Конспект лекций по математике для студентов инженерно-технических специальностей. В 3 ч. Ч. 2

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9.

....................................................................................................... 4

9.1.

,

................................................. 4

9.2.

............................................................................................................. 8

9.3.

.................................................... 25

9.4.

........................................................................................................ 28

9.5.

........................... 37

10.

(

,

,

I

) ....................................................................

78

10.1.

,

...........

............................................................................................................................................

78

10.2.

................................... 83

10.3.

.

.

.......................................................................... 84

10.4.

................. 87

10.5.

.

.

..................................................................... 90

10.6. ......................................

(

)

94

10.7.

............................................................................ 101

10.8.

.......................................................... 106

11.

............. 109

11.1.

.

.................................. 109

11.2.

.

..............................

...............................................................................................................................

110

11.3.

.

........................................................

112

11.4.

...................................................... 115

11.5.

....................................................... 119

11.6.

...................................................................... 122

11.7.

.

.

.....

.....................................................................................................

124

12.

130

12.1.

..................................................................................................................... 130

12.2.

........................... 130

12.3.

............................ 135

12.4.

............ 141

13.

................................................................................................................................... 143

13.1.

...................................................................................................................... 143

13.2.

...................................................................................................................... 147

13.3.

......................................................................... 155

13.4.

.......................................................... 166

14.

.................................................................................... 173

14.1.

............................................................................................ 173

14.2.

.......................................................................................... 178

14.3.

.............................................................................................................. 180

,

,

y f x ,

a;b ,

x a,

x

b, y 0 ( . 9.1)

y

y

f

x

x

0

a

b

. 9.1

.

,

.

.

a;b

x0 a x1 ...

xn b.

xi 1 , xi

i 1, n

pi .

xi

xi 1

xi i

1, n ,

max xi (1 i n)

.

,

,

,

[xi 1 , xi ]

f (x)

T .

[t0 ,T ]

n

[t0,t1 ], [t1 ,t2 ], ..., [tn 1 ,T ]

t k tk tk 1 tn T .

f (tk ) tk .

f (tk ) tk ,

n

S lim

f (tk ) tk ,

(9.1)

0

k 1

max tk

k 1, 2,...,n .

,

(9.1)

v f (t)

t0 ,T .

.

,

[a;b]

( x) lim

([a, b]

O ( x))

.

([a;b]

0

O (x))

(

),

x a

x b .

,

X

,

.

a;b

... xn 1 xn

b.

[xi

1 , xi ] (i

x 0

a

x i

1, n)

max xi

pi .

xi xi 1

xi

i

1, n

,

(1

i

n )

.

(

),

,

xi 1 , xi

.

,

( pi )

xi

xi 1 , xi .

,

n

( pi ) xi

i 1

n

lim

( pi ) xi ,

0

i 1

,

,

,

1.

a;b

y f x .

2.

a;b

3.

xi 1 , xi

pi (i 1, n) .

4.

n

n

f ( pi )(xi xi 1 )

f ( pi ) xi ,

i 1

i 1

n-

y f x

a;b ,

n

lim

f ( pi ) xi ,

0

i 1

.

9.1.

f

a;b .

(

)

f

a;b

8

n

lim

f ( pi ) xi ,

0

i 1

a;b

[xi 1 , xi ]

pi [ xi 1, xi] (i 1, n) .

b

def

n

f (x)dx

lim

f ( p i ) x i.

(9.2)

a

0

i 1

,

f x

f x dx

, f x

:

f x

a;b

(9.2)

,

,

,

,

,

.

,

(9.2)

,

X,

S1

S3

+

+

0

a

_ S2

x

b

. 9.4

v t , t0 t

T.

t k tk tk 1

v( pk ) tk ,

n

pk [tk 1 ;tk ].

v( pk ) tk

k

1