2
.
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f(x) (a, b). .
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f(t), t- , f(t)- ( ) .
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f(x) = 0, . , . - 0, - , 0, .
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.
f(x) = u, g(x) = v - , .
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3) , v ¹ 0
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4) 12)
5) 13)
6) 14)
7) 15)
8) 16)
.
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y = uv. , :
lny = vlnu
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.
= f(x) , x = g(y) , .
x = g(y) :
.. g¢(y) ¹ 0
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. arctg.
arctg , tg, .. :
,
:
.. :
, , .
.
y = f(x) :
: , a0, D0.
: .
aDx- , f¢(x)Dx, .. f¢(x)Dx- D.
. f(x) .
dy df(x).
, dy = f¢(x)Dx
dy = f¢(x)dx.
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.
y
f(x)
K
dy
M Dy
L
a
x x + Dx x
DMKL: KL = dy = tga×Dx = y¢×Dx
, f(x) .
.
u = f(x) v = g(x)- , , :
1) d(u v) = (u v)¢dx = u¢dx v¢dx = du dv
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2) d(uv) = (uv)¢dx = (u¢v + v¢u)dx = vdu + udv
3) d(Cu) = Cdu
4)
.
.
y = f(x), x = g(t), . - .
dy = f¢(x)g¢(t)dt = f¢(x)dx.
, dy , - , .
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dx = Dx,
t, D ¹ dx.
dy = f¢(x)Dx .
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.
.
.
.
(1685-1731)
. 1) f(x) = (n+1) .{ .. n }.
2) - , ¹ .
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.
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(1)
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= :
(3)
= , :
.
Ci (2), :
, f(x), .. . Rn+1(x). :
f(x) = Pn(x) + Rn+1(x)
.
Rn+1(x).
y ,
= -
f(x) Rn+1(x) -
.
Pn(x) , -
= - .
0 a x x
Rn+1(x). .. eÎ(a, x), q 0 < q < 1, e = a + q(x a).
:
, a = x0, x a = Dx, x = x0 + Dx, :
0 < q < 1
n =0, : f(x0 + Dx) f(x0) = f¢(x0 + qDx)×Dx . ( (1736-1813) ).
. , , ..
.
.
(1698-1746) .
= 0:
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, , , .. , - , , , .
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.
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, . , (, , 10 20 , ) 4-10 .
, .
f(x) = ex.
: f(x) = ex, f(0) = 1
f¢(x) = ex, f¢(0) = 1
f(n)(x) = ex, f(n)(0) = 1
:
: .
= 1.
8 : e = 2,71827876984127003
10 : e = 2,71828180114638451
100 : e = 2,71828182845904553
.
, , , 6-7 .
f(x) = sinx.
f(x) = sinx; f(0) = 0
f¢(x) = cosx = sin(x + p/2); f¢(0) = 1;
f¢¢(x) = -sinx = sin(x + 2p/2); f¢¢(0) = 0;
f¢¢¢(x) = -cosx = sin(x + 3p/2); f¢¢¢(0)=-1;
f(n)(x) = sin(x + pn/2); f(n)(0) = sin(pn/2);
f(n+1)(x) = sin(x + (n + 1)p/2); f(n+1)(e) = sin(e + (n + 1)p/2);
:
f(x) = cosx.
cosx, , :
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f(x) = (1 + x)a.
(a - )
..
:
a = n, n- f(n+1)(x)=0, Rn+1 = 0,
, .
: .
.
. 1.
. 2.
. 3.
. 4.
, , .
sin200.
200 : 200 = p/9.
, :
0,3420.
. , , 0,0002.
, 0 sinx . , , , , .. sinx @ x.
: sin28013¢15¢¢.
, , :
10 = ; 280 ;
1¢ ; ;
; ;
, : sinx = .
,
sin = 0,472869017612759812,
, , 0,000002, .
f(x) = ln(1 + x).
: f(x) = ln(1 + x); f (0) = 0;
f¢(x) = ;
:
( ) . ln1,5. , , . 0,0003.
ln1,5 = 0,405465108108164381
, , , .
, .
.
y = f(x) D D.
.
, , , .
: Maple (Ó Waterloo Maple Inc.) , MapleV Release 4.
.
.
( (1652-1719)- )
f(x) [a, b], (, b) f(a) = f(b), (, b) e, a < e < b, f(x) ,
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f¢(e) = 0.
, (a, b) e , y = f(x) . , .
. , f(x) [a, b] . m . = m M ¹ m.
M = m. f(x) [a, b] . e .
= m. , m [a, b]. e, a < e < b , f(e) = M. - , D ( , e + D ) :
Df(e) = f(e + Dx) f(e) £ 0
e , .
.. , :
.
:
1) f(x) [a, b] , f(a) = f(b) = = 0, e, a < e < b, , f¢(e) = 0. .. , .
2) (, b) f(x) (n-1)- n , , (n 1) .
.
( (1736-1813) )
f(x) [a, b] (, b), e
a < e < b, , .
, , .
.
.
0 e b x
f(x) , (, b) e , y = f(x) , . , .
.
F(x) = f(x) y
:
F(x) . , [a, b] (, b). e, a < e < b, F¢(e) = 0.
.. , ,
.