der
TRANSCRIPT
-
.. | | MathUs.ru
1 2
1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.6 . . . . . . . . . . . . . . . . . . . . . 91.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.12 . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2 232.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.4 - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
. -, : -, ; . , - !
, -, , , . : .
. - . , , - , .
1
-
1 , . . , .
1.1
:
1,1
2,1
3,1
4, . . . ,
1
n, . . .
(. 1).
0 112
13
14
. 1. 1/n (n N)
, ( -). n = 10 1/10 ; n = 100 1/100 ; n = 1000 1/1000 . .
, 1/n , , . :
limn
1
n= 0.
, 0. .
an = 3 +1
n(n N)
3.
limn
an = limn
(3 +
1
n
)= 3.
, a , a , . , an a : , an, , a .
, 1, 1, 1, 1, . . . : . , , 1 -? ( , , 1), 1 2. , , 1.
, . -, y = f(x) , - x y. x , y .
2
-
. . y = f(x) , x .
, . 2 y = x2. x = 2 A(2, 4).
X
Y
2
4 A
. 2. y = x2
, x 2 ( ). A, . , 4, :
limx2
x2 = 4. (1)
? . , x 2, x2 22 = 4. , - ?
. . 3.
X
Y
1
pi
. 3. y =sinx
x
f(x) =sinx
x.
: x = 0 ( f(0) ), (0, 1). , f(0) , x 0 1. , :
limx0
sinx
x= 1. (2)
3
-
. (2), . -
:
sin 0,1
0,1,
sin 0,01
0,01,
sin 0,001
0,001, . . .
, .
1.2
(1) (2) . - (1) , 2, f(x) = x2 : lim
x2x2 = 22 = 4. (2) -
0 f(x) = sinx/x (, , ).
a x f(x)
limxa
f(x) = f(a), (3)
f(x) a. a.
, f(x) = x2 x = 2 ( ). , .
f(x) = sinx/x x = 0. , (0, 1) .
. :
y =x2 4x 2 .
, , . .
1.3
60 /. ? : , 60 .
, , . , , , - 60 /. . , 60 /?
. , s, , t :
s(t) = t2,
, . , t = 0 , t = 1 s(1) = 1, t = 2 s(2) = 4, t = 3 s(3) = 9, .
, . -: 1; s(2)s(1) = 3; s(3)s(2) = 5, .
4
-
. , , . ? , t = 3?
: v = s/t ( - ). , .
. t = 3 t, s, , s t. t, .
, . t = 1.
s = s(4) s(3) = 42 32 = 7, :
s
t=
7
1= 7 (4)
(, , /). t. t = 0,1:
s = s(3,1) s(3) = 3,12 32 = 0,61,s
t=
0,61
0,1= 6,1. (5)
t = 0,01:
s = s(3,01) s(3) = 3,012 32 = 0,0601,s
t=
0,0601
0,01= 6,01. (6)
t = 0,001:
s = s(3,001) s(3) = 3,0012 32 = 0,006001,s
t=
0,006001
0,001= 6,001. (7)
(4)(7), , s/t 6. , t = 3 6 /.
, t s , - s/t v, t:
v = limt0
s
t. (8)
:v(t) = lim
t0s(t+ t) s(t)
t. (9)
s(t) = t2 , . :
s = s(t+ t) s(t) = (t+ t)2 t2 = t2 + 2tt+ t2 t2 = t(2t+ t), :
v(t) = limt0
s
t= lim
t0t(2t+ t)
t= lim
t0(2t+ t) = 2t. (10)
, t = 3 (10) : v(3) = 2 3 = 6, .
.
5
-
1.4
. - , . .
y = f(x). , x . X x, Y f(x) (. 4).
x
f(x)
x+ x
f(x+ x)
x
f
y = f(x)
X
Y
. 4.
x , x. x + x. f(x+ x).
f = f(x+ x) f(x) (11)
, x. ? x
t, f s, t. . .
. f (x) f(x) x , :
f (x) = limx0
f
x= lim
x0f(x+ x) f(x)
x. (12)
(8) (9). , ? , .
. - . , - (12).
6
-
1.5
, : f(x) = c. :
f = f(x+ x) f(x) = c c = 0., :
f (x) = limx0
f
x= lim
x00
x= lim
x00 = 0.
, :
c = 0.
, f(x) = xa. f(x) = x. :
f = f(x+ x) f(x) = x+ x x = x.
:
f (x) = limx0
f
x= lim
x0x
x= lim
x01 = 1.
,x = 1.
f(x) = x2. s(t) = t2, . ( ) , (10).
:
f = f(x+ x) f(x) = (x+ x)2 x2 = x2 + 2xx+ x2 x2 = x(2x+ x).
:
f (x) = limx0
f
x= lim
x0x(2x+ x)
x= lim
x0(2x+ x) = 2x.
, (x2)
= 2x.
f(x) = x3. :
f = f(x+ x) f(x) = (x+ x)3 x3 == x3 + 3x2x+ 3xx2 + x3 x3 = x(3x2 + 3xx+ x2).
:
f (x) = limx0
f
x= lim
x0x(3x2 + 3xx+ x2)
x= = lim
x0(3x2 + 3xx+ x2) = 3x2.
, (x3)
= 3x2.
7
-
, : (x4)
= 4x3,(x5)
= 5x4,
. . .
(xn) = nxn1.
, n, a:
(xa) = axa1, a R. (13) ,
f(x) =x: (
x)
=(x
12
)=
1
2x
121 =
1
2x
12 =
1
2x.
, . : .
. , : (1
x
)= 1
x2.
: ) ( ); ) - (13).
. f(x) = sinx. :
f = sin(x+ x) sinx., :
sin sin = 2 sin 2
cos +
2.
:
f = 2 sinx
2cos
(x+
x
2
),
f (x) = limx0
2 sin x2
cos(x+ x
2
)x
.
:
f (x) = limx0
sin x2
x2
cos
(x+
x
2
). (14)
(14) . -, .
. t = x/2. , t 0 x 0. :
limx0
sin x2
x2
= limt0
sin t
t= 1
( (2) ). , .
8
-
x + x2, , x 0 x.
, ( ). , (3) , x = 0:
limx0
cos
(x+
x
2
)= cosx. (15)
(14) 1f (x) = 1 cosx = cosx.
,(sinx) = cosx.
. , (cosx) = sinx.
:
cos cos = 2 sin 2
sin +
2.
(, , ) -. , . , .
( ) -, . , : f(x) = x7 sin 3
4x2 5x? (12) -
. , .
.
1.6
, f(x) a, f(x) a : lim
xaf(x) = f(a). ,
. -: f(x) a,
limx0
f(a+ x) = f(a). (16)
, ? x 0, a + x a, a + x a. (15).
(16) . f(a + x) f(a), f(a + x) f(a) . f(a + x) f(a)? f f(x) a. :
1, a, b, - ab. , .
9
-
f(x) , x 0:
limx0
f = 0. (17)
. , - ( (12) x ).
X
Y
y = |x|
. 5. y = |x|
-? , . -: f(x) = |x| x = 0. . 5.
:
f = |x+ x| |x|.
x = 0 :
f = |x|.
? : x ( ), - ( ).
1. x 0, x > 0. f = x, f
x= 1.
2. x 0, x < 0. f = x, f
x= 1.
, f/x . , x , x (, x 1; 0,1; 0,01; 0,001; . . . ). f/x 1 1, , , ., f(x) = |x| x = 0.
, . , -. (0, 0) y = |x|.
? , - , . , .
, f(x) x. f/x x 0. x ; , . ( ? , , , , - .) , f 0 x 0; (17) x.
10
-
1.7
, . , . , , (12).
u v , . - :
u(x+ x) = u(x) + u, v(x+ x) = v(x) + v,
u v u v x. (11) .
.
0. . c , (cu) = cu.
2 . , , .
, :
(5x2) = 5(x2) = 10x,
(3 sinx) = 3(sinx) = 3 cosx.
1. . (u + v) = u + v ( - ).
, f(x) = u(x) + v(x). :
f = f(x+ x) f(x) == u(x+ x) + v(x+ x) u(x) v(x) == u(x) + u+ v(x) + v u(x) v(x) = u+ v.
:
f (x) = limx0
f
x= lim
x0u+ v
x= lim
x0
(u
x+
v
x
).
u/x v/x x 0 u(x) v(x). 2 u(x) + v(x):
f (x) = u(x) + v(x).
., 0 1, :
(sinx+ cosx) = (sinx) + (cosx) = cosx sinx,(x3 + 4 cosx 10) = (x3) + (4 cosx) + (10) = 3x2 4 sinx
2, a, b, a+ b. , .
11
-
( 10 !).2. . (uv) = uv + uv.
, . f(x) = u(x)v(x). :
f = f(x+ x) f(x) = u(x+ x)v(x+ x) u(x)v(x) == (u(x) + u)(v(x) + v) u(x)v(x) == u(x)v(x) + v(x)u+ u(x)v + uv u(x)v(x) == v(x)u+ u(x)v + uv.
:
f (x) = limx0
v(x)u+ u(x)v + uv
x= lim
x0
(u
xv(x) + u(x)
v
x+
u
xv
).
u(x)v(x). u(x)v(x). u
xv? u
x u(x), v
, v(x) x . . :
f (x) = u(x)v(x) + u(x)v(x),
. :
(x2 sinx) = (x2) sinx+ x2(sinx) = 2x sinx+ x2 cosx.
0:
(cu) = cu+ cu = cu,
c = 0.
3. .(uv
)=uv uv
v2.
.
f(x) =u(x)
v(x).
:
f =u(x+ x)
v(x+ x) u(x)v(x)
=u(x) + u
v(x) + v u(x)v(x)
=v(x)u u(x)vv(x)(v(x) + v)
,
f
x=v(x)u
x u(x)v
x
v(x)(v(x) + v),
f (x) = limx0
v(x)ux u(x)v
x
v(x)(v(x) + v)=u(x)v(x) u(x)v(x)
v2(x).
v - v(x).
12
-
, .
(tg x) =(
sinx
cosx
)=
(sinx) cosx sinx(cosx)cos2 x
=cos2 x+ sin2 x
cos2 x,
(tg x) =1
cos2 x.
,
(ctg x) = 1sin2 x
.
. , , - , , .
, , u(x) = sin x v(x) =x. x (
x v), ( v(x) u). :
u(v(x)) = sinx.
, u v. : x v, v(x) u.
, , u , v . x , . :
v(u(x)) =
sinx.
. () () . :
(sinx) = cos
x (x) = cosx 1
2x,
(
sinx) =1
2
sinx (sinx) = 1
2
sinx cosx.
:
[(4x2 + 3x+ 2)5] = 5(4x2 + 3x+ 2)4 (4x2 + 3x+ 2) = 5(4x2 + 3x+ 2)4 (8x+ 3),[A sin(x+ )] = A cos(x+ ) (x+ ) = A cos(x+ ).
, ? .
4. . [u(v(x))] = u(v(x))v(x).
f(x) = u(v(x)). :
f = u(v(x+ x)) u(v(x)) = u(v(x) + v) u(v(x)),
f
x=u(v(x) + v) u(v(x))
x=u(v(x) + v) u(v(x))
v
v
x.
13
-
x 0 v 0, u(v(x) + v) u(v(x))
v u(v(x)).
,
f (x) = limx0
u(v(x) + v) u(v(x))v
v
x= u(v(x))v(x),
.
1.8
y = f(x) (. 6) : A (x0, f(x0)) B (x0 + x, f(x0 + x))., f(x) A.
y = f(x)
A
B
C
X
Y
x0
f(x0)
x0 + x
f(x0 + x)
x
f
y = kx+ b
. 6. : f (x0) = tg = k
AB . AB X . -, [0, 180); .
AC BC X Y . , BAC = , AC = x BC = f ,
f
x=BC
AC= tg. (18)
x . f/x f (x0). ? B A, A.
, , , tg tg ( )., :
f (x0) = limx0
f
x= lim
x0tg = tg. (19)
14
-
, , k y = kx+ b. :
f (x0) = tg = k. (20)
, . , f(x) ? . 7.
y = f(x)
A
B
C
X
Y
x0
f(x0)
x0 + x
f(x0 + x)
x
f < 0
. 7. f (x0) = tg = k
, , -: f = BC. , BAC = 180 , tgBAC = tg. :
f
x= BC
AC= tgBAC = tg.
, (18). - (19) (20). , .
. x0 , x0, , , k .
1.9
, y = f(x) - (x0, f(x0)). - , f(x) x0.
y = kx + b, k b. k : k = f (x0). :
y = f (x0)x+ b, (21)
b.
15
-
, (x0, f(x0)) , . (21):
f(x0) = f(x0)x0 + b,
b = f(x0) f (x0)x0.
b (21):
y = f (x0)x+ f(x0) f (x0)x0,
y = f (x0)(x x0) + f(x0). (22)
(22) . y = x2
x0 = 3. : f(x0) = 9, f (x) = 2x, f (x0) = 6. (22):
y = 6(x 3) + 9 = 6x 9.
. y = 1/x . , -, , 2.
1.10
, , . , , - .
, , - . : f(x) = |x| x = 0.
X
Y
A
B
y = f(x)
x1 x2
. 8.
, . , , . 8, x1 x2.
, - x1 x2. , A B - ( - ). A B . , , , , - .
( ) . , . , , .
16
-
, , , . , , , , . . 9.
X
Y
C
x0
y = f(x)
. 9. ,
C , X. = 90, tg . , f (x0).
(22), - . 9 . : x = x0.
. y = 3x (0, 0).
1.11
, , . .
. 10 y = f(x) x1 x2.
X
y = f(x)
x1
1x2
2
. 10. : f (x1) > 0, f (x2) < 0
x1 . , x1 1 X. ; , - x1:
f (x1) = tg1 > 0.
x2 . x2 2 X. , x2:
f (x2) = tg2 < 0.
17
-
.
. -, .
. , .
, , . , :-)
. ,,
f(x) = x3 3x., , .
:f (x) = 3x2 3 = 3(x+ 1)(x 1).
(. 11).
X11+ +
. 11. f(x) = x3 3x
, (,1] [1,+) - [1, 1]. . - : f(1) = 2,f(1) = 2, (. 12):
X
Y
1
2
1
2
y = x3 3x
. 12. f(x) = x3 3x
1 1 -? . , - , .
18
-
, - .
.
1. . . 13 x = a f(x): a , .
X
Y
y = f(x)
M
a
. 13.
M , X. f (a) = 0. a : -. , (+) () .
2. . . 14 x = b f(x): b , .
X
Y
y = f(x)
N
b
. 14.
N . f (b) = 0. b : . () (+) . . -, ; .
19
-
3. . . 15. S , f (c) = 0.
X
Y
y = f(x)
S
c
. 15.
x = c : , . (+), . , , ., x = c .
. .
, - .
. , x = 0 :
) f(x) = 2x;) f(x) = x2 + 5;) f(x) = 1/x;) f(x) = |x|;) f(x) = x3;) f(x) =
x;
) f(x) = 3x.
. sinx cosx. , tg x .
:
f(x) = x4 4x3. , , , .
: D(f) = R. :
f (x) = 4x3 12x2 = 4x2(x 3). x : x = 0 x = 3. - , .
20
-
. (. 16).
X30
+
. 16. f(x) = x4 4x3
x = 0 : (, 0], [0, 3]. x = 0 .
x = 3 () (+). - [3,+) , x = 3 . : f(3) = 27.
X:
x4 4x3 = 0 x3(x 4) = 0., X x = 0 ( ) x = 4. (. 17):
X
Y
3
27
y = x4 4x3
40
. 17. f(x) = x4 4x3
1.12
(2) :
limt0
(1 + t)1/t = e = 2,718281828459045 . . . (23)
. :
1,110, 1,01100, 1,0011000, . . .
, (23), e .
21
-
e , . : 2,7, , :-)
. - . (23) , t (1 + t)1/t e;, 1 + t et; , et 1 t; , (et 1)/t 1. ,
limt0
et 1t
= 1. (24)
e , , . e ? , ex ( ) :
(ex) = ex. (25)
. :
(ex) = limx0
ex+x exx
= limx0
ex(ex 1)x
= ex limx0
ex 1x
= ex,
, (24).
e. ln:
lnx = loge x.
- ax. , a = eln a:
(ax) = (ex ln a) = ex ln a(x ln a) = ax ln a.
:
(ax) = ax ln a.
? . y = lnx. x:
x = ey,
x :
1 = eyy.
y =
1
ey=
1
x.
,
(lnx) =1
x.
(13). :
(xa) = (ea lnx) = ea lnx(a lnx) = xa ax
= axa1.
22
-
2 , , .
-, . , ? , ., x(t) v(t) :
x(t) = 1 + 12t 3t2, (26)v(t) = 12 6t. (27)
, t, x .
-, . , :
x(t) x(t). (28)
, , :
x(t) dx
dt(29)
( ). (29).
:dx
dt= lim
t0x
t= lim
t0x(t+ t) x(t)
t, (30)
, dt, dx x(t). , ; .
, , (29) . dx dt. dt , dx/dt (30) .
, , , dx, dt, . . .
(26) , (28) (29):
x(t) = 1 + 12t 3t2 x(t) = ddt
(1 + 12t 3t2) = 12 6t.
( ddt
.)
, (27). , .
23
-
2.1
, (27) , -. , t < 2, t = 2 t > 2.
? : , vx X. (27) :
vx = 12 6t. (31)
, , . , vx X:
vx > 0 X ;vx < 0 X.
(, vx = 3 /, , 3 / , X.)
(31) : t < 2 X ; t = 0 ; t > 2 , , X.
, v. - .
1. X, dx , dt.
x =dx
dt= v.
2. X, dx < 0. dt dx, dx/dt = v
x =dx
dt= v.
, vx = v, vx = v. :
x = vx, (32)
: .
, () . :
x > 0 vx > 0 X x ;x < 0 vx < 0 X x .
2.2
. . , .
24
-
, , v0 = 2 / v = 14 / t = 3 . :
a =v v0t
, (33)
:
a =14 2
3= 4
2.
, 4 /. , , , v0 = 14 / v = 2 /
t = 3 c? (33) :
a =2 14
3= 4
2.
, , 4 /. , ? , ,
, . - : (33) t dt, v v0 dv dt, - :
a =dv
dt= v. (34)
, , . (34), , , . -
, , (34) a = v = 0. , , -. (34) .
C , . -, . , .
X. : X X .
1. ~a X (. 18). X -: ax > 0.
X
~a
. 18. ax > 0
X. :
(vx > 0), : . vx .
(vx < 0), : -. , vx, , .
25
-
, ax > 0, vx , .
2. ~a X (. 19). X : ax < 0.
X
~a
. 19. ax < 0
X. :
(vx > 0), : . vx .
(vx < 0), : -. vx, , .
, ax < 0, vx , - - , .
ax (-) vx (34):
ax =dvxdt
= vx. (35)
ax vx. , vx x. ax - x:
ax = x. (36)
. (26):
x = 1 + 12t 3t2
( , ). , :
vx = x = 12 6t,ax = vx = 6.
, 6 /2. , X.
, - (, , ~a = const). .
, - , .
. :
x = 2 + 3t 4t2 + 5t3.
26
-
:
vx = x = 3 8t+ 15t2,ax = vx = 8 + 30t.
: .
. X :
x = 5 sin 2t.
, , 5 5. , .
:
vx = x = 5 cos 2t 2 = 10 cos 2t,ax = vx = 20 sin 2t.
, .
2.3
(32) (35) , . , , .
, ~u(t), . , .
() . - ~u t :
~u = ~u(t+ t) ~u(t). , . - ~u . 20.
~u(t)
~u(t+ t)
~u
. 20.
t , ~u (, , ). t 0 ~u/t , ~u:
d~u
dt= lim
t0~u
t= lim
t0~u(t+ t) ~u(t)
t. (37)
, ~u ; , (37). , , .
27
-
, d~u/dt . , - ~u, dt. , ; .
d~u/dt , - dt, d~u ~u. dt (37) , .
, OXY . : , .
6.1 , ~u - ~i, ~j - OXY :
~u = ux~i+ uy~j.
ux uy ~u , ~u (. 21).
ux
uy
~i
~j
~u
O X
Y
. 21. ~u = ux~i+ uy~j
~u , , ux uy :
~u(t) = ux(t)~i+ uy(t)~j.
t+ t :
~u(t+ t) = ux(t+ t)~i+ uy(t+ t)~j.
~u :
~u = ~u(t+ t) ~u(t) = (ux(t+ t) ux(t))~i+ (uy(t+ t) uy(t))~j = ux ~i+ uy ~j.
, :~u
t=
uxt
~i+uyt
~j.
28
-
t 0, ~u :d~u
dt= ux~i+ uy~j. (38)
, ~u (ux, uy), d~u/dt ~u, (ux, uy).
2.4 -
, M OXY (. 22). - M , , .
X
Y
O
M
~r
~v
x
y
~i
~j
. 22.d~r
dt= ~v
- M ~r =OM , -
, M . - .
, - , . , -.
M x y. , - ~r (x, y), :
~r = x~i+ y~j.
(38):d~r
dt= x~i+ y~j.
, x = vx y = vy. :d~r
dt= vx~i+ vy~j.
~v M . :d~r
dt= ~v. (39)
: - -.
, . . .
29
-
2.5
, ~v. :
~v = vx~i+ vy~j,
(38):d~v
dt= vx~i+ vy~j.
, vx = ax vy = ay. :
d~v
dt= ax~i+ ay~j.
~a M :
d~v
dt= ~a. (40)
, .
, (39) (40) . ; , , - .
, , . - , , . ( -, z-).
30
-