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Национальный горный университет

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[НЕСОРТИРОВАННОЕ]

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Высшая математика. Том 2

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= 0 , # - y=f(x) %

% lim α

∆x→0 ∆x

dy=A·dx x, % x f '(x)='.

, %

∆y = A +

→ 0 +x60, lim

∆y = A.

∆x

∆x→0

∆x

, , %, - %, -

% ' , # .

& dy y = x

x2 − 1 + ln

x + x2 − 1

2 dy = (x′

x2 − 1 + x ( x2 − 1)′ + (x + x2 − 1)′ )

dx =

x +

x2 − 1

= ( x2 −1 +

2x2

2 x2 −1

)dx = (

x2 −1+ x2

x + x2 −1

)dx =

x2 −1

x2 −1 (x + x2 −1)

2 x2 −1 x + x2 −1

2x2

−

2x2 − 1 + 1

2x2

(

)dx =

dx .3

−

x2 − 1

3.18. " # !$ " % 3.8

/ ’ , # (

& dy :

y = tg(2arccos 1− 2x2 ), x > 0.

y = xarcsin(1 x)+ ln

x +

x2 −1

, x > 0.

y = x2 arctg

x2 − 1

x2 − 1.

y = 1 + 2x − ln

x + 1 + 2x

−

y = arccos(1

+ 2x2 ),

x2 + 3

x2 + 3.

x > 0.

y = xln

x +

−

8. y = arccos

((x2 − 1)

2 )).

y = arctg (sh x)

+ (sh x)lnch x.

(x2

y = ln (cos2 x +

1 + cos4 x ).

10. y = ln(x+ 1+ x2 )−

1+ x2 arctg x.

11.

y =

−

12.

y = ln (ex + e2x − 1)+ arcsinex .

1 + x2

+ x2

13.

y = x

4 − x2 + aarcsin (x 2).

14.

y = lntg (x 2)− x sin x.

131

15. y = 2x + ln sin x + 2cos x .

17.	y = ln	x +	x2 + 1
				.

19.y = arctg x2 − 1. x

21.y = arctg tg x + 1 .

23.

y = ln

cos

x tg

25.

y = x(sin ln x − cos ln x).

27.

y = cos x lntg x − lntg

(

)

29.

y =

+ x

arctg

x − 1

31.

y = x

x2 − 1 + ln

x +

x2 − 1

16.	y =	ctg x −				tg3 x					3.
18.	y = 3	x + 2		.
18.	y = 3			.
		x − 2
20.	y = ln		x2 − 1		−				1			.
									1

							x		2 − 1
							x		2 − 1
22.	y = ln		2x + 2			x2 + x + 1							.
24.	y = ex		(cos 2x + 2sin 2x)										.
24.	y = ex		(cos 2x + 2sin 2x)										.
26.	y =		x − 1 −			1		e2			x−1 .
26.	y =		x − 1 −					e2			x−1 .
						2

28.	y =	3 + x2 − xln								x + 3 + x2				.

30.	y = xarctg x − ln									1 + x2 .

3.19. .' ( ) * !

y=f(x)

. $

M(x; ), -

, %

x +x,

= NM1.	*. 3.8. '
& x++x
y++y = f(x) -

% M1(x++x; y++y).

& +MNT NT=MN·tg 7.

tg 7 = f '(x), MN = +x, NT = f '(x)·+x. <

dy=f '(x)·+x, dy = NT.

, , f(x) % -

y=f(x) .

132

3.20. 4 ' " ( &($ ) * !

* ( #, - u % ,

y=f '(u) % dy = f '(u)du.

', - # % , u % - %, # . ) y=f(u), u=g(x) # = f(g(x)). , :

′

= f

′

(x).

(u) g

u (

)

g′

(

)

dx, g'(x)dx=du, dy= f '(u)du.

dy = y′

dx = f ′

4 ' 3. y=f(u), u=g(x), %

dy= f '(u)du, #, # u # -

( , , %

# % ( . 5

% .

y = sin x . & dy.

2 $ ,

dy = cos x d x = cos x	1	dx .3

2	x

3.21. 5 &( " ) * ! 7! 6 7 &!$

) y0=f(x0) y0' = f '(x0)

x0. ', x # x0. 0 ' +y

+y=dy+$·+x, # % -

. ,, +x

# #, # +y > dy #

+y > f '(x0)·+x.

, , +y = f(x) – f(x0), f(x) – f(x0)>f '(x0)·+x.

& f (x) ≈ f (x0 )+ f ′(x0 )∆x .

# # y = x2 + 5, x = 1,97.

21. $# % x0 , # x , -# #-

f (x0 ) f ′(x). 5 x0 = 2 .

133

# % ∆x = x − x0 ,

∆x = 1,97 − 2 = −0,03.

& x0 = 2 :

f (x0 )= f (2)= 22 + 5 = 9 = 3 .

& x0 = 2 :

′

= .

f (x)=

2 x2 + 5

x2 + 5

, f (x0 )= f

(2)=

2 22 + 5

3 3

& f (x) ≈ f (x0 )+ f ′(x0 )∆x ≈ 3 +

(−0,03) = 3 − 0,02 = 2,98 .3

3.22. " # !$ " % 3.9

/ ’ , # (

# # .

1.	y = 3 x,	x = 7,76.	2.	y = 3 x3 + 7x, x = 1,012.
3.	y = (x +	5 − x2 ) 2, x = 0,98.	4.	y = 3 x,	x = 27,54.
5.	y = arcsin x, x = 0,08.		6.	y = 3 x2	+ 2x + 5,	x = 0,97.
7.	y = 3 x,	x = 26,46.	8.	y = x2	+ x + 3,	x = 1,97.
9.	y = x11,	x = 1,021.	10. y = 3 x, x = 1,21.

11.	y = x21,		x = 0,998.	12.	y = 3 x2 ,			x = 1,03.
13.	y = x6 ,		x = 2,01.	14.	y = 3 x,			x = 8,24.
15.	y = x7 ,		x = 1,996.	16.	y = 3 x,			x = 7,64.
17.	y =	4x − 1, x = 2,56.		18.	y = 1		2x2 + x + 1, x = 1,016.
19.	y = 3 x,		x = 8,36.	20.	y = 1		x ,	x = 4,16.
21.	y = x7 ,		x = 2,002.	22.	y =	4x − 3,			x = 1,78.
23.	y =	x3 ,	x = 0,98.	24.	y = x5,		x = 2,997.
25.	y = 5 x2 ,		x = 1,03.	26.	y = x4,		x = 3,998.
27.	y =	1 + x + sin x, x = 0,01.		28.	y = 3 3x + cos x, x = 0,01.
29.	y = 4 2x − sin (π x 2), x = 1,02.			30.	y =	x2 + 5,			x = 1,97.
31.	y = 1	2x + 1, x = 1,58.

134

3.23. " 2 "

) y=f(x) % [ ; b]. &-

f '(x) x, # f '(x) % %

x. ) % . , -

f(x). ' ( %

# y=f(x) % y' # f ''(x).

, y'' = ( ')'. , - = 5, y'= 5x4, y''= 20x3. <, , -

. ' %

# % y''' # f '''(x).

$ , n- f(x) % ( - () (n – 1) - % (n) # f(n)(x):

(n)= ( (n-1))'.

, , - -

&'% ( (

– ().

) % s=s(t), s – (, - ( t. , ( v % v= s'(t)= v(t), # .

4t ( % v=v(t). * ( -

t++t. 9 % ( v1 = v(t++t). , -

+t % ( +v=v1– v = v(t + +t) – v(t). $(

∆v = a		% +t.
∆t	.
∆t	' t %
	' t %
+t>0: a = lim a .			∆v	′
+t>0: a = lim a .			= lim	= v (t).
		∆t→0	∆x→0 ∆t

, , % (-

. < ( % ( s t: v = s'. $

, % a = v'(t) = (s')' = s''(t), #

( a = s''(t).

' 1: & n– -

y = log3 (x + 5).

2 % y = f (x) ,

n– :

1				− 1			1 2		y(	4	) =
y′ =	(x + 5)ln3	,	y′′ =		,	y′′′ =		,
				(x + 5)2 ln3			(x + 5)3 ln3

−1 2 3

(x + 5)4 ln 3

135


&, - y		(n)	=	(−1)n−1 (n − 1)!		.3
				(x + 5)n ln 3
' 2: & n– -
y = 32 x+5.			y = f (x)
2 %					,
n– :
y′ = 2 ln 3 32 x+5 ,	y′′ = 22 ln2 3 32 x+5 ,				y′′′ = 23 ln3 3 32x+5 .
&, - : y(n)			= 2n lnn 3 32 x+5 .3
' 3: & 4- y = (x3 + 3)e4 x+3 .
2 %			y = f (x)
.	(x3 + 3) e4 x+3 = (4x3 + 12 + 3x2 ) e4 x+3 ;
y′ = 3x2 e4 x+3 + 4
y′′ = (12x2 + 6x) e4 x+3 + (16x3 + 48 + 12x2 ) e4 x+3 =
(16x3 + 24x2 + 6x + 48) e4 x+3;
y′′′ = (48x2 + 48x + 6) e4 x+3 + (			64x3 + 96x2 + 24x + 192) e4 x+3 =
= (64x3 + 144x2 + 72x + 198) e4 x+3;

yIV = (192x2 + 288x + 72) e4 x+3 + (256x3 + 384x2 + 96x + 768) e4 x+3 = = (256x3 + 576x2 + 384x + 840) e4 x+3 = 8e4 x+3 (32x3 + 72x2 + 48x + 105).

$ : yIV = 8e4 x+3 (32x3 + 72x2 + 48x + 105).3

3.24. " # !$ " % 3.10

/ ’ , # (

& n– .

1.	y = xeax .			2.	y = sin 2x + cos(x + 1).
3.	y = 5 e7 x−1.			4.	y =	4x + 7		.

	y = lg(5x + 2).					2x + 3
5.				6.	y = a3x.
7.	y =	x		8.	y = lg(x + 4).
			.
		2(3x + 2)
9.	y =	x.		10. y =			2x + 5
									.
							13(3x + 1)

136

11.	y = 23x+5.
13.	y = 3 e2x+1.
15.	y = lg(3x + 1).
17.	y =			x
						.
			9	(4x + 9)
19.	y =		4	.

			x
21.	y = a2x+3.
23.	y =			e3x+1.
25.	y = lg(2x + 7).
27.	y =			x	.
			x
				+ 1
29.	y =	1 + x			.

		1 − x
31.	y = 32x+5.


12.	y = sin (x + 1)+ cos 2x.
14.	y =	4 + 15x	.

		5x + 1
16.	y = 75x.
18.	y = lg (1 + x).
20.	y =	5x + 1
					.
		13(2x + 3)
22.	y = sin (3x + 1)+ cos5x.
24.	y =	11 + 12x		.

		6x + 5
26.	y = 2kx.
28.	y = log3 (x + 5).
30.	y =	7x + 1
						.
		17(4x + 3)

3.25. " # !$ " % 3.11

/ ’ , # (

& .

y = (3 − x2 )ln2 x,

y = (2x2 − 7)ln (x − 1),

yV = ?

yIII = ?

y = xcos x2,

yIII = ?

y =

ln (x − 1)

yIII

= ?

x − 1

y =

log2 x

yIII =

y = (4x3 + 5)e2x+1,

yV = ?

ln x

y = x2 sin (5x − 3),

yIII = ?

y =

, yIV = ?

y = (2x + 3)ln2 x,

yIII

= ?

10.

y =x(1 + x2 )arctg x,

yIII = ?

11.

y =

ln x

yIV = ?

12.

y = (4x + 3) 2− x,

yV = ?

ln (3 + x)

13.

y = e1−2x sin (2 + 3x),

yIV = ?

14.

y =

yIII

= ?

3 + x

137

15.

y = (2x3 + 1)cos x,

yV = ?

16.

y = (x2 + 3)ln (x − 3),

yIV = ?

17.

y = (1 − x − x2 )e(x−1) 2,

yIV = ?

18.

y =

sin 2x,

yIII = ?

19.

y = (x + 7)ln (x + 4),

yV = ?

20.

y = (3x − 7) 3− x,

yIV = ?

21.

y =

ln (2x + 5)

yIII = ?

22.

y = ex 2 sin 2x,

yIV

= ?

2x + 5

23.

y =

ln x

, yIII = ?

24.

y = xln (1 − 3x),

yIV

= ?

25.

y = (x2 + 3x + 1)e3x+2 ,

yV = ?

26.

y = (5x − 8) 2− x,

yIV = ?

27.

y =

ln (x − 2)

= ?

28.

y= e−x (cos2x−3sin2x),

yIV = ?

x − 2

log3 x

29.

y = (5x − 1)ln2 x,

yIII

= ?

30.

y =

yIV

= ?

y = (x3 + 3)e4x+3,

yIV

31.

= ?

3.26. ) * ! " 2 "

) % y=f(x), x – . ,

% dy=f '(x)dx x, x

( f '(x), dx = +x x (

x # % ). * dy

x, % .

y=f(x) %

# % %

d2y: d(dy)=d2y.

& . dx x ,

d2y = d(dy) = d[f '(x)dx)] = [f '(x)dx]'dx = f ''(x)dx·dx = f ''(x)(dx)2.

' (dx)2 = dx2. , d2 = f ''(x)dx2.

< # -

% :

d3y=d(d2y)=[f ''(x)dx2]'dx=f '''(x)dx3.

$ n- % (

(n – 1)- dn(y)=d(dn–1y) # dn(y) = f (n)(x)dxn.

&, , #-

(

d2 y

dn y

′

′′

(

)

(x) = dx

(x) = dx2

, … ,

(x) = dxn .

138

3.27. ) *-" )# * +, ' (

) x=x(t), y=y(t), t [T1, T2].

1 t [T1, T2] x . 0-

x - xOy,

t - . 1 t % T1 T2,

- % . , -

% , t % , #

% .

', - x=x(t) % t=t(x). ,, , %

% x: y=y[t(x)]. , x, -

, - x % .

' , ,

. 4 # ( t -

# x .

) : x = R cost , 0 ≤ t ≤ 2π .

y = R sin t

' # % - , -

t -

. ' t =0 -

M(R, 0). 0- t = π ,

M R

; R

, , %

, R.

, t % , , -

*. 3.9. 1, -

(x, ),

0- t, ,

- x . &

, x2+ y2=R2(cos2t + sin2t) x2+ y2=R2.

' , -
′		yt′		y′(t)			dy
′	= xt′		= x′(t) .
% yx	= xt′		= x′(t) .				dx
, :					dy	= f (t), x = x(t).
, :						= f (t), x = x(t).

$ , - -

, . & y′′ .

139

′

′′

′

dyx

$ , - yx' %

yxx

= (yx )

f ′(t)

′

t, # yx'=f(t), %:

(yx )

x′

t .

dx2

x′(t)

' 1: & yx' , :

x = ln (t + 1 + t2

)

1 + 1 + t2

y = 1 + t2 − ln

1 +

2 xt′ =

2 1 + t

1 + t

+ t

1 + t

+ 1

t +

t + 1 + t

t −

1 + t

y′

−

2 1 + t2

1 + 1 + t2

−

t2 − 1 + t2 − 1 − t2 =

1 + t2 + 1 =

1 + t2

1 + 1 + t2

t 1 + t2

1 + 1 + t2 t

1 + t2

t2 + 1

+ 1 t2 + 1

t2 +

t2 + 1

y′

t2 + 1 t2 + 1

t2 + 1

+ 1

y′

. =

= t

$ : y′ = t +

xt′

' 2: &

x = ln t,

y = arctg t.

2 & xt′

yt′ =

y′x =

yt′

;

1 + t2

xt′

′

1 + t2 − 2t2

1 − t2

(y′x ) t

(1 + t2 )2

y′′ , -

= t . 1 + t2

140

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