-1677): pi P , - .

. pi

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Leibniz (1646-1716): pi. f (a) f pi x a:

df (x)dx

a

= limh0hf (a)

h = limh0f (a+h)f (a)

h =: f(a).

h 0, f (a + h) f (a)

h

pi pi P := (a, f (a)).

. pi

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-pi (325-265 pi..), (287-212 pi..), pi- ( ) (262-190 pi..) ..

Bhaskara (1114-1185) () pi Rolle.

Sharaf al-Din al-Tusi (1135-1213) () pi pi pi pi3 .

. pi

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pi pi pi pi. f (a) f : R, a , pi pi- pi P := (a, f (a)).

pi P pi

y = mP(x a) + f (a),pi mP = f (a). . pi

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;

(;): pipi P ,

pi pi P . pi P pi.

pi pi pi pi pi pi .

. pi

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pi P pi.

pi P pi( pi ) pi .

pi P pi .

pi P pi pi pi .

. pi

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:

PQ pi, Q pi P . pi M, pi M. pi y = f (x) x = a, pi (a, f (a)) f (a).

. pi

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: pi P P .()

pi P pi P pi pi P .()

. pi

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pi (pi ) pi (, - ) pi pi- .

pi pi pi

. pi

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pi :

(1731-1800)

. pi

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, pi pi , .

. pi

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I. C. Bivens (What a tangent line is when it isnt a limit,Mathematical Association of America, (1986) 133-143.)

f : R , pi . y = M(x) = mx + bpi y = f (x) P = (a, f (a)) R, y = K (x) = kx + c , pi pi P ,pi > 0, f (x)M(x) f (x) K (x)x (a , a + ).

. pi

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: pi pi, .

: pi pi L1(x),L2(x), pi > 0, f (x)L1(x) = f (x)L2(x), x (a , a+ ). f (x) = 12

[L1(x) + L2(x)

], x (a , a + ),

pi pi a, f pi . pi - pi (a, f (a)).

. pi

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: pi pi P := (a, f (a)), pi m , k , pi > 0:f (x) f (a)

x a m f (x) f (a)

x a k,

x (a , a + ).: pi pi P = (a, f (a)), pi f (a) .

. pi

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pi- f : R, (pi ,) x = a ,

Y (h) = mPh, h R,pi mP = f (a).

f (a + h) f (a)mPh =: a(h) pi 2 . . pi

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Leibniz,

df (a)(h) = f (a)h. (1)

g(x) = x , g (x) = 1, pi

dx(h) = dg(x)(h) = 1h = h.

(1)

df (a)(h) = f (a)dx(h), h R,

df (a) = f (a)dx .

. pi

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-

f : (,+] pi- a , pi > 0 k R

f (x) f (a) k(x a), x (a , a + ). pipi f a Df (a) k.

f (a) := {Y = kh, h R, k Df (a)} pi f a.

. pi

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f (x) :=

{(x 1)2 cos(x 1), < x < 1(x 1) cos(x 1) + 1, 1 x < +.

Df (x), x R f (x), x R

. pi

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f (x) := |x12|+1, x R

Df (x), x Rf (x), x R

. pi

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f (x) :=

0, x [1, 1]|x | 1, x [2,1) (1, 2]+, x (,2) (2,+).

Df (x) =

{0}, x (1, 1),[1, 0], x = 1,[0, 1], x = 1,{1}, x (2,1),{1}, x (1, 2),(,1], x = 2,[1,+), x = 2,, x (,2) (2,+).

f (x) = x sin 1x , x R

Df (0) = ,

Df (x) = {sin 1x 1x cos 1x }, x 6= 0.

. pi

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: 1. f pi a, 0 Df (a).2. f Df (x) 6= , x , .3. (Rockafellar) f1, f2 : R ,

D(f1 + f2)(a) = Df1(a) +Df2(a).

4. f a f (a) , f (a)pi f (a) = {Y = f (a)h : h R}.

. pi

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: pi ,y = f (x), x (a , a + ) Graph(f ) = P = (a, f (a)) .

y = mx + b pi pi P ,

f (x)mx b = 0 a 2.

. pi

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:

f pi-, n, y = mx + b pi P , pi

f (x)mx b(x a)2

pi n 2.

. pi

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: pi pi pi f (x) = x3 + 3x 5 P = (1, f (1)) = (1, 1).

pi y = mx + b. pi f (x) (mx + b) (x 1)2. f (x) (mx + b)

(x 1)2 = x + 2 +(6m)x (7 + b)

x2 2x + 1 .

m = 6 b = 7.

. pi

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: f (x) = 3x4 6x2 + 3x + 1 P = (1, f (1)) = (1,5).

f (x) (mx + b)(x (1))2 =

=3x4 6x2 + 3x + 1mx b

(x + 1)2=

= 3x2 6x + 3 + (3m)x 2 b(x + 1)2

,

pi m = 3 b = 2. . pi

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DESCARTES: Descartes pi , (pi, , ,) pi pi pi -pi .

. pi

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Kh ( x x ,) pi pi P := (a, f (a)) Ph := (a+ h, f (a+ h)). (f (a) = 0, f (x) + 1 f (x), f (a) 6= 0.)

K = limh0Kh.

pi. PK , pi m = 1/.

. pi

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: f (x) = 3x2 2x + 1 P = (1, f (1)) = (1, 2).

h 6= 0pi Ph = (1 + h, f (1 + h)) == (1 + h, 2 + 4h + 3h2). Kh = (kh, 0), P ,Ph.

. pi

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pi |Kh P| ==(

(1 kh)2 + 22)1/2

=(k2h 2kh + 5

)1/2

|Kh Ph| =(1 + h kh)2 + (2 + 4h + 3h2)2

)1/2=(

(1 kh)2 + h2 + 2h(1 kh)+

+(4h + 3h2)2 + 4(4h + 3h2))1/2

,

, pi

h + 2(1 kh) + h(4 + 3h)2 + 4(4 + 3h) = 0. . pi

30 2014

kh = 1 +h+h(4+3h)2+4(4+3h)

2 , pi

k = limh0kh = 1 +162

= 9.

=(9,0) pi pi pi P := (1, 2). KP 14 . , pi m = 4,pi pi y = 4x 2. . pi

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: pi pif (x) = sin(x). pi

P = (pi

4, sin(

pi

4)) = (

pi

4,

22

).

h 6= 0, Ph = (

pi

4+ h, f (

pi

4+ h)) = (

pi

4+ h, sin(

pi

4+ h)

= (pi

4+ h,

22

(sin(h) + cos(h)).

K (h) = (kh, 0) P ,Ph.

. pi

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(khpi4 )2+(

22

)2 = (khpi4h)2+(

22

(sin(h)+cos(h)))2.

pi

kh =h2

+pi

4+

12sin(2h)2hpi

k := limh0

kh =pi

4+

12

=pi + 24

.

KP , pi K = (k, 0),

:= [22

]/[pi + 24 pi

4] =

2.

. pi

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pi

m = 12 =22,

pi pi

y22

=

22

(xpi4

),

y =

22x+

2(4 pi)

8.

. pi

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LEIBNIZ: f P = (x , f (x)),An = (x + hn, f (x + hn)),n = 1, 2, , pipi lim hn = 0. n = 1, 2, , PAn

mn =f (x + hn) f (x)

hn.

. pi

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pi pi pi PAn, n = 1, 2, (pi ) . pi pi P ,

pi (hn).

. pi

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pi, m = limmn,

m = limf (x + hn) f (x)

hn.

pi , pipi- pi . -, ,

m = limh0

f (x + h) f (x)h

.

. pi

30 2014

: f (x) = 4x5 3x sin(x) + 1 P = (a, f (a)), pi a R. h 6= 0 f (a + h) f (a)

h=

= 4[(a + h)4 + (a + h)3a + (a + h)2a2 + (a + h)a3+

+ a4] 3asin(h2)

h2

cos(a +h2

) 3 sin(a + h),

pi

m = 20a4 3a cos(a) 3 sin(a).

. pi

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a = pi4 , pi

m =5pi4

64 32(pi + 4)8

,

pi y = mx + b pi pi P , pi

b = 4a53a sin(a)+1ma

=4pi5

256+

3pi22

32+ 1.

. pi

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: [ pi .] (zoom) pi pi.

:

. pi

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: 3.000 pipi.

. pi

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:

. pi

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: 1.000.000 pipi.

. pi

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: : 100.000.000 pipi.

. pi

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. pi

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. pi

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f (x) = x tan[cos(ln1/3

[ x2 + 12+ cos(x)

]+ x sin2

[|2x5 3x4|+ exp

( x3 3x2 + xsin(x4 + 1)

)])]

. pi

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:

. pi

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. pi

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: 5.000.000.000 pipi

. pi

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:

. pi

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. pi

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. pi

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: 2 10111=200.000.000.000 10100=200.000.000.000 Googols pipi.

. pi

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pi

. pi

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