1. , , ..
lim (u1 + u2 + + un) = lim u1+ lim u2+ + lim un
2. , ..
lim (u1 × u2 × × un) = lim u1 × lim u2 × × lim un
3. , , .. lim V ¹ 0.
4. u = u(x), z = z(x), v = v(x) u £ z £ v u(x) v(x) ( ¥ ) b, z = z(x) ( ¥) .
4 , . (2.1)
(2.1) sin x: sin x ~x.
. . 2.3 = = sin. , , ( ) , = 0 . ( sin x).
, : (2.2)
( p) = 2,71828; , , , , ln x = logex. = ( ). : .
:
. = f() :
1. ;
2. , .. . . 0 D = 0 + D. D = f(0 + D) f(0).
f() 0, , ..
(2.3) (2.3`)
: , . , , . , , (2.4), .. (2.4) ( ( ) ): .
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:
:
.
, f(x) (, b), a < b, . , , . , b1, b2 f(a) , . , b1 ¹ b2 ( b2 - b1) , b1 = b2. , , . ( : ).
( ).
1. f(x) [a, b], = 1 , f(x1) ³ f(x), , 2 ,
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(, (, b) . : = (, b) , .. b!)
3. f(x) [a, b] f(a) = A f(b) = B , m, , = , a b, f(c) = m ( , 2 3).
: f(x) , , .
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1.4. :
1) ; 2) ; 3) ; 4) .
1.5. :
1) ; 2) ; 3) ; 4) .
1.6. ( ) :
1) ; 3) ;
2) ; 4) .
1.7. :
1) ; 3) ;
2) ; 4) .
1.8. = -1:
1) ;
2) 2 ;
3) 1 .
1.9. [ ; ] , :
1) ;
2) ;
3) .
.
= f(x) . D. + D , , f (x + D), ..
D = f (x + D) f (x). . , y` ( f`(x) dy / dx). ` , .
(3.1) (3.1`)
y = f(x) D D, . . (f`(x) = j(x)). = f `() `/ = . .
( ) . : , V = s / t (s = s (t) , t ). , , ( ). .
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.
: f`(x) f(x) (, ). , (1.36), = f(x) (0, 0) 0 = f `(x0)(x x0) (3.3).
, y = f(x) = 0, .. , . (), , ().
. = f() , . , , , g , .. . D = f `(x0) Dx + gDx, , D 0 D 0 f(x) 0. , . , 0, . ,
. [a, b]. , .. ( ) , .
= 0. , .. = 0 , .
. = 2. D = (x + Dx)2 2 = 2xDx + D2 , .. = 2, ` = 2. , , = n, n , nxn1, .. = n, ` = nn1 (3.4). , , n. :
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= sinx, y` = cosx (3.5) = cosx, y` = sinx (3.6)
, .. = , , ` = 0 (3.7)
, = c f(x), c = const, y` = cf `(x) (3.8).
, .. , (3.9)
( () , i , n .. .)
, .. = uv,
y` = u`v + uv` (3.10).
( ) , , , .. y = u / v, (3.11).
. , .
= logax, (3.12). , (3.12`)
y = tg x, (3.13) y = tg x, (3.14)
= (a > 0), ` = ln a (3.15) ()` = ex (3.15`)
- = F(u), u = f(x), = F(f(x). u .
: u = f(x) ux` = f `(x), y = F(u), u y`u = F(u), = F(f(x)) y` = F`u(u)f `(x) y`x=y`uu`x (3.16)
( ).
: y = sin x2 => y = sin u, u = x2, (3.16). (3.5) (3.4) : y`u = cos u, u`x = 2x, y`x = 2xcos x2.
.. , F(x, y) = 0 (3.17).
(, (3.17) = f (), . ).
: F(x, y) = sin (x + y) e(x y) = 0. , , :
, , , . = n. , ln y = n ln x, n. . (, - ).
. y = f(x) , (a, b), (a < b). ( (f(x2) > f(x1) x2 > x1) . f(x2) < f(x1) x2 > x1 ). ( ) . , . , , = j(). y = f(x), y = f(x) = j(). = j() y = f(x) . , :
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1. () [a, b], f(a) = c, f(b) = d, [c, d];
2. y = f(x) , , ().
: = 2 (¥, ¥) , : (0 £ < ¥) (- ¥ < < 0).
: y = f(x) = j(), j`() , y = f(x) f `(x) 1 / j`(), ..
f`(x) = 1 / j`() (3.18).
, :
y = arcsin x, (3.19) y = arccos x, (3.20)
y = arctg x, (3.21) y = arcctg x, (3.22)
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(3.11) :
(, sin2x + cos2x=1; sin2x = 2sinx cosx)
= . ln = xlnx. , (lny)` = (xlnx)` => y`/ = lnx + 1 => y` = xx (lnx + 1).
(3.19). , y = arc sin x => sin y = sin arc sin x => x = sin y. (3.18):
(, ) : ( ) t, ( , ); .. ( - 1.7.1)
( , t - (1) t (2)). , , :
. y = f(x) ( ), y` = f`(x) = dy / dx ( f(x)) . , , , .. f(x); (y`)`=(f`(x))`. , y`` = f ``(x) = d2y / dx2 () . , n (n Î Z), y(n) = f(n)(x) (n , ). .
.
y = f(x) , . D/D D 0 a 0 D 0. D D = f `(x) Dx + aDx. f`(x) ¹ 0 f `(x) D D, aD . (f`(x) D) , D, dy = f `(x) D.
= . , dy = dx dy = f`(x)dx (3.24).
f`(x) = dy / dx .
, D = dy + aDx , :
D f `()D => f (+D) f () @ f `(x) D => f (x + D) @ f(x) + f `(x) D (3.25.),
, D.
: sin460; 460 = 450 + 10 = p/4 + p/180; (3.25) , sin(x + D) sin x + D cosx sin 460 = sin (p/4 + p/180) @ sin p/4 + (p/180)cos p/4 0,7194.
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d(u + v) = du + dv (3.26), d(uv) = vdu + udv (3.27) ..
. 3.2. = f(x) (, ) . D D 1( + D, + D). NT NT = MN tg a = D f `(x) = dy ( ), .. (,).
, , . ( ) d(dy) = dy2. d2y = [f `(x) dx]`dx = f``(x)(dx)2, dx . , dny = f(n)(x)(dx)n; , .
dny = f(n)(x)dxn (3.24').
1.10. , :
1) ;
2) ;
3) .
1.11. , :
1) ;
2) ;
3) .
1.12. , ' =
1) ; 2) ; 3) .
1.13. , ' =
1) ; 2) ; 3) .
1.14. , ''' =
1) ; 3) ;
2) ; 4) .
1.15. ; ' =
1) ; 3) ;
2) ; 4) .
1.16. ; d 3 x =
1) ; 3) ;
2) ; 4) .