Hesaban f1[Www.riazisara.ir]
-
Upload
pardise4000 -
Category
Documents
-
view
217 -
download
0
description
Transcript of Hesaban f1[Www.riazisara.ir]
-
www.riazisara.ir
-
2
: ::::
1 : 2 3 ... n+ + + ++ + + ++ + + ++ + + +
:
1 2 3 12 1 2 1 1 2 1
1 1 2
A ... n n(n )A ( n) ( (n )) ... (n ) A n( n) A
A n (n ) ...
++++= + + + += + + + += + + + += + + + + ++++ = + + + + + + = + = = + + + + + + = + = = + + + + + + = + = = + + + + + + = + =
= + + += + + += + + += + + +
100n ! ====
:
(((( )))) (((( ))))
2 1 1 2 3 1n
n
S a (a d) (a d) ... a (n )d na ... (n ) d= + + + + + + + = + + + + + = + + + + + + + = + + + + + = + + + + + + + = + + + + + = + + + + + + + = + + + + +
: 1
1 2 12
(n )(n)... (n )
+ + + =+ + + =+ + + =+ + + =
1 2 1 2 1 1
2 2 2 2 2
nn
n(a a )(n )n na n(n )d n( a (n )d) n(a a (n )d)S na ( )d
++++ + + + + + + + + + + + + + + + + = + = = = = = + = = = = = + = = = = = + = = = =
: 2
2 1
2 2
nn
n(a a )n( a (n )d)S
+++++ + + + = == == == =
1: 2 3n nS a a a ... a= + + + += + + + += + + + += + + + 1n : + n na S S = = = =
1 1 2 11
1 2
n nn n n
n n
S a a ... aS S a
S a a ... a
= + + += + + += + + += + + + = = = == + + += + + += + + += + + +
7: 16 20a a+ =+ =+ =+ . 22 =
:
7 16 6 15 2 21 20a a a d a d a d+ = + + + = + =+ = + + + = + =+ = + + + = + =+ = + + + = + = 20
2222 2 22 1
11 2 21 2202
[ a ( )d]S [ a d]
+ + + + = = + == = + == = + == = + =
12 : 10, , ...
:
12 2 1 20 24 2 1 1 12 13
2n
n[ (n ) ( )]S (n ) n n
+ + + + = = = = == = = = == = = = == = = = =
7. 30 5: 12a 5a ====
:
55 2 4
30 12 2 4 2 62
[ a d]S a d a d
++++= = = + + == = = + + == = = + + == = = + + =
75
6 12 34 6 3 4 9
22 6
a a dd d a a a d
a d
= + == + == + == + = = = = = + = = = = = + = = = = = + = = = = = + =
+ =+ =+ =+ =
www.riazisara.ir
-
3
nnSn n : 57 2 ====++++
:
: t
(((( ))))141226
26122547
1257
12
122
11
============
====++++========
====++++========
ttd
SSt
St
2 n: 2nS n n= += += += . +
:
:
(((( ))))2 2 2 21 2 1 2 1 2 2 2 1 2 2 1n n na S S (n n) (n ) (n ) (n n) (n n n ) n= = + + = + + + = += = + + = + + + = += = + + = + + + = += = + + = + + + = + 7 8 9a a a+ + =+ + =+ + =+ + =
2 7 1 2 8 1 2 9 1 51( ) ( ) ( ) + + + + + = + + + + + = + + + + + = + + + + + = :
2 29 6 7 8 9 9 2 9 6 2 6 51S S a a a ( ) = + + = + + = = + + = + + = = + + = + + = = + + = + + =
1 3 : 2
10 .
:
5
10
5
3 110 2 5 9
1 9 5525 10
2 2 2 2
4 3 5
a[ ]
d S [ ]
a a d a
==== + + + +
= = = == = = == = = == = = == + = == + = == + = == + = =
8. 4 :
:
a a
a d (a d) a d a d
d d a a
a a d
+ = + + = + =+ = + + = + =+ = + + = + =+ = + + = + =
= = + = = = = + = = = = + = = = = + = = = + == + == + == + =
2 8
7
7 2 8 0 4 0
2 4 2 12 4 8
6 4
88 2 8 7 2
82
[ ( ) ]S
+ + + + = = = = = = = =
101 3 :
:
( )a ( k ) k k ...
+ + + +< = + < < + + + + = = =< = + < < + + + + = = =< = + < < + + + + = = =< = + < < + + + + = = =
17 3 990 3 2 1 101 0 2 1 33 0 16 3 9 99 17 51 867
2
www.riazisara.ir
-
4
5 5 53 17 :
:
: 53 17
a d d d+ = + = = =+ = + = = =+ = + = = =+ = + = = =36
6 53 17 6 53 66
. :47 41 35 29 23 :
1752
47235====
++++ )(
::::
nS a aq= += += += + aq++++2
aq++++ 3 n... aq + ++ ++ ++ + 1
nqS aq==== aq++++2 n
... aq+ ++ ++ ++ + 1
nn n
n nn
a( q )( q)S a aq a( q ) S
qaq
= = = = = = = = = = = =++++
11 1
1
1q . >>>>
2 1
1
n aa aq aq ... aq ...
q
+ + + + + =+ + + + + =+ + + + + =+ + + + + =
2 : 2 2 22 31 23 9
13 3
...+ + + = = =+ + + = = =+ + + = = =+ + + = = =
19 : 27
6
6 3 33 36
3 3 33
1
19 19 1 19 1 1 19 8 211
27 27 27 27 27 31 1 1
1
a( q )
S q ( q )( q )qq q q
S a( q ) q q
q
+ + + +
= = = = + = = == = = = + = = == = = = + = = == = = = + = = =
12: 2
, , 5 5 ...
:
5 56
1 12
4 512a aq ( )= = == = == = == = =
5
1 5
1 5
5 6 10
6 10
12 1
41
124
102411 1
1512512 4
11
4
( ( ) )
S
S
S( ( ) )
S
====
= = = = = = = =====
: 3127 . ,
:
1 3
: 27
q
aq q q q q
>>>>= = = = == = = = == = = = == = = = =
014 4 427 27 81 3 33
1 : 3
. 13 3 27 9 3 1
www.riazisara.ir
-
5
32 16 : 3
.
:
16a ==== 3 3
432 16 1
1 1 5 0 5 16 21 3 1 2
aS q / q / a aq ( )
q q= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
. . 1:
. .
99
:
. :
1 : 1
2S ====
2 : 1 1
2 4S = += += += +
3 : 1 1 1
2 4 8S = + += + += + += + +
: n
1 11
1 1 1 1 12 21
12 4 8 2 21
2
n
n nn n
( ( ) )
S ... ( ) S ( )
= + + + + = = = + + + + = = = + + + + = = = + + + + = =
99: 100
nS . = < > = <
.% 99 7
: 2 4
a aa , , , 1 ...
2.
:
21
12
aS a= == == == =
2 141 2 2
1 1 2 21 1
2 2
a
aq aS a S a= = = = = = == = = = = = == = = = = = == = = = = = =
www.riazisara.ir
-
6
. 1 :
:
1= 3
3
2 2=
3
4 3=
3 3 33 6
12 41
2
...+ + + = =+ + + = =+ + + = =+ + + = =
21 3
4 1 ====
21 3 1 3
2 4 4 4( )= = = = = = = = 2
21 3 1 3
4 4 16 4( )= = = = = = = = 3
3 3
34 41 3 3
14 4
S = = == = == = == = =
:
. 5 .
. nh n :
1
2
3
1
5
5
4
5
16
5
4n n
h
h
h
h
====
====
====
====
:
1
10 10 1010
4 16 4n, , , ... , , ...
.
1 4
n
: 10 40
11 31
4
aS
q= = == = == = == = =
www.riazisara.ir
-
7
1000 . :
50 . 9/0
. 10000
:
: nS n
1 1000S ====
29
1000 100010
S = + = + = + = +
nn
S ( ) ( )
S ( ) ( ) ( )
= + + = + + = + + = + +
= + + + + = + + + + = + + + + = + + + +
23
2 1
9 91000 1000 1000
10 10
9 9 91000 1000 1000 1000
10 10 10
:
50
50
91000 1
109948 46
91
10
( ( ) )
S /
= = = =
:
1000 1000091
110
aS
q= = == = == = == = =
: .21
21 21 21 211000 21
1100 10000 1000000 991
100
/ ...= + + + = == + + + = == + + + = == + + + = =
(
.
:
0 21 2199 21
99100 21 21
x /x x
x /
==== = = = = = = = =
====
5
38 5 34710000 385 0 38 0 005 0 0005 0 00005 0 38
1 100 900 9001
10
/ / / / / ... /= + + + + = + = + == + + + + = + = + == + + + + = + = + == + + + + = + = + =
(
:
x / x /x x
x /
= == == == = = = = = = = = =
====
0 385 100 38 5 347900 347
9001000 385 5
www.riazisara.ir
-
8
::::
:
.
:
1( 3 2 2
3 2
2
2
2 3 1 2 1
2 72 4 2
7 3 1
7 14 7
17 8
n n n n n
nn n n
n n
n n
n
+ + + + + + + + + + + +
+ + + +
+ + + +
2( 4 3 2 2
24 3 2
3 2
3 2
3 1 2 1
2
2 1
2
1
n n n n n
n nn n n
n n
n n n
n
+ + + +
+ + + +
+ + + + + + + +
: : : :
: (r(x (q(x (B(x (P(x
P(x) B(x)P(x) B(x).q(x) r(x)
q(x)
r(x)
= + = + = + = +
.
:
P(x) (x a) q(x) r= += += += + x a==== x a==== :
P(a) (a a) q(a) r r P(a)= + == + == + == + =
7 : 4 32 6 3 1P(x) x x x= + = + = + = + 2 4x . ++++
:
2 4 0 2x x+ = = + = = + = = + = = 7 4 32 2 2 6 2 3 2 1 256 96 24 1 377P( ) ( ) ( ) ( ) = + = = = + = = = + = = = + = =
. :
5) 5a b++++ a b++++ 5 50 0a b a b ( b) b+ = = + =+ = = + =+ = = + =+ = = + =
4) 4a b a b++++ 4 40 0a b a b ( b) b+ = = =+ = = =+ = = =+ = = =
www.riazisara.ir
-
9
7) 7a b a b 7 70 0a b a b a a = = = = = = = = = = = =
n :
0n na b (a b)q(x) = + = + = + = + 2 1 2 1 0n na b (a b)q(x)+ ++ ++ ++ ++ = + ++ = + ++ = + ++ = + +
:
1 2 11
1
1 2 2
2 2
1
1
n n
n n nn n
n n
n n
n n
n n
n n
a b a b
a a b ... ba a b
a b b
a b a b
a b b
ab b
ab b
+ + ++ + ++ + ++ + +
nn : ax ax ++++ : ++++ 1 (n 2 (n ) 4 ) 3 . 1 :
====++++====
========++++
++++====
na
na)a()x(R
axax
ax)x(P
nnnax
nn
2
0
nn n ax ax ++++ . ++++
3 : 22 1x x x + + + + 2 3 2x x + + + +
. :3 2 22 1 3 2 2 1P(x) x x x (x x )q(x) (ax b) (x )(x )q(x) (ax b)= + = + + + = + += + = + + + = + += + = + + + = + += + = + + + = + +
1 1 22 3
2 2 1 3
P( ) a b ar(x) x
P( ) a b b
= + = == + = == + = == + = = = = = =
= + = = = + = = = + = = = + = =
x : x x + + + + 4 3 22 x 3 x + + + +2 5 4
. :
.
(((( )))) (((( ))))
r(x) ax b
x x x x x ,
P( ) a b
P( ) a b
a b aa r(x) x
a b b
= += += += +
+ = = = + = = = + = = = + = = =
= + = + = = + = + = = + = + = = + = + =
= + = + == + = + == + = + == + = + =
+ = =+ = =+ = =+ = = = = = = = = = =
+ = = + = = + = = + = =
2 5 4 0 4 1 0 1 4
1 1 2 1 3 3
4 4 256 128 16 3 141
4 141 483 144 48 51
3 51
www.riazisara.ir
-
10
4 : 33 2x x x + + + + 2 1x . ++++
:
x r(x) ( ) x( ) x x= = + = = = + = = = + = = = + = 2 21 1 3 1 2 4 1 2 x 1x = = = = 2 : x 1x = = = = 4x 2x
.
.
6 : 33 1x x + + + 2 + 1x x+ ++ ++ ++ . +
:
2 2 31 0 1 1 0 1 0 1 1 1 3 1 1x x (x )(x x ) x x P( )+ + = + + = = = = + = + + = + + = = = = + = + + = + + = = = = + = + + = + + = = = = + = 3 1x 2 1x x+ ++ ++ ++ . +
:
x x x x
x x xx x x
x x x
x x x
x
x x x
x x
x x
+ + + + + ++ ++ ++ +
+ + + ++ ++ ++ ++ +
+ + + +
+ + + +
+ ++ ++ ++ +
+ ++ ++ ++ +
6 3 2
4 36 5 4
5 4 3
5 4 3
3
3 2
2
2
3 1 1
2 2
3 1
2 1
2 2 2
2 2 1
2 2 2
1
3 : 2 1ax bx x+ ++ ++ ++ 1x + 2x 4 b a ++++
:3 21 1 1 1 4a( ) b( ) ( ) a b+ + = + =+ + = + =+ + = + =+ + = + =
3 22 2 2 1 8 4 3 0a( ) b( ) ( ) a b + + = + + = + + = + + = + + = + + = + + = + + = 19 19 29
8 4 4 3 0 12 19 0 412 12 12
a ( a) a a b + + = + = = = = + + = + = = = = + + = + = = = = + + = + = = = =
)))) : ))))xf 2x 5(((( )))) (((( ))))123 ++++++++++++==== xfxxxP 1++++x :
(((( )))) (((( ))))(((( )))) (((( ))))
(((( )))) (((( ))))(((( ))))
3522111
1
152
1
23
====++++====++++============
++++++++++++====
====
f?PR
xfxxxP
f
1234 : ++++++++++++++++ xbxaxx 12 x . b
:
1234 ++++++++++++++++==== xbxaxx)x(P
1204201
03
============++++
====++++++++
====++++++++
abb
ba
ba
====
====
01
01
)(f
)(f)x)(x( 11 12 ++++ x
www.riazisara.ir
-
11
123 : 3 ++++++++==== axx)x(f 322 xx abx x++++2 x(f(
x====2
:
)abx()x(Q)xx(axx)x(f ++++====++++++++==== 32123 23
========++++====
====++++====++++++++====
201231
82733016813
baaba:x
ababa:x
138319 3 ++++======== xx)x(fa
531762422 ====++++++++========++++++++==== )(fRR)x(Q)x()x(f . 2 2 x
262 ==== )(Q53222176242 ++++++++====++++==== )(Q)(:x
123: 23 ++++++++==== xxx)x(f 32 2 ++++x
:
2
3032 22
========++++ xx
2x x(f( 2
: 3
2
4112
2
1113
2
91
2
32
2
33
++++====
====++++
====++++
++++
====
xxxxx)()(xR
)))) : )))) (((( )))) nnn xxaxxpINn 21222 ++++++++==== a x++++3 ++++
: (((( ))))xp 3++++x (((( ))))xp 3++++x (((( )))) 03 ====p :
(((( )))) (((( )))) (((( )))) (((( )))) (((( )))) (((( )))) (((( )))) (((( )))) (((( ))))
(((( ))))
(((( ))))27
2
3
2901
3
1903
013
193
03333303333
2
2
2122221222
============++++
====
++++
====++++++++====++++++++==== ++++
aaa
a
aap
n
n
nnnnnn
www.riazisara.ir
-
12
::::
:
(a b)
(a b) a b
(a b) a ab b
(a b) a a b ab b
(a b) a ab a b ab b
+ =+ =+ =+ =
+ = ++ = ++ = ++ = +
+ = + ++ = + ++ = + ++ = + +
+ = + + ++ = + + ++ = + + ++ = + + +
+ = + + + ++ = + + + ++ = + + + ++ = + + + +
1
2 2 2
3 3 2 2 3
4 4 3 2 2 3 4
1 1
1 1
2 1 2 1
3 3 1 3 3 1
4 6 4 1 4 6 4 1
.
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
:
1 2 2
1 2
n n n n nn n n(a b) a a b a b ... b
n
+ = + + + ++ = + + + ++ = + + + ++ = + + + +
n n!i (n i)! i!
====
:
n(a -1 b)++++ 1n . ++++
k)1 -2 n ++++( k k nn
a b (a b)k
++++
.
: : 52 3( x y)++++)
5 4 3 2 2 3 4 55 5 5 5 52 2 3 2 3 2 3 2 3 31 2 3 4 5
( x) ( x) ( y) ( x) ( y) ( x) ( y) ( x)( y) ( y)
+ + + + ++ + + + ++ + + + ++ + + + +
5 4 3 2 2 4 532 240 720 1080 810 243x x y x y x y xy y= + + + + += + + + + += + + + + += + + + + +
4 4 3 1 2 2 3 44 4 4 42 2 2 2 21 2 3 4
(x y) x (x) ( y) x ( y) x( y) ( y)
= + + + + = + + + + = + + + + = + + + +
(
4 3 2 2 3 48 24 32 16x x y x y xy y + + + + + + + +
)921: : )x
. ++++
:
( ) : ( ) ( )x x x x x
+ = = =+ = = =+ = = =+ = = =
3
9 5 4
4 4 4
92 2 9 8 7 6 16 16 20161 1 3 42
5 4 3 2
www.riazisara.ir
-
13
x)81 : x: )x
. ++++
:
8 8 28 81k
k kx x
k kx
====
8 x 2 0 4k k = = = = = = = : =
8 8 7 6 570
4 4 3 2
= == == == =
42: 3( x y)++++
:
( x y) ( x) ( x) ( y) ( x) ( y) ( x)( y) ( y) x x y x y xy y+ = + + + + = + + + ++ = + + + + = + + + ++ = + + + + = + + + ++ = + + + + = + + + +4 4 3 2 2 3 4 4 3 2 2 3 42 3 2 4 2 3 6 2 3 4 2 3 3 16 96 216 216 81 .
1x y= == == == 1 y x =
.42 3 625( )+ =+ =+ =+ : =
2: 3 362 3n( ) b+ = ++ = ++ = ++ = . n b +
2 3 362 3n( ) b = = = = . b
:
:
1 2 22 3 2 2 3 2 3 31 2
n n n n nn n( ) ( ) ... ( )
+ = + + + ++ = + + + ++ = + + + ++ = + + + +
n n n n n
A
n n n n( ...) ( ...)
= + + + + + += + + + + + += + + + + + += + + + + + +
2 4 2 1 32 2 3 2 3 2 3 2 32 4 1 3
1 1 2 22 3 2 2 3 2 31 2
n n n nn n( ) ( ) ( ) ...
= + + + = + + + = + + + = + + +
n n n n
A
n n n( ...) [ ...] b
= + + + + = = + + + + = = + + + + = = + + + + =
2 2 1 32 2 3 2 3 2 3 362 32 1 3
: :
2 2 2 2 2 2 24 3 362 3 3 362 1 361 363 361 121 19 11 19 11 209n( ) ( ) b b ( ) ( )( ) b b = = = = = = = = = = = = = = = = = = = = = = = = = = = =
www.riazisara.ir
-
14
::::
. .
.
. :
. .
1 .. . 7 5 3 2
.
. 480 :
:
5480 2 3 5= = = =
480 2
240 2
120 2
60 2
30 2
15 5
3 3
1
.. . . .
)))) b a.. ))))a,b . . : ..
2 2
2 1
2
36 2 336 60 2 3
60 5 3 2( , )
= = = = = = = =
= = = =
22 . 60 36 3
. .. ..
.
a,b b a.. .
..
: .
2 236 60 2 3 5[ , ] = = = = .. .: ..
36 225 405, ( ,
( , , ) , [ , , ]
= = = = = = = = = = = = = = = =
= = = = = = = =
2 2
2 2 2 4 2 2
4
36 2 3
225 5 3 36 225 405 3 36 225 405 3 5 2
405 81 5 3 5
www.riazisara.ir
-
15
140 15 35, ( ,
( , , ) , [ , , ]
= = = =
= = = = = = = = = = = = = = = =
2
2
140 7 2 5
15 5 3 140 15 35 5 140 15 35 5 3 7 2
35 7 5
156 236 78, ( ,
( , , ) , [ , , ]
= = = = = = = = = = = = = = = = = = = = = = = =
2 2
2 2
156 39 2 13 3 2
236 59 2 156 236 78 2 156 236 78 59 13 3 2
78 13 3 2
36 40 :
.
d 36 40 d :
d .
. 36 40
( , )
32
2 2
40 2 540 36 2 4
36 3 2
= = = = = = = = = = = =
= = = =
d . . : d , ,1 2 4====
4d. : . . d ====
.
48 40 72 :
d d 48 40 72 d : 40 72 . . d .
. 484 3 3 2 348 40 72 2 3 2 5 2 3 2 8d ( , , ) ( , , )= = = == = = == = = == = = =
6 5 9 8
4 . 20 .
.
24 18 :
. 32
.. . 32 24 18 : : ..
2 5 3 5 218 32 24 3 2 2 2 3 2 3[ , , ] [ , , ]= = = = = = = = 5 .. : 22 2 3
. 288..
xx
www.riazisara.ir
-
16
, : , , ...4 7 10 , , , ...1 5 9
:
:
4 7 10
4 3 1 3 1n
, , , ...
b (n ) n= + = += + = += + = += + = +
1 5 9
1 4 1 4 3 4 1n
, , , ...
a (n ) n n= + = = += + = = += + = = += + = = +
12. 1 3 4 nb na
:. 1
,c qc k c k
c q
==== = = = = = = + = = + = = + = = + = = = =
3 4 121 41 12 12 1
1 3
k k / ... k / ... k + + + + 100 12 1 999 99 12 998 8 83 9 83 . ( 75 + = + = + = + =83 9 1 75(
:::: .. .. .. .. .. .. .. ..
(q(x (p(x
q(x),p(x) .
(q(x ( ) ,p(x)
. (q(x (p(x
. ..
..
.
:
p(x) x (x )(x x ) (x , x ) (x ) , [x ,x ] (x )(x )(x x )q(x) x (x )(x )
= = + += = + += = + += = + + = = + + + = = + + + = = + + + = = + + +
= = += = += = += = +
3 23 2 3 2 2
2
1 1 11 1 1 1 1 1 1 1
1 1 1
. : ... .. 3 22 2x x x+ + + + 3 24 5 2x x x + + + + (
(x )(x x ) (x )(x x )
(x )(x )(x ) (x )(x )(x )
(x )(x )(x )(x ) (x )
+ + + + + + + + + + + +
+ + + + + + + +
+ + + + + + + +
2 2
2
1 3 2 1 3 2
1 2 1 1 2 1
1 2 11 2
(((( ))))(x ) (x ),(x )(x )(x ) (x )[(x ) (x ),(x )(x )(x )] (x ) (x )(x )
+ + = + + = + + = + + =
+ + = + + + = + + + = + + + = +
2
2 2 2
1 2 1 2 1 1
1 2 1 2 1 1 4 1
3 1x 6 ++++ 1x 24 4x (
(x )(x ) (x )(x )(x x ) (x )(x x ) + + + + + + + + + + + + + + + + + + + + + + + + + + + +4 2 24 1 1 1 1 1 1 1
(((( ))))4 2 24 1 1 1 1 1 1 1 1(x )(x ),(x )(x )(x x ),(x )(x x ) (x ) + + + + + + + = + + + + + + + + = + + + + + + + + = + + + + + + + + = + [ (x )(x ),(x )(x ) ,x ] (x )(x ) (x ) + + + = + = + + + = + = + + + = + = + + + = + = 3 3 3 3 3 64 1 1 1 1 1 4 1 1 4 1
www.riazisara.ir
-
17
. :
x x x x
x x x x
+ + + + + + + +
+ + + +
2 2
2 3 2
5 4 2
4 2 (
x
(x )(x ) (x )(x ) x xxx(x ) x (x )
x
+ + + +
= = = = = = = = ++++22
14 1 2 1
14 2
x
x x x x
+++++ + + +
+ + + + + + + + 21 1 8
1 3 2 3 (
x (x )(x ) (x ) (x x ) (x )(x ) x
x x (x )(x ) (x )(x ) (x )(x ) (x )(x ) x
+ + + + + + ++ + + + + + ++ + + + + + ++ + + + + + += + = = = == + = = = == + = = = == + = = = =
+ + + + + + + + + + + + + + + + + + + + + + + +
21 1 8 1 3 1 8 5 6 6 1 6
1 3 3 1 3 1 3 1 3 1 3
www.riazisara.ir
-
18
::::
f (x) g(x)==== x0 . f (x) ====
2 . 5 4 0x x + = + = + = + =
:
4 1 0 1 4(x )(x ) x , = = = = = = = =
: : : :
0ax : b+ =+ =+ =+ bx : =a
= = = =
2 : 3 0x + =+ =+ =+ 3 : =2
x
====
y :::::::: ax b= += += += +
::::
ax b+ >+ >+ >+ > bx : 0a
> > > >
30 :
. 400 20
:
: x
20 30f (x) x= += += += + : 400
x x x /+ =+ =+ =+ =38
20 30 400 30 380 12 63
. 12 ( )
. 3200000 :
. 30000 20000
: x :
30000 20000 10000P(x) x x x= == == == =
3200000
:
10000 3200000 320x x> >> >> >> > . 321
++++ bxa
= = = = x
y a a
www.riazisara.ir
-
19
: : : :
2 0ax bx c+ + =+ + =+ + =+ + 0a = .
:
. .
:
: 2 -1
20 :
:
: x
2
271 1 20 1 20 7 19 12 12 19 123 4 12 12
xx x x x( )( ) x x (x )(x ) x
>>>>+ + = + + = + = + = =+ + = + + = + = + = =+ + = + + = + = + = =+ + = + + = + = + = =
4. 10:
. 22 39. .
.
:
)10x : . x x)++++
40 4
210 . : 40 10 40x( x) x x+ = + + = + + = + + = +
: 39 22 x 0
2 210 40 39 22 29 62 31 2 0 31x
x x x( ) x x (x )(x ) x>>>>
+ = + = + = =+ = + = + = =+ = + = + = =+ = + = + = = . 1231 1271 41 31
: -22 2 2
2 2 2
4 4 2 4
a a a ax ax x ax (x )+ = + + = + + = + + = + + = + + = + + = + + = +
2 : 3 4 0x x+ =+ =+ =+ =
:
x x
x x (x ) (x ) x
x x
+ = =+ = =+ = =+ = =+ = + = + = + = + = + = + = + = + = + = + = + = + = + = + = + = + = = + = = + = = + = =
2 2 2
3 51
3 9 3 25 3 5 2 23 4 0 4 0
3 52 4 2 4 2 24
2 2
2 m : 1x x m + = + = + = + x =
:
( x ) m x m x m = = = = = = = = = = = = 21 2 1 2 1 2 2 m > > > 2m < : >>>>
m m m m m
-
20
2 - 3 4( b ac) : = = = = :
2 22 2 2
2 2
40
2 4 4
b c b b c b acax bx c a(x x ) (x )
a a a aa a
+ + = + + = + = =+ + = + + = + = =+ + = + + = + = =+ + = + + = + = =
2 2
1 22
4 4
2 2 2 2 2 24,
b b ac b ac b bx x
b a a a a aa
+ = = = = = + = = = = = + = = = = = + = = = = =
2 : 3 4 0x x+ =+ =+ =+ =
:
, ,x( )( )
x x x xx
= = = = + = = = + = = = + = = = + = = = ====
212
1 2 1 22
43 3 4 4 1 3 253 4 0
12 2
:P S -4
1 : 21 2
S x x
P x .x
= += += += +
==== :
1 22 2
b b bS x x
a a a
+ + + + = + = + = = + = + = = + = + = = + = + =
2 2 2
1 2 2 2
4
2 2 4 4
b b b b (b ac) cP x x ( )( )
a a aa a
+ + + + = = = = == = = = == = = = == = = = =
1 2x xa
= = = =
a)0 : )
2 20 0b cx x x Sx Pa a
+ + = + =+ + = + =+ + = + =+ + = + =
2 : P S 0x Sx P + = + = + = + =
2 . 1x ,x 2 0ax bx c+ + =+ + =+ + =+ + : =
1 2 0(x x )(x x ) = = = =
2 1 2 1 2 0x (x x )x x x + + = + + = + + = + + 2 = 0ax bx c+ + =+ + =+ + =+ + : . =
1 2S x x= += += += 1 + 2P x x==== :2 0x Sx P + = + = + = + =
2 0ax bx c+ + =+ + =+ + =+ + b = cx xa a
+ + =+ + =+ + =+ + =2 . . 0
2f (x) ax bx c= + += + += + += + 2 + b cg(x) x xa a
= + += + += + += + 0f . + (x) (0g(x ==== . ====
:
2 23 2 32 2
xy x ,y x x= + = + = + = + = + = + = + = +
.
. P S
.
. 2
. P S
y x x= + = + = + = + 22 3
xy x= + = + = + = + 2
3
2 2
www.riazisara.ir
-
21
1: 3
4 3
:
1 21
3S x x= + = = + = = + = = + =
1 24
3P x x= = = = = = = =
2 2 1 40 03 3
x Sx P x x + = + = + = + = + = + = + = + =
12
2
4
3 4 0 3 4 1 0 3
1
xx x ( x )(x )
x
= = = = + = + = + = + = + = + = + = + =
====
28 22:
:
1 2 1 2 12 21 2 2
2 22 11 411 28 0 7 4 0
28 7
(x x ) x x S xx Sx P x x (x )(x )
x x P x
+ = + = = =+ = + = = =+ = + = = =+ = + = = = + = + = = + = + = = + = + = = + = + = = = = == = == = == = =
. : 7 4
2 27 4 65d = + == + == + == + = 1 : 3 1 3,+ + + + .
:
1 3 1 3 2S = + + = += + + = += + + = += + + = +
1 3 1 3 1 3 2P ( )( )= + = = = + = = = + = = = + = = 2 20 2 2 0x Sx P x x + = = + = = + = = + = =
. P S
: 2 2 2 21 2 1 2 1 22 2x x (x x ) x x S P+ = + = + = + = + = + = + = + = (
3 3 3 31 2 1 2 1 2 1 23 3x x (x x ) x x (x x ) S PS+ = + + = + = + + = + = + + = + = + + = (
2 2 21 2
2 2 2 2 21 2 1 2
1 1 2x x S P
x x x x P
++++ + = =+ = =+ = =+ = ( =
21 2 1 2 1 22 2 2x x A A x x x x S P A S P = = + = = = = + = = = = + = = = = + = = (
x x x x S
x x x x P
+++++ = =+ = =+ = =+ = =1 2 1 2
2 1 1 2
(
22 : 5 1 0x x + = + = + = + =2 2
2 2 ++++
) (
:
3
3 3 3 3
2 2 2 2 2 2 2 2
5 5 12 3
2 2 2 2 2 3 2 3 2 2 295
1
2
[( ) ( )( )][( ) ( )] (S SP)
P ( )
+ + + + + + + + + + + + + = = = = =+ = = = = =+ = = = = =+ = = = = =
d4
7
www.riazisara.ir
-
22
2 : 1 0x x = = = 2. = 22 + + + + (
)
:
:
1 1b c
S , Pa a
= + = = + = = = = + = = + = = = = + = = + = = = = + = = + = = =
2 2
2 2 2 2 2 2 2
1 0 1
2 2 1 1 2 1 1 4(S) P ( ) ( )
= = = = = = = =
+ = + + = + = + = + = + + = + = + = + = + + = + = + = + = + + = + = + =
2 : 3 1 0x x + = + = + = + 2 = 3 1( ) (
)
:
2 : 23 1 0 3 1 + = = + = = + = = + = =
( ) P = = = = = = = = = = = = = = = =2 2 23 1 1
2 : 5 1 0x x+ + =+ + =+ + =+ + 1 = 2 1 2x x x x + + + 1 + 2x x>>>>
:
1 21 2 1 2
1 2
25 421
21 11
1
x xx x x .xa
x .x
= = = = = = = = = = = =
+ = + + = + + = + + = +====
2: 1x ,x 2 2 1 2 1 0x (m )x m+ + + =+ + + =+ + + =+ + + 2x m x1 m =
:
m x .x P m m m (m ) m= = = = = == = = = = == = = = = == = = = = =2 2 21 2 2 1 2 1 1 0 1
2 2x 1x: 4 1 0x x + = + = + = + 1 = 2x x
:A
A x x A x x x x S P A S P>>>>= = + = = = == = + = = = == = + = = = == = + = = = =
021 2 1 2 1 22 2 2 4 2 1 2
2 2x 1x: 1 0x x+ =+ =+ =+ 2 = 1x x>>>> 2 21 25 3x x++++
:2 2 2 2 2 2 2 2 2 21 2 1 2 1 2 1 2 1 25 3 4 4 4x x x x x x (x x ) (x x )+ = + + = + + + = + + = + + + = + + = + + + = + + = + +
(x x )(x x )
(S P) ( ) (S) ( ) ( )( )a
++++
= + = + + = += + = + + = += + = + + = += + = + + = +
1 21 2
2 54 2 4 1 2 1 12 51
www.riazisara.ir
-
23
20 ::::2222 0(a )ax bx c + + = + + = + + = + + =
a)0 2 ) 1 b c )+ + =+ + =+ + =+ + c 1 =a
.
a c b+ =+ =+ =+ c -1 =a
.
. : 2 2 0x x+ =+ =+ =+ ( =
11 2 02
x(x )(x )
x
==== + = + = + = + =
= = = =
23 4 0x x+ =+ =+ =+ ( =
1
1 3 4 0 4
3
x
(x )( x )x
====
+ = + = + = + = = = = =
23 8 5 0x x+ + =+ + =+ + =+ + ( =
1
1 3 5 0 5
3
x
(x )( x )x
= = = =
+ + =+ + =+ + =+ + = = = = =
2 4 3 0x x+ + =+ + =+ + =+ + ( =
11 3 03
x(x )(x )
x
= = = = + + =+ + =+ + =+ + =
= = = =
20 ) 2 0a , ax bx c + + = + + = + + = + + : =
0
0
0
0
0 0 0
0
0 0
0
c
b
c a
ba
a
c bx , x
a a
b
a
c b
a a
b
a
====
>>>> > > > > = = = >>>> <
< < < <
= = = =
0 = = = =
0 . < < < <
a)0 a)0 > > > . >>>
-
24
2 : 2 0x mx m + = + = + = + m =
:
b . a
c
a
> > > >
> > > >
> > > >
0 .
0 .
0 .
m
m m
mS m m m(m ) m m
P m
>>>>
= > = > = > = >
= > > > > > >= > > > > > >= > > > > > >= > > > > > >
= >= >= >= >
2
0
4 4 0
20 2 0 0 4 1 0 1 0 1
1
0
22 : 3 5 0x x = = = =
:
5. 02
c
a= < = < = < = <
3 . . 02
b
a
= > = > = > = >
2 m: 21 9 2 0(m )x m(m )x+ + =+ + =+ + =+ + =
:
2 090 0 3
13
mm(m )
S mm
m
====
= = = == = = == = = == = = =++++ = = = =
2 04 0 4 1 2 0 1 0 13
mb ac (m )( ) m m
m
==== = = + > + > > = = + > + > > = = + > + > > = = + > + > >
====
)3
.
:
P) , S 2 : P S P S 0X S X P + = + = + = + =
X x )
. X x .
2 : 0ax bx c+ + =+ + =+ + =+ + : =
: ) 2 1x ,x 2 . 2X x= = = = 1 1X x= = = = .
: b b
S X X x x (x x ) S ( )a a
= + = = + = = == + = = + = = == + = = + = = == + = = + = = =1 2 1 2 1 2
cP X X ( x )( x ) x x P
a = = = = == = = = == = = = == = = = =1 2 1 2 1 2
:
2 2 20 0 0b c
x S x P x x ax bx ca a
+ = + = + = + = + = + = + = + = + = + = + = + =
www.riazisara.ir
-
25
: . x :
X x x X= = = = = = = = :
2 20 0a( X) b( X) c aX bX c + + = + = + + = + = + + = + = + + = + = : )
: :
1 21 2
1 1X X
x x= == == == =
1 21 2
1 2 1 2
1 1b
x x S baS X Xcx x x x P c
a
++++ = + = + = = = == + = + = = = == + = + = = = == + = + = = = =
1 21 2
1 1 1 1 aP X X .
cx x P c
a
= = = = == = = = == = = = == = = = =
2 2 2 20 0 0 0b a b a
x S x P x ( )x x x cx bx ac c c c
+ = + = + + = + + = + = + = + + = + + = + = + = + + = + + = + = + = + + = + + =
:
1 1X x
x X= == == == =
2 21 1 0 0a( ) b( ) c cX bX aX X
+ + = + + =+ + = + + =+ + = + + =+ + = + + =
11 0n nn nP : a x a x a+ + + =+ + + =+ + + =+ + + =1
1 0n n
nP : a x a x a + + + =+ + + =+ + + =+ + + =
. P P .
11 1
1 1 1 10 0
nXn n n
n n n nX x a ( ) a ( ) a a a X a Xx X X X
= = + + + = + + + == = + + + = + + + == = + + + = + + + == = + + + = + + + =
.
22 : 2 0(m )x x m + = + = + = + 23 = 2 1(m )x ( m)x+ + =+ + =+ + =+ + M . =
:
:
2
2
1 12 2 0
2 2 0
(m )( ) ( ) mx x
mx x (m )
+ = + = + = + =
+ + = + + = + + = + + =
. (
.)
mm m( m) m
mm m
==== = = = = = = = = = = = = = = = =
====+ + + +
2 12 1 2 2 1 2 133 2 1
2 0ax bx c+ + =+ + =+ + =+ + 2 = 0a x b x c + + =+ + =+ + =+ + : =a b c
a b c= == == == =
2 0ax bx c+ + =+ + =+ + =+ + 2 = 0kax kbx kc+ + =+ + =+ + =+ + =
( ).
-
26
6 : 310 1 0x x + = + = + = + =
:
: P, S 33
X x x X= == == == = 3 36 3 210 1 0 10 1 0( X) ( X) X X + = + = + = + = + = + = + = + =
2 : 2 3 6 0x ( )x + + = + + = + + = + + =
:2
x X x X= = = = = = = = 2 2 3 6 0( X) ( )( X) + + = + + = + + = + + =
2 3 6 0X ( )( X) + + = + + = + + = + + = 2 2
2 3 6 2 3 2 6 6( ) X X ( )X (X ) + = + + + = + + = + + + = + + = + + + = + + = + + + = +
2 25 2 6 6 2 6 5 6 0( )X (X X) X X + = + + + = + = + + + = + = + + + = + = + + + =
2 : 3 5 0x x+ =+ =+ =+ . =
.
:
:
1 : 12 2
2
2
x x
x x
= += += += + = += += += +
1 2 1 22 2 4 4 3 4 1S (x ) (x ) x x S = + + + = + + = + = + == + + + = + + = + = + == + + + = + + = + = + == + + + = + + = + = + =
1 2 1 2 1 22 2 2 4 5 2 3 4 7P (x )(x ) x x (x x ) ( ) = + + = + + + = + + = = + + = + + + = + + = = + + = + + + = + + = = + + = + + + = + + = :
2 20 7 0x S x P x x + = = + = = + = = + = = :
2 2 2
2 2
2 3 2 5 0 4 4 3 6 5 0 7 0
X x x X
(X ) (X ) X X X X X
= + = = + = = + = = + =
+ = + + = = + = + + = = + = + + = = + = + + = =
2 : 0ax bx c+ + =+ + =+ + =+ + = k a b c
:
1 2x kx====
2x . 1 2 2 2 2
21 2 2
1S x x kx x (k )x
P x x kx
= + = + = += + = + = += + = + = += + = + = +
= == == == =
22 2 2 2 2 2 22
22
1 1 1b
( )S (k ) x S (k ) b (k )acP k ac kP kx ( )a
= += += += + + ++ ++ ++ + = = = = = = = = = = = =====
www.riazisara.ir
-
27
-4 + + + + .
:
2S = + + = = + + = = + + = = + + = 2P ( )( )= + = = + = = + = = + =
.
1++++ : . 5
:
1 . 5
S ( ) ( )= + + == + + == + + == + + =1 5 1 5 2
P ( )( ) x Sx P x x= + = = + = == + = = + = == + = = + = == + = = + = =2 21 5 1 5 1 5 4 0 2 4 0 0x -5 = = = . =
.
. 2
2 : . ++++3
:
:
2 3x = += += += + 22 2 2 2 4 2
2 3 2 6 5 2 6 5 4 6 10 25 24x x (x ) (x x )= + + = = + == + + = = + == + + = = + == + + = = + =
4 210 1 0x x + = + = + = + = .
: 4:
2 2 2 21 3 1 3 1 2 3 2 2x (x ) ( ) x x x x= = + = == = + = == = + = == = + = = x . 2
.
-
28
::::::::
2y ax bx c= + += + += + += + +
2y ax bx c= + += + += + += + : +
a >
a >>>>
> > > >
= = = =
= = = =
-
29
21 a : 2 2f (x) (a )x x a= + += + += + += + x +
:
x > ) 2x ( + > = < + < > + > = < + < > + > = < + < > + > a a
aa a
> < > < > < > < > > > >
> > > > > > > >
2 12
1 1
21 : 2 4f (x) (m )x mx= += += += m +
a : ,> > > > 0 : 0
m m > > > > > > > >1 0 1
m (m ) m (m ) (m ) = = = = 2 2 24 16 1 0 4 1 0 2 0
m = . 2
-
30
::::::::
:
: : 2 5 6x x + + + + (
3 2 2 3(x )(x ) x 1
1 xx
+ < > >< > >< > >
. x
1 5 1 5
2 2( , ) ( , )
+ + + += = = =
21 2 2 1( x)(x ) x + > + > + > + > (
x x x x x x + + + +
+ > + < < + < < + < < + < < >>> > > > > > > > >
-
31
::::::::
2y ax bx c= + += + += + += + : +
a >
a >>>> > > > >
= = = =
= = = =
-
32
:
2 2
2 2 4 4 4
amax
b ba(x ) a(x ) y
a a a a a
> + = + + = + = + + = + = + + = + = + + =
60 :
:
60 : x 2x
2 2
2
6060 2 60 2 2 60 15
2 2
2 15 60 15 450max
S x( x) x x x x x( )
S
= = = + = == = = + = == = = + = == = = + = =
= + = = + = = + = = + =
. :
:
x)2 y) P+ = =+ = =+ = =+ = =
2 42 2 22 2 2 1 4 2 4
Px
max
P
P P P P PS xy x( x) x x x x y P x y y x y
( )
===== = = = = + = + = = == = = = = + = + = = == = = = = + = + = = == = = = = + = + = = =
2f: (x) ax bx c= + += + += + += + . c b a +
() () ()
2f (x) ax bx c= + += + += + += + + f ( ) c==== .
2
4
b
aS
a
.
:
2
4
b
aS b
a
= = == = == = == = =
1 12 1 1 1( , ) f
f ( ) c y ax a a= = = = == = = = == = = = == = = = =
)
4 4 82 2
2 2 84 4
b bb a
a a
aa a
= = = = = = = = = = = =
= = = = = = = = = = = =
x
y
x
1
1
3
2
4
2
2 4
www.riazisara.ir
-
33
3f: ( ) 3c : ==== ====
.ab a
b a( ) a ( a) a a a a( a )ab a
==== ==== = = = = = = = = = = = = = = = = = = = = ========
22 2 2204 3 8 8 20 16 5 16 5 5
816
a ==== a . 0 ==== 516
25 5 5 58 316 2 16 2
b y x x= = = + += = = + += = = + += = = + +
f) : ( ) c ==== ====
a 2 >>>> 2 4f ( ) , f ( ) = = = = = = = . =2
2
2 2 2 4 2 2 2 1 1 16 1 2 1
6 64 4 4 4
a( ) b( ) c a b a ba a ( ) b
f ( ) a( ) b( ) a b
+ + = = = + + = = = + + = = = + + = = = = = = = = = = = = = = = = + = + == + = + == + = + == + = + =
21 2 1 213 3 6 3
b y x x = + = = + = + = = + = + = = + = + = = +
2y : xx
= += += += . x +
:2
2 22 2 2x
y x xy x (y)xx
++++= + = + == + = + == + = + == + = + =
. y . ) y (
. y x
. x
2 24 2 8
4 4
( y) y ( ) y yx
= == == == =
2 x 8y . x
2 8y 2x x . . x x
. x ++++
y x y y 2. 8y !
2 2
2 2
y
y
.
x 2x + 2 < 2y .
.
: : :
12
12
x : xx
x : xx
> + > + > + > +
< + < + < + < +
2 2
2 2
-
34
:
2
2
ax : x a
x
ax : x a
x
> + > + > + > +
< + < + < + < +
: 2 22
2a x a ax (x a )
x : x ax x x
+ + + + > + > + > + > +
.
:
22 2x
x+ + + +
www.riazisara.ir
-
35
::::::::
4 2ax bx c+ + =+ + =+ + =+ + =
4 2ax bx c+ + =+ + =+ + =+ + = . 2x X==== 2
aX bX c+ + =+ + =+ + =+ + . =
:
2 2 2 4aX bX c X x b ac+ + = = = + + = = = + + = = = + + = = =
4
2
3
b
c a
ba
a
c:
a
b
c b a
ba a
a
> > > >
> > > > < < < <
> < > < > < > <
> > > > = = = =
1
< < < <
2
b
a:
b
a
> > > > = = = =
< < < <
: . + + > > + + > > + + >
2 4 4 4 20 0m m m+ + > + + > + + > + + >
(*)mm mm
>>>> > > > > > > > >
< < < <
2 416 0 44
)I (50 0 51
mP m
++++> > > > > > > > > > > >
)II (20 0 21
(m )S m
+ + + +> > > > > > > > > > > >
(**)2(I) (II) m > > > >
4(*) (**) m > > > > .
x X x= = = = = = = = = = = = 21 1 1
x X x= = = = = = = = = = = = 22 2 2 . 4
-
36
2 : 4 2 21 1 2(x ) (x ) + = + = + = + . =
2 : 1X x= = = = :
XX X (X )(X )
X
+ = + = + = + = + = + = + = + = + = + = + = + = = = = =
24 2 2 2
2
2 2 2 1
1
x xX x
x x
= = = = = = = = = = = = = = = = = == == == =
22
2
2 21 1 1 3
: .4 23 2x x + = + = + = + ( =
22 2
2
2 23 2 2 1
1 1
x xx X X X (X )(X )
x x
= = = = = = = = = + = = = + = = = + = = = + = = = = = = = = = =
. 4 2 2
22 7 2 63 3
x x( ) ( ) + = + = + = + ( =
xX x x
xX X X (X )(X )
xX x x
= = = = = = = = = = = = = = = =
= + = = = + = = = + = = = + = =
= = = = = = = = = = = = = = = =
22
22
22
6 2 6 24 2 63
2 7 6 6 13
1 2 1 9 33
. 4 2 2 24 2 4 15( x ) ( x ) = = = ( =
2 22 2
2 2
5 4 5 14 2 15 5 3
3 4 3 7 7
X x xx X X X (X )(X )
X x x x
= = = = = = = = = = = = = = + = = = + = = = + = = = + = = = = = = = = = = = = = = = = =
. 2
www.riazisara.ir
-
37
::::::::
.
.
.. ( ..
. .)
: . 1 1 1
15 18x x+ =+ =+ =+ =
++++ (
2 2
2
15 1 2 15 115 15 36 270 21 270 9 30
15 18 1815
x x xx , : x x x x x x ,
x(x ) x x
+ + ++ + ++ + ++ + + = = + = + = = = = + = + = = = = + = + = = = = + = + = =
++++ ++++
. .
2
2 3 5 2 3
2 2 2 21
x x
x xx
+ + + + = = = =
+ + + + (
2 2
2
2 3 5 2 3 1 5 2 2 3 2 5 3 10 2 5 7 2 3
2 2 2 1 1 2 2 2 1 1 2 1 1 2 11
x ( x )(x ) ( ) x x x x x x
x (x )(x ) x (x )(x ) (x )(x ) (x )x
+ + + + + + + + + + + + + + + + + + + + + + + + = = = = = = = = = = = = = = = =
+ + + + + + + + + + + + + + + + + + + +
1 11 2 7 2 7 2 32 7 2 3 7 3
2 1 1 2 1 2 1
x ,(x )( x ) x xx x
(x )(x ) (x ) (x )
+ + + + + + + + = = + = = = = + = = = = + = = = = + = = + + + + + + + + + + + +
.
2
3 2
2 3
x
x x x
++++====
(
32 2 23 3 9 2 4 13 13 13 13x ,
x , : x x x x x x x(x ) x , x = + = = = = = + = = = = = + = = = = = + = = = =
2
2 3 12
3 3 9x x x = = = =
+ + + + (
2
2 3 3 3 15 123 3 15 12 3
3 3 3 3 9
(x ) (x ) xx , : x x
(x )(x ) (x )(x ) x
+ + + + = = = = = = = = = = = = = = = =
+ + + + + + + +
. 3
13
x
x
++++====
++++ (
3x }3 : . }
. 5 :
. 4 200
:
5 :
:
4200 8
100kg = = = 200: =
8 : ( x . 5
)
8 5 8 1 4020 160 200
200 100 200 20 19
x xx x x kg
x x
+ ++ ++ ++ += = + = + == = + = + == = + = + == = + = + =
+ ++ ++ ++ +
-
38
: 5 y :
y y yy
= = = == = = == = = == = = =
8 51000 5 800 5 200 40
200 100
. 40
12000 :
. 4 100
:
: y x
12000xy ====
12000
120004 100 12000 4 100 400 12000 4 100 4 4 100 4(x )(y ) xy y x y (x ) y ( )
y+ = + = = + = ++ = + = = + = ++ = + = = + = ++ = + = = + = +
y y y y ( y ) y y= + = = = == + = = = == + = = = == + = = = =2 2 24 1200000 400 100 300000 50 302500 50 550 600
www.riazisara.ir
-
39
::::::::
. .
2 : 1y x= += += += 3 +1
A 11
B
.
xM :y
B A .
. 2 2 2 2 2 2 2 21 1 3 1 1 1 3 1MA MB (x ) (y ) (x ) (y ) (x ) (y ) (x ) (y )= + + + = + + + + = + = + + + = + + + + = + = + + + = + + + + = + = + + + = + + + + = +
2x 2y++++ 22 2 2x y x+ + + =+ + + =+ + + =+ + + = 2y++++
2 16 2 10 8 4 8 1 1
2 2
y xx y x y x x
++++ + + = + = + = + + = + = + = + + = + = + = + + = + = + =
1
1 1 41 2
32 2
2
x
x x x
y
==== + + = = + + = = + + = = + + = =
====
AB :
.
1 3
1 1 1
2 2
A BO
++++++++
= = == = == = == = =
2 12 2 1 2 2
4 2ABm m y (x ) y x= = = = = += = = = = += = = = = += = = = = +
2 2 3 12 3
2 1 2 4
y xy y , x
y x
= += += += + = = = = = = = = = = = =
= += += += +
: :
15 2 80 5x+ + =+ + =+ + =+ + (=
15 2 80 25 2 80 10 2 80 100 10x x x x+ + = + = + = =+ + = + = + = =+ + = + = + = =+ + = + = + = =
2 3 4x x= += += += (+
4 3 4 4x x x= + == + == + == + =
x x = = = = (21
2 2 2 2 1 11 2 12 2
x x x x x = = = = = = = = = = = = = = = =
x 1
-
40
: :
1 2
2
11 1 1 1 1
21
x x xxx ( x ) x
x( x )
= == == == = = + = + = = + = + = = + = + = = + = + = = = = = ++++
1x . 2 ====
:
x x
( x)( ) xx( x )
x( x )
= = = = = = = = = = = = ==== = + = = + = = + = = + = ++++
= = = = ++++
22
1 11
1 1 11 1 11
21
1 2 3x x + + = + + = + + = + + (=
:
1 2 3 3x x + + + + + + + + . .
x x + = + = + = + =2 (1
: 2
2 5 254 1 4 1 5 4 1 4 1 5 1 14 16
( ( x) x ) ( x ) x x x x x x x+ + + = = + + = + = + = + =+ + + = = + + = + = + = + =+ + + = = + + = + = + = + =+ + + = = + + = + = + = + =
25 91
16 16x = = = = = = = =
3 : 2 4 3x x + = + = + = + =
:
2 2 73 2 4 3 4 9 12 4 3 9 16 7 9 7 1 19
x x x x x x x ( x )(x ) x , + = + = + = = = + = + = + = = = + = + = + = = = + = + = + = = =
:3
4 34
x x
:2
4 3 3 2 2 33
x x x x + + + +
.
www.riazisara.ir
-
41
::::::::
y f (x)==== y g(x)==== f (x) g(x)==== x x
y .
y f (x)==== y g(x)==== .
: .
2x x= = = = (
1x . ====
.
2 3x x= = = = (
1x . ====
xx
+ = + = + = + =21
1 ( 0
2x Sinx = = = ( =
2
xSinx====
21 : 2 1x x x+ = ++ = ++ = ++ = . +2 2 4 31 2 1 1 1 1 1 1 1x x x (x ) x (x ) (x )((x ) )+ = + + = + + = + + + =+ = + + = + + = + + + =+ = + + = + + = + + + =+ = + + = + + = + + + =
-
42
::::::::
.
.
a aa
a a
====
(5 3
5 33 5>>>>
(
. : 3 12
3 14 2 3 1 3 1 3( )>>>>
= = = = = = = = (2
214 2 2 2 212 1 1 1
xxx x (x ) x
+ >+ >+ >+ >+++++ + = + = ++ + = + = ++ + = + = ++ + = + = (+
::::::::
a a a a
a a ab a b
= == == == =
= == == == =
2
22
2 2
2 2
x a x a a x a
x ax a x a
x a
a a a
a b a b+ ++ ++ ++ +
a b a b
a b a b= = = = = = = =
::::::::
f (x) x==== :
x xf (x)
x x
====
-
43
4 2y x= += += += ( +
1 3y x x= + = + = + = + (
1 3 3 2 4 3
1 3 1 3 2 1 3
1 3 1 4 2 1
x x x x x
y x x x x
x x x x x
+ + + +
= + < < = <
-
44
::::::::
.
. ( .
.)
3: 1 2y x x x= + += + += + += + . +
:
1 1y( ) = + = = + = = + = = + =
1 3 2 1y( ) = = = = = = = =
1 3 2 2 3y( ) = + == + == + == + =
2 6 1 4 1y( ) = = = = = = = =
y f(x)==== y f (| x |)====::::::::
y f(x)==== y f (x)=
-
45
: . y x====)
1
2y
x====
(
-
46
: : : : : : : :
.
f (x) g(x)==== f (x) g(x)
f (x) g(x)
====
= = = = .
.
. :
2 5 1x x = = = = (
2 5 1 42 5 1
2 5 1 3 6 2
x x xx (x )
x x x x
= = = = = = = = = = = =
= = = = = = = = = = = =
4 2 1x x+ = ++ = ++ = ++ = ( +
:
14 2 1
1 31 4 2 4 2 1
324 2 1
5
x x x
x x x x (x )
x x x
+ = + = + = + = + = + = + = + = + + + = + + + + = + + + + = + + + + = +
+ = = + = = + = = + = =
x 12
x = = = = 35
1 3
x = = = = .
1 5x = = = ( =
1 5 6 4 6x
x x , x
= = = = = = = = = = = = 6x = = = = .
2 1 7x x + = + = + = + ( =1 8
2 1 7 3 82 3
1 1 2 7 62
1 2 7 3 6 2
x x x x x
x x x x
x x x x x
+ = = = + = = = + = = = + = = =
< < + = = < < + = = < < + = = < < + = =
= = = = = = = = = = = =
4x x = = = = ( 2
2 2
4
4 4 2 2x
x x
x x x x x
-
47
: : : : : : : :
. :
2x x= = = = (
1
2Sin x (====
2 1 2x x x = = = = (
:
x
x
x
x
x : x x x x x x x
x : x x x x x x x x x
x : x x x x x x x x
x : x x x x x x x
<
-
48
1 2 3 4x x x + + = + + = + + = + + (=
4 12y( ) = = = =
3 6y( ) = = = =
2y( ) = = = =
1 2y( ) = = = =
2y( ) ====
4y . ====
4y ====
1 2 3y x x x= + += + += + += + +
www.riazisara.ir
-
49
: : : : : : : :
.
:
k : x k k x k
x kx k
x k
> > > >
2
.
: . 2 1 9x (
:
9 2 1 9 8 2 10 4 5x x x : :
2 2 2 22 1 81 4 4 1 81 4 4 80 20 5 4 4 5( x ) x x x x x x (x )(x ) x + + + + + + + + . .
3 1 4x + + + + (
:
51
3( , ] [ , ) + + + + :
3 1 4 1
53 1 4
3
x x
x x
+ + + +
+ + + +
:
2 2 2 21
3 1 16 9 6 1 16 9 6 15 3 2 5 1 3 5 5
3
x
( x ) x x x x x x (x )( x )x
+ + + + + + + + + + + + + + + + + + + + + + + +
1 1x x + + + ( +
: 12 2 21 1 2 1 1 3 3 3
x(x ) x x x x x x x(x ) x
+ + + + + + + + + + + + .
2 1 2x x < < < < (
.x
: 2 2 2 2 2 2 22 1 2 4 4 1 4 4 3 3 1 1 1( x ) (x ) x x x x x x x < + < + < < < > < + < + + < + + > > .
-
50
2 2 3x x (
x: x x= == == == =22 : 22
2 3 3 1x x ( x )( x ) + + + + 1x 3x : ++++
3 3 3x x
: 1 2
2 1 5
x
x x
: (I) 1 2 2 1 2 1 3x x x < < < < > > > > > > > > > > > > > < > >< > >< > >< > >
2 : x II I 3x<
-
51
: : : : : : : :
(g(x f (x)>>>> f g .
.
x
). g f ( g f
.
: .
1 1x x+ > + > + > + > (
2 2 21 1 2 1 3 3 3x (x ) x x x x x(x ) x+ > = + < < < = + < < < = + < < < = + < < < <
>3x + > (+
x < < < < x 1 x > > > > > > > >2 1 0 1
2 2 21 1 1 2 2 1 21
xx : x x x x (x )(x ) x
x
>>>> > + > + > > > + > + > > > + > + > > > + > + > >
< < < <
2 21 1 1 1 11
xx : x x x x x(x ) x
x
>>>> < > + > + > < < > + > + > < < > + > + > < < > + > + > <
< < < <
1 2( , ) ( , )= + = + = + = +
2 3 1x x (2 2 22 2 2 3 1 2 2 2 4 4
xx x x x x x ( x) x x
= = + = = + = = + = = : +
2 3 2 1 2 1 2x x (x )(x ) x + + + +
1
xx
(>>>>2
3
2
11 1
xx x x
x
>>>>< > >< > >< > >< > >
1
1
21y x= += += += +
3
1 1 2
1y x= += += += +3
2 1y x= = = =
1
1
y x====
1y
x====
1
1
22y x= = = =
2
3 1y x= = = =
www.riazisara.ir
-
52
2 2xx (
1 5x x+ + + + (
1 1 2
4
4
16
2
2
1
5y ====
322 1
1y x x= + = + = + = +
www.riazisara.ir