ﻓﺼﻞ 2
رﻳﺎﺿﻲ ﭘﺎﻳﻪ
(1ﺑﺴﻂ دوﺟﻤﻠﻪاي ﻏﻴﺎثاﻟﺪﻳﻦ ﺟﻤﺸﻴﺪ ﻛﺎﺷﺎﻧﻲ
(2ﺑﺨﺶﭘﺬﻳﺮي
(3ﺗﺎﺑﻊ ﻧﻤﺎﻳﺶ و ﺗﺎﺑﻊ ﻟﮕﺎرﻳﺘﻤﻲ
(4ﺗﺎﺑﻊ درﺟﻪ 2و ﺣﻞ ﻣﻌﺎدﻻت درﺟﻪ ،2ﮔﻮﻳﺎ ،اﺻﻢ
(5ﺗﺎﺑﻊ
ﺑﺴﻂ دوﺟﻤﻠﻪاي ﻏﻴﺎثاﻟﺪﻳﻦ ﺟﻤﺸﻴﺪ ﻛﺎﺷﺎﻧﻲ
(1ﻣﻲداﻧﻴﻢ
(2
n n
= =1
0 n
و
n n
= =n
n −1 1
و
n n
!n
!) = = r !(n − r
n −r r
n
n
n
n n −1 n n
(a + b )n = a n + na n −1b + ... + nab n −1 + b n = a n b 0 + a n −1b + a n −2b 2 + ... +
+ b
ab
0
1
2
n −1
n
(3ﺗﻌﺪاد ﺟﻤﻼت ﺑﺴﻂ
(4ﺟﻤﻠﻪي ) ( k + 1ﺑﺴﻂ
(a + b )n
ﺑﺮاﺑﺮ
(a + b )n
اﺳﺖ.
) (n + 1
ﺑﺮاﺑﺮ اﺳﺖ ﺑﺎ
n n −k k
b
a
k
ﻛﻪ ﺟﻤﻠﻪي ﻋﻤﻮﻣﻲ ﺑﺴﻂ ﻧﺎﻣﻴﺪه ﻣﻲﺷﻮد.
(5ﺟﻤﻼت ﻛﻪ از دو ردﻳﻒ ﺑﻪ ﻳﻚ ﻓﺎﺻﻠﻪاﻧﺪ ،ﺿﺮﻳﺐﻫﺎي ﻣﺴﺎوي دارﻧﺪ زﻳﺮا
(6ﺑﺮاي ﻳﺎﻓﺘﻦ ﻣﺠﻤﻮع ﺿﺮاﻳﺐ ﺑﺴﻂ دوﺟﻤﻠﻪاي
ﻋﺒﺎرﺗﺴﺖ ا
(a + b )n
n n
=
r n −r
ﺑﻪ ﺟﺎي ﻣﺘﻐﻴﺮﻫﺎي aو bﻋﺪد ) (1ﻣﻲﮔﺬارﻳﻢ .ﺑﻨﺎﺑﺮاﻳﻦ ﻣﺠﻤﻮع ﺿﺮاﻳﺐ ﺑﺴﻂ
(a + b )n
2n
(7
n n n
n n n
+ =2
+ + + ... +
n −1 n
0 1 2
n n
n n n
− =0
− + − ... +
0 1 2
n −1 n
(8در ﻣﺜﻠﺚ ﺧﻴﺎم -ﭘﺎﺳﻜﺎل اﻋﺪاد ردﻳﻒ
(9ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﻣﺜﻠﺚ ﺧﻴﺎم -ﭘﺎﺳﻜﺎل دارﻳﻢ:
) (n + 1
ام ﺿﺮاﻳﺐ ﺑﺴﻂ دو ﺟﻤﻠﻪاي
n n n +1
+ =
r −1 r r
(a + b )n
)ﺗﺴﺎوي ﭘﺎﺳﻜﺎل(
ﻣﻲﺑﺎﺷﻨﺪ.
94ﺟﺰوه ﻛﺎر در ﻛﻼس ﺣﺴﺎب و دﻳﻔﺮاﻧﺴﻴﻞ و اﻧﺘﮕﺮال و رﻳﺎﺿﻲ ﭘﺎﻳﻪ
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(a + b )n
(10اﮔﺮ ﺿﺮﻳﺐ ﺟﻤﻠﻪي pام ﺑﺎ ﺿﺮﻳﺐ ﺟﻤﻠﻪاي qام در ﺑﺴﻂ
ﺑﺮاﺑﺮ ﺑﺎﺷﻨﺪ آﻧﮕﺎه
p +q = n + 2
(11ﺑﺮاي ﻳﺎﻓﺘﻦ ﺟﻤﻠﻪي ﺛﺎﺑﺖ )ﺑﺪون ﻣﺘﻐﻴﺮ( ﻳﻚ ﺑﺴﻂ ﻣﻲﺗﻮاﻧﻴﻢ در ﺻﻮرت اﻣﻜﺎن ﺑﻪ ﺟﺎي ﻣﺘﻐﻴﺮﻫﺎ ﻋﺪد ﺻﻔﺮ ﺑﮕﺬارﻳﻢ.
(12ﺑﺮاي ﻳﺎﻓﺘﻦ ﻣﺠﻤﻮع ﺿﺮاﻳﺐ ﺟﻤﻼﺗﻲ ﻛﻪ ﻓﺎﻗﺪ ﻣﺘﻐﻴﺮ ﺧﺎﺻﻲ ﻫﺴﺘﻨﺪ ﺑﺎﻳﺪ ﺑﻪ ﺟﺎي اﻳﻦ ﻣﺘﻐﻴﺮ ﺧﺎص در ﺻﻮرت اﻣﻜﺎن ﻋﺪد ﺻﻔﺮ و ﺑﻪ ﺟﺎي ﺑﻘﻴﻪ ﻣﺘﻐﻴﺮﻫﺎ
ﻋﺪد ) (1ﻗﺮار دﻫﻴﻢ.
.13اﮔﺮ nﻓﺮد ﺑﺎﺷﺪ ﺑﺴﻂ (a + b )n + (a − b )nداراي
.14ﻣﺠﻤﻮع ﺗﻮان ﻣﺘﻐﻴﺮﻫﺎ در ﺑﺴﻂ (a + b )nﺑﺮاﺑﺮ
n +1
2
n2 +n
ﺟﻤﻠﻪ اﺳﺖ و اﮔﺮ nزوج ﺑﺎﺷﺪ n + 2ﺟﻤﻠﻪ دارد.
2
اﺳﺖ.
.15در ﺑﺴﻂ دوﺟﻤﻠﻪاي ، (a + b )nاﮔﺮ nزوج ﺑﺎﺷﺪ ،ﺟﻤﻠﻪي ﺷﻤﺎره
n
n
2
اﺳﺖ و اﮔﺮ nﻓﺮد ﺑﺎﺷﺪ ﺟﻤﻼت ﺷﻤﺎرهﻫﺎي
(16ﺗﻌﺪاد ﺟﻤﻼت ﺑﺴﻂ
.17
(a 1 + a 2 + ... + ak )n
n +1
2
و
ﺑﺮاﺑﺮ اﺳﺖ ﺑﺎ
n +2
2
n +3
2
ﻛﻪ ﺟﻤﻠﻪي وﺳﻂ اﺳﺖ .ﺑﺰرﮔﺘﺮﻳﻦ ﺿﺮﻳﺐ ﻋﺪدي را دارد ﻛﻪ ﻣﻘﺪار آن
ﻛﻪ ﺑﺮاﺑﺮ
n
n +1
2
ﻫﺴﺘﻨﺪ ،ﺑﺰرﮔﺘﺮﻳﻦ ﺿﺮﻳﺐ ﻋﺪدي را دارﻧﺪ.
n +k −1 n +k −1
=
k −1
n
n n n
n n n
n n −1
+ + + ... + = + + ... +
=2
0 2 4
n 1 3
n −1
ﻣﺠﻤﻮﻋﻪ ﺗﺴﺖﻫﺎي ﺑﺴﻂ دو ﺟﻤﻠﻪاي
.1در ﺑﺴﻂ ﻋﺒﺎرت ، (2x − 3)6ﺿﺮﻳﺐ ﺟﻤﻠﻪ ﺷﺎﻣﻞ
2160 (1
.2ﺣﺎﺻﻞ
(1
1620 (2
1080 (3
540 (4
17 17 18
?= + +
4 4 5
18
5
.3ﺟﻤﻠﻪي ﭼﻬﺎرم ﺑﺴﻂ دوﺟﻤﻠﻪاي
5 (1
x4
ﻛﺪام اﺳﺖ؟
(2
2 n
)
x
19
6
(1 +
6 (2
ﺑﺮاﺑﺮ
(3
160x −3
18
11
(4
19
14
اﺳﺖ اﻳﻦ ﺑﺴﻂ ﭼﻨﺪ ﺟﻤﻠﻪ دارد؟
7 (3
8 (4
ﻓﺼﻞ :2رﻳﺎﺿﻲ ﭘﺎﻳﻪ 95
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.4ﻣﺠﻤﻮع ﺿﺮاﻳﺐ ﺳﻪﺟﻤﻠﻪاي اول و ﺳﻪ ﺟﻤﻠﻪاي آﺧﺮ ﺑﺴﻂ دوﺟﻤﻠﻪاي
112 (2
211 (1
114 (3
.5ﺿﺮﻳﺐ ﺟﻤﻠﻪي ﺷﺎﻣﻞ a 10b 9در ﺑﺴﻂ دوﺟﻤﻠﻪاي
(2
(1ﺻﻔﺮ
.6اﮔﺮ nﻋﺪد ﻃﺒﻴﻌﻲ ﺑﺎﺷﺪ ﺣﺎﺻﻞ
(1
(a + b )18
2n − 2
18
10
(3
1 9
)
x
(1ﺷﺸﻢ
2n − 2n
(2x 2 +
) (−1 )n (2n − 2n
(4ﺻﻔﺮ
b
(2a 3 + )7
4
(4ﻫﺸﺘﻢ
ﻛﺪام اﺳﺖ؟
672 (3
336 (4
ﭼﻨﺪ ﺟﻤﻠﻪ دارد؟
7 (1
.10ﺟﻤﻠﻪي ﺛﺎﺑﺖ ﺑﺴﻂ دوﺟﻤﻠﻪاي
10 (1
ﻛﺪام اﺳﺖ؟
(3ﭘﻨﺠﻢ
42 (2
(a + b )15 + (a − b )15
18
9
1 (4
ﻣﺴﺘﻘﻞ از xاﺳﺖ؟
.8ﺿﺮﻳﺐ ﻋﺪدي ﺟﻤﻠﻪي ﺷﺎﻣﻞ a 15در ﺑﺴﻂ دوﺟﻤﻠﻪاي
.9ﺑﺴﻂ
(3
(2ﻫﻔﺘﻢ
21 (1
411 (4
ﻛﺪام اﺳﺖ؟
n
n n n n
n
− + − + ... + (−1 )
0 1 2 3
n
(2
.7ﺟﻤﻠﻪي ﭼﻨﺪم ﺑﺴﻂ دوﺟﻤﻠﻪاي
(a + b )n
ﻛﺪام ﻋﺪد ﻣﻲﺗﻮاﻧﺪ ﺑﺎﺷﺪ.
8 (2
1
x
3
(x x +
11 (2
14 (3
16 (4
ﻛﺪام اﺳﺖ؟
55 (3
(4ﺟﻤﻠﻪاي ﺛﺎﺑﺖ ﻧﺪارد.
96ﺟﺰوه ﻛﺎر در ﻛﻼس ﺣﺴﺎب و دﻳﻔﺮاﻧﺴﻴﻞ و اﻧﺘﮕﺮال و رﻳﺎﺿﻲ ﭘﺎﻳﻪ
.11ﻣﺠﻤﻮع ﺗﻮان ﻣﺘﻐﻴﺮﻫﺎ در ﺑﺴﻂ
(a + b )n
110 (1
ﻛﺪام ﻋﺪد ﻧﻤﻲﺗﻮاﻧﺪ ﺑﺎﺷﺪ؟
420 (2
.12ﺗﻌﺪاد ﺟﻤﻼت ﺑﺴﻂ
(a + b + c )3
4 (1
640 (3
2 (1
930 (4
ﭼﻨﺪﺗﺎﺳﺖ؟
9 (2
.13ﺟﻤﻠﻪي ﺛﺎﺑﺖ ﺑﺴﻂ دوﺟﻤﻠﻪاي
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(3a + 2 )8
(2
10 (3
12 (4
ﻛﺪام اﺳﺖ؟
28
(3
58
(4
38
.14در ﺑﺴﻂ ، (2a + 3b − c )7ﻣﺠﻤﻮع ﺿﺮاﻳﺐ ﺟﻤﻼﺗﻲ ﻛﻪ bﻧﺪارﻧﺪ ﻛﺪام اﺳﺖ؟
(2
1 (1
47
.15ﺿﺮﻳﺐ ﺟﻤﻠﻪﻫﺎي دوازدﻫﻢ و ﺑﻴـﺴﺖ و ﻧﻬـﻢ دوﺟﻤﻠـﻪاي
ﺟﻤﻠﻪاي
(1ﻫﺸﺘﻢ
.16ﺣﺎﺻﻞ
(1
2 12
(a + b )n
(3
(a + b )2n +3
37
(4ﺻﻔﺮ
ﺑـﺎ ﻫـﻢ ﺑﺮاﺑﺮﻧـﺪ در اﻳـﻦ ﺻـﻮرت ﺿـﺮﻳﺐ ﺟﻤﻠـﻪي دوازدﻫـﻢ ﺑـﺴﻂ دو
ﺑﺎ ﺿﺮﻳﺐ ﺟﻤﻠﻪ ﭼﻨﺪم ﺑﺮاﺑﺮ اﺳﺖ؟
(2ﻧﻬﻢ
(3ﻫﻴﺠﺪﻫﻢ
6 6 6
6 6
6
6
+ 3 + 9 + 27 + 81 + 243 + 729
0 1 2
3 4
5
6
(2
36
(3
(4ﻫﻔﺪﻫﻢ
ﻛﺪام اﺳﺖ؟
56
(4
4 10
ﻓﺼﻞ :2رﻳﺎﺿﻲ ﭘﺎﻳﻪ 97
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.17اﮔﺮ b ∈Zو
n ∈N
و
(2 + 3 )n = 362 + b 3
184 (1
ﺑﺎﺷﻨﺪ آﻧﮕﺎه b + nﻛﺪام اﺳﺖ؟
189 (2
.18ﺟﻤﻠﻪي ﭼﻨﺪم ﺑﺴﻂ
3 7
)
x
209 (3
(4x 2 − 12x + 9 )3 (2 −
(1ﺷﺸﻢ
ﻣﺴﺘﻘﻞ از xاﺳﺖ؟
(2ﻫﻔﺘﻢ
.19ﻣﺠﻤﻮع ﺗﻮان ﻣﺘﻐﻴﺮﻫﺎ در ﺑﺴﻂ
(a + b )n
110 (1
214 (4
(3ﻫﺸﺘﻢ
(4ﻧﻬﻢ
ﻛﺪام ﻋﺪد ﻧﻤﻲﺗﻮاﻧﺪ ﺑﺎﺷﺪ؟
640 (3
420 (2
930 (4
.20در ﺑﺴﻂ ، (a + b )nﺟﻤﻠﻪي ﻳﺎزدﻫﻢ ﺑﺰرﮔﺘﺮﻳﻦ ﺿﺮﻳﺐ ﻋﺪدي را دارد n ،ﻛﺪام ﻋﺪد ﻧﻤﻲﺗﻮاﻧﺪ ﺑﺎﺷﺪ؟
19 (1
.21ﺣﺎﺻﻞ ﻋﺒﺎرت
(1
20 (2
4 19 − 2 19
(2
.22در ﺟﻤﻠﻪي ﭼﻨﺪم ﺑﺴﻂ دوﺟﻤﻠﻪاي
(1ﭘﻨﺠﻢ
.23ﺿﺮﻳﺐ ﺟﻤﻠﻪ ﮔﻮﻳﺎي ﺑﺴﻂ
(1
4 20 − 2 20
2 9
)b
3
(2 3 x + x 3 )10
.24ﺑﻪ ازاي ﻛﺪام ﻣﻘﺪار nدر ﺑﺴﻂ
(2
(3x 5 − 2 )n
x 3n
ﻛﺪام اﺳﺖ؟
4 19
(4
4 20
ﺗﻮانﻫﺎي aو bﺑﺎ ﻫﻢ ﺑﺮاﺑﺮﻧﺪ:
(3ﻫﻔﺘﻢ
(4ﻫﺸﺘﻢ
در ﺻﻮرت وﺟﻮد ﻛﺪام اﺳﺖ؟
10
32
5
6 (2
(3
(3a 2 +
(2ﺷﺸﻢ
10
64
4
15 (1
21 (3
20 20
20 2 0 20
20
+ + ... + + + ... +
2 4
20 1 3
19
22 (4
(3
10
16
6
(4ﺟﻤﻠﻪ ﮔﻮﻳﺎ ﻧﺪارد.
ﺟﻤﻠﻪي ﻫﻔﺘﻢ ﻣﺴﺘﻘﻞ از xاﺳﺖ؟
20 (3
12 (4
98ﺟﺰوه ﻛﺎر در ﻛﻼس ﺣﺴﺎب و دﻳﻔﺮاﻧﺴﻴﻞ و اﻧﺘﮕﺮال و رﻳﺎﺿﻲ ﭘﺎﻳﻪ
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ﺑﺨﺶﭘﺬﻳﺮي
(1اﮔﺮ ﭼﻨﺪ ﺟﻤﻠﻪاي ) P (xرا ﺑﺮ ﭼﻨﺪﺟﻤﻠﻪاي
ار درﺟﻪ
) B (x
درﺟﻪ
(2اﮔﺮ ﺑﺎﻗﻲﻣﺎﻧﺪه ﺗﻘﺴﻴﻢ
) P (x
) R (x
) B (x
ﺗﻘﺴﻴﻢ ﻛﻨﻴﺪ ﻳﻚ ﺧﺎرجﻗﺴﻤﺖ ) Q (xو ﺑﺎﻗﻲﻣﺎﻧﺪهي
) R (x
ﺑـﻪ دﺳـﺖ ﻣـﻲآﻳـﺪ ﺑـﻪ ﻃـﻮري ﻛـﻪ
ﻛﻢﺗﺮ ﺧﻮاﻫﺪ ﺑﻮد.
) B (x
) Q (x
) P (x
) ⇒ P (x ) = Q (x
) B (x ) + R (x
ﺑﺎﻗﻲﻣﺎﻧﺪه
ﻓﺎﻛﺘﻮر
) P (x
اﺳﺖ و ﻣﻲﻧﻮﻳﺴﻴﻢ
(3ﺑﺮاي ﺗﻘﺴﻴﻢ ﭼﻨﺪﺟﻤﻠﻪاي
اوﻟﻴﻦ ﺟﻤﻠﻪ
) P (x
.4اﮔﺮ
) P (x
) B (x
، x nﻋﺪد
.6اﮔـــﺮ
ﺑﺨﺶﭘـﺬﻳﺮ اﺳـﺖ و ﻣـﻲﺗـﻮان ﮔﻔـﺖ
b
)
a
) P (x
(−
ﺑﺮ
) B (x
) B (x
ﺗﻘﺴﻴﻢ ﻛﺮده و ﻋﺒﺎرت ﺣﺎﺻﻞ را در ﺗﻤﺎم ﺟﻤﻼت ) B (xﺿﺮب ﻣﻲﻛﻨﻴﻢ و ﺗﻘﺴﻴﻢ را اداﻣﻪ داده
ﺷﻮد.
ﺑﺮ ax n + bﻣﻲﺗﻮان
) f (x
) f (x
) (m −1
ﺧﻮاﻫﺪ ﺑﻮد.
را ﺗﺎ ﺟﺎﻳﻲ ﻛﻪ اﻣﻜﺎن دارد ﺑﺮﺣﺴﺐ ﺗﻮانﻫﺎ
xn
ﻣﻲﻧﻮﻳﺴﻴﻢ ،ﺳﭙﺲ ﺑﻪ ﺟﺎي
را ﺑﮕﺬارﻳﻢ.
را ﺑـــﺮ
) P (b ) − P (a
b −a
) B (x
ﻳـﻚ ﻋﺎﻣـﻞ ﻳـﺎ
اﺑﺘﺪا ﻫﺮ دو را ﺑﺮﺣﺴﺐ ﺗﻮانﻫﺎي ﺑﻴﺸﺘﺮ ﺑﻪ ﻛﻢﺗﺮ ﻣﺘﻐﻴﺮ ﻣﺮﺗﺐ ﻣﻲﻛﻨﻴﻢ )ﺑﻪ ﺗﺮﺗﻴﺐ ﻧﺰوﻟﻲ( .ﺳـﭙﺲ
را ﺑﺮ ﻳﻚ ﭼﻨﺪ ﺟﻤﻠﻪاي درﺟﻪ mام ﺗﻘﺴﻴﻢ ﻛﻨﻴﻢ ،ﺑﺎﻗﻲﻣﺎﻧﺪه ﺣﺪاﻛﺜﺮ از درﺟﻪ
.5ﺑﺮاي ﻳﺎﻓﺘﻦ ﺑﺎﻗﻲﻣﺎﻧﺪه ﺗﻘﺴﻴﻢ
ﻣﻘﺴﻮم
) R (x
) P (x ) = B (x )Q (x
را ﺑﺮ اوﻟﻴﻦ ﺟﻤﻠﻪ
ﺗﺎ درﺟﻪ ﺑﺎﻗﻲﻣﺎﻧﺪه از درﺟﻪ
) P (x
ﺑﺮ
) B (x
ﺑﺮاﺑﺮ ﺻﻔﺮ ﺑﺎﺷﺪ ﻳﻌﻨﻲ
) P (x
ﺑﺮ
) B (x
ﻣﻘﺴﻮم ﻋﻠﻴﻪ
ﺧﺎرج ﻗﺴﻤﺖ
x −a
b
)
a
(ax n + b = 0 ⇒ x n = −
ﺗﻘـــﺴﻴﻢ ﻛﻨـــﻴﻢ و ﺧـــﺎرجﻗـــﺴﻤﺖ ) Q (xﺑﺎﺷـــﺪ ،ﻣﻘـــﺪار ﺧـــﺎرجﻗـــﺴﻤﺖ ﺑـــﻪ ازاي
ﺑﺮاﺑـــﺮ اﺳـــﺖ ﺑـــﺎ:
x =b
= ) Q (b
.7در ﻳﻚ ﭼﻨﺪ ﺟﻤﻠﻪاي ﻣﺎﻧﻨﺪ an x n + an −1x n −1 + ... + a1x 1 + a0ﺑﺮاي ﻳﺎﻓﺘﻦ ﻣﺠﻤﻮع ﺿﺮاﻳﺐ ﻛﺎﻓﻲ اﺳﺖ ﺑﻪ ﺟﺎي xﻋﺪد ) (1ﻗﺮار دﻫﻴﻢ و ﺑﺮاي
ﻳﺎﻓﺘﻦ ﺟﻤﻠﻪ ﻣﺴﺘﻘﻞ از ) xﺟﻤﻠﻪ ﺛﺎﺑﺖ( ﻛﺎﻓﻲ اﺳﺖ ﺑﻪ ﺟﺎي xﻋﺪد ﺻﻔﺮ ﻗﺮار دﻫﻴﻢ.
.8اﮔـــﺮ ﺑـــﺎﻗﻲﻣﺎﻧـــﺪه ﺗﻘـــﺴﻴﻢ
) P (x
ﺑـــﺮ
) (a 1x 1 + b1 )(ax 2 + b2 )....(an x n + bn
.9ﺑﺮاي ﻳﺎﻓﺘﻦ ﺑﺎﻗﻲﻣﺎﻧﺪه ﺗﻘﺴﻴﻢ
اﺗﺤﺎدﻳﻪﻫﺎي
) P (x
) (x ∓ a )(x 2 ± ax + a 2
ﺑﺮ
, ax 1 + b1
an xn + bn ,...,ax 2 + b2ﺑﺮاﺑـــﺮ Rﺑﺎﺷـــﺪ آﻧﮕـــﺎه ﺑـــﺎﻗﻲﻣﺎﻧـــﺪه ﺗﻘـــﺴﻴﻢ
) P (x
ﺑـــﺮ
ﺑﺮاﺑﺮ Rاﺳﺖ.
x 2 ± ax + a 2
اﺳﺘﻔﺎده ﻛﺮد.
ﻣﻲﺗـﻮان ﺑـﺎ ﺻـﻔﺮ ﻗـﺮار دادن ﻣﻘـﺴﻮم ﻋﻠﻴـﻪ و ﺿـﺮب ﻃـﺮﻓﻴﻦ ﺗـﺴﺎوي در
x ±a
از
ﻓﺼﻞ :2رﻳﺎﺿﻲ ﭘﺎﻳﻪ 99
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.10اﮔﺮ ﺑﺎﻗﻲﻣﺎﻧﺪه
ﺑﺎﻗﻲ ﻣﺎﻧﺪه
) f (x
ﺑﺮ
) h (x
ﺑﺮاﺑﺮ
) R1 (x
ﺑﺮ
) h (x
اﺳﺖ.
) R1 (x )R2 (x
و ﺑﺎﻗﻲﻣﺎﻧﺪه
.12اﻟﻒ.
ب.
xm −yn
xm −yn
زﻣﺎﻧﻲ ﺑﺮ
زﻣﺎﻧﻲ ﺑﺮ
x p +yp
ﺑﺨﺶﭘﺬﻳﺮ اﺳﺖ
ج.
xm + yn
زﻣﺎﻧﻲ ﺑﺮ
x p + gq
د.
xm + yn
ﻫﻴﭻﮔﺎه ﺑﺮ
x p − yq
ﺑﺨﺶﭘﺬﻳﺮ ﻧﻴﺴﺖ.
m n
=
p q
m n
=
p q
ﺑﺨﺶﭘﺬﻳﺮ اﺳﺖ ﻛﻪ
ﺑﺨﺶﭘﺬﻳﺮ اﺳﺖ ﻛﻪ
.13
) g (x
) h (x
) a n + 1 = (a + 1 )(a n −1 − a n −2 + ... − a + 1
.11ﺑﺎ ﺗﻘﺴﻴﻢ a n + 1ﺑﺮ a + 1ﺑﻪ ﺗﺴﺎوي زﻳﺮ ﻣﻲرﺳﻴﻢ.
x p − yq
ﺑﺮ
ﺑﺮاﺑﺮ
) R (x
ﺑﺎﺷﺪ ،ﺑﺎﻗﻲﻣﺎﻧﺪهي
) f (x )g (x
ﺑﺮ
) h (x
ﺑﺮاﺑﺮ ﺑـﺎ
m n
=
p q
)ﻋﺪد زوج(
)ﻋﺪد ﻓﺮد(
) x n − y n = (x − y )(x n −1 + x n −2y + x n −3y 2 + ... + xy n −2 + y n −1
n .14زوج ) x n − y n = (x + y )(x n −1 − x n −2y + x n −3y 2 − ... + xy n −2 − y n −1
) x n + y n = (x + y )(x n −1 − x n −2y + x n −3y 2 − ... + xy n −2 + y n −1
Nﻓﺮد
ﻣﺠﻤﻮﻋﻪ ﺗﺴﺖﻫﺎي ﺑﺨﺶﭘﺬﻳﺮي
.1ﻣﻘﺪار mﻛﺪام ﺑﺎﺷﺪ ﺗﺎ ﭼﻨﺪ ﺟﻤﻠﻪاي
13/5 (1
P (x ) = x 3 − mx 2 − x + 4
17/5 (3
-13/5 (2
.2در ﭼﻨﺪ ﺟﻤﻠﻪاي ، P (x ) = x 3 + ax 2 + x + bﻣﻘﺪار
- 4 (1
P (x ) = 5x 2 − 2x 3 − x + 4
3 (1
.4اﮔﺮ
6 (1
a
b
(3
ﺑﺮ
B (x ) = 2x − 3
1 (2
ﻳﻚ ﻓﺎﻛﺘﻮر )ﻋﺎﻣﻞ(
f (x ) = x 3 + 2x 2 − 5x − 6
2 (2
-17/5 (4
ﻛﺪام ﺑﺎﺷﺪ ﺗﺎ ﺑﺎﻗﻲﻣﺎﻧﺪه ﺗﻘﺴﻴﻢ آن ﺑﺮ
4 (2
.3اﮔﺮ ﭼﻨﺪ ﺟﻤﻠﻪاي
x −2
ﺑﺮ
2x + 1
ﺑﺨﺶﭘﺬﻳﺮ ﺑﺎﺷﺪ؟
x −1ﺑﺮاﺑﺮ 4ﺑﻮده و ﺑﺮ x + 2
1
4
(4
1
4
ﺗﻘﺴﻴﻢ ﻛﻨﻴﻢ ﻣﻘﺪار ﺧﺎرج ﻗﺴﻤﺖ ﺑﻪ ازاي
-1 (3
ﺑﺎﺷﺪ ،ﻣﺠﻤﻮع رﻳﺸﻪﻫﺎي ﻣﻌﺎﻣﻠﻪي
-6 (3
−
x = −1
-3 (4
f (x ) = 0
ﺑﺨﺶﭘﺬﻳﺮ ﺑﺎﺷﺪ؟
ﻛﺪام اﺳﺖ؟
-2 (4
ﭼﻘﺪر اﺳﺖ؟
100ﺟﺰوه ﻛﺎر در ﻛﻼس ﺣﺴﺎب و دﻳﻔﺮاﻧﺴﻴﻞ و اﻧﺘﮕﺮال و رﻳﺎﺿﻲ ﭘﺎﻳﻪ
.5اﮔﺮ ﻳﻜﻲ از ﺟﻮابﻫﺎي ﻣﻌﺎدﻟﻪي
x 3 − 2x 2 + ax + 2 = 0
(1ﺻﻔﺮ
(1
P (x ) = ax 2 + bx + c
ﺑﺮ
x −1
5 (1
ﺑﺮ
x 2 − 5x + 16
m = −8 , n = 24
و
x −2
4 (1
.9ﺑﻪ ازاي ﻛﺪام ﻣﻘﺪار ،nﭼﻨﺪﺟﻤﻠﻪاي
2 (1
-1 (4
ﺑﺨﺶﭘﺬﻳﺮ ﺑﺎﺷﺪ.
(4
m =2 , n =4
c −b
a
ﭼﻘﺪر اﺳﺖ.
ﺑﺮ
- 2 (2
و
در ﺗﻘـﺴﻴﻢ ﺑـﺮ
) (x −1
ﻫـﻢ ﺑـﺎﻗﻲﻣﺎﻧـﺪه ﺑﺎﺷـﻨﺪ ،ﺑـﺎﻗﻲﻣﺎﻧـﺪه
ﻛﺪام اﺳﺖ؟
-4 (3
P (x ) = (k − 2 )x n − 3kx n −1 + 54
3 (2
-1 (4
g (x ) = 2x 6 + bx 3 − 2x + 4
) (x + 1
m =1 , n =6
a ≠0
1 (3
f (x ) = ax 5 − 3x 2 + x + 1
h (x ) = 4x 7 + ax 3 + bx 2 − x − 4
(3
ﺑﺨﺶﭘﺬﻳﺮ ﺑﺎﺷﺪ ﺣﺎﺻﻞ
- 5 (2
.8اﮔﺮ ﭼﻨﺪﺟﻤﻠﻪايﻫـﺎي
ﺗﻘﺴﻴﻢ
-2 (3
x 4 − 3x 3 + mx + n
(2
m = −2 , n = 12
.7اﮔﺮ
ﺑﺮاﺑﺮ 2ﺑﺎﺷﺪ ﻣﺠﻤﻮع دو ﺟﻮاب دﻳﮕﺮ ﻣﻌﺎدﻟﻪ ﻛﺪام اﺳﺖ؟
1 (2
m .6و nﻛﺪام ﺑﺎﺷﻨﺪ ﺗﺎ ﭼﻨﺪ ﺟﻤﻠﻪاي
saati.com
ﺑﺮ
-12 (4
2x − 6
4 (3
ﺑﺨﺶﭘﺬﻳﺮ اﺳﺖ؟
(4ﺑﻪ kﺑﺴﺘﮕﻲ دارد.
.10اﮔﺮ ﺑﺎﻗﻲﻣﺎﻧﺪه ﺗﻘﺴﻴﻢ ) f (xﺑﺮ x 2 − 1و x 2 − 4ﺑﻪ ﺗﺮﺗﻴﺐ x + 3و x −2ﺷﻮد آن ﮔﺎه ﺑﺎﻗﻲﻣﺎﻧﺪه ﺗﻘﺴﻴﻢ ) f (xﺑﺮ x 2 + 3x + 2
(1
7x − 9
(2
−7x − 9
(3
−7x + 9
(4
7x + 9
ﻛﺪام اﺳﺖ؟
ﻓﺼﻞ :2رﻳﺎﺿﻲ ﭘﺎﻳﻪ 101
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.11اﮔﺮ ﺑﺎﻗﻲﻣﺎﻧﺪه ﺗﻘﺴﻴﻢ ) P(xﺑﺮ
(1
x −2
(2
−3x + 1
.12اﮔﺮ ﭼﻨﺪﺟﻤﻠﻪاي
(3
−3x − 1
x2 +1
ﺑﺮ
x −1
−2x 2 + x + 4
.14اﮔﺮ ﺑﺎﻗﻲﻣﺎﻧﺪه ﺗﻘﺴﻴﻢ ) p(xﺑﺮ
2x − 1
2 (1
(2
.16ﺑﺎﻗﻲﻣﺎﻧﺪه ﺗﻘﺴﻴﻢ
(3
p (x ) = x 2n + 2x 2n −1 + 2x − 2
x −2
x 17 + x 2 − x + 1
(1ﺻﻔﺮ
0/25 (1
وxو
2x + 1
ﺑﺮ
ﺑﺮ
x 2 + 3x + 2
x3 −x
0 / 5 (2
(4
4x 3 − x
ﻛﺪام اﺳﺖ؟
ﭼﻴﺴﺖ؟
(4
x2 +1
)(n ∈N , n ≤ 10
(4
x −4
ﻛﺪام اﺳﺖ؟
1 (2
t 10 − 1
) p (x
ﺑﺮ
ﺑﺮ
x 3 + x 2 − 2x
2x 2 + x + 4
x2 −1
x (3
x +3
) (1 + t )(1 − t + t 2 − t 3 + t 4
−2x 2 − x + 4
ﺑﺮاﺑﺮ 2ﺷﻮد ،ﺑﺎﻗﻲﻣﺎﻧﺪه ﺗﻘﺴﻴﻢ
(2ﺻﻔﺮ
.15ﺑﺎﻗﻲﻣﺎﻧﺪه ﺗﻘﺴﻴﻢ
.17ﺣﺎﺻﻞ ﻋﺒﺎرت
2x 2 − x + 4
ﺑﺮ
x3 +1
(3
=A
ﺑﻪ ازاي
t =5 5
x2 +1
(4
ﻛﺪام اﺳﺖ؟
0/75 (3
ﻛﺪام اﺳﺖ؟
-5 (4
x −2ﺑﻪ ﺗﺮﺗﻴﺐ 7و 4و 10ﺷﻮد ،ﺑﺎﻗﻲﻣﺎﻧﺪه ﺗﻘﺴﻴﻢ )f(x
(3
3x − 1
g (x ) = ax 9 + bx 6 − 2
5 )3
وxو
(2
(4
3x + 1
ﺑﺨﺶﭘﺬﻳﺮ ﺑﺎﺷﺪ ،ﺑﺎﻗﻲﻣﺎﻧﺪه ﺗﻘﻴﻢ
- 1 (2
.13اﮔﺮ ﺑﺎﻗﻲﻣﺎﻧﺪه ﺗﻘﺴﻴﻢ ) f(xﺑﺮ
(1
ﺑﻪ ﺗﺮﺗﻴﺐ ﺑﺮاﺑﺮ 5و -10ﺑﺎﺷﺪ ،ﺑﺎﻗﻲﻣﺎﻧﺪه ﺗﻘﺴﻴﻢ ) p(xﺑﺮ
x +3
f (x ) = ax 4 + bx 2 + 3
1 (1
(1
و
x2 +x −6
ﻛﺪام اﺳﺖ؟
1 (4
−x 2 + 1
ﻛﺪام اﺳﺖ؟
102ﺟﺰوه ﻛﺎر در ﻛﻼس ﺣﺴﺎب و دﻳﻔﺮاﻧﺴﻴﻞ و اﻧﺘﮕﺮال و رﻳﺎﺿﻲ ﭘﺎﻳﻪ
.18ﻋﺒﺎرت
(1
.19اﮔﺮ
x 45 + y 75
ﺑﺮ ﻛﺪاﻣﻴﻚ از ﻋﺒﺎرات زﻳﺮ ﺑﺨﺶﭘﺬﻳﺮ ﻧﻴﺴﺖ؟
x 3 + y5
x 9 + y 15
(2
(3
f (x ) = x (x + 2 )(x + 4 )(x + 5 )(x + 6 ) + k
63 (1
p (x ) = x 95 + x 55
- 1 (1
.21اﮔﺮ ﭼﻨﺪﺟﻤﻠﻪاي
x 2 + 7x + 3
ﺑﺮ
-63 (2
.20ﺑﺎﻗﻲﻣﺎﻧﺪه ﺗﻘﺴﻴﻢ
ﺑﺮ
p (x ) = x 8 + 64
12 (1ﻳﺎ 4
ﺑﺮ
x 2 −x + 1 = 0
(3
x 4 + mx 2 + n
?= k
-54 (4
ﺑﺨﺶﭘﺬﻳﺮ ﺑﺎﺷﺪ؟
(4
x −1
ﺑﺮ
x 9 + x 8 + ... + x + 1
x 8 + x 7 + ... + x + 1
x +1
? = m +n
12 (3ﻳﺎ -4
p (x ) = x 99 + x 98 + ... + x + 1
(1ﺻﻔﺮ
ﺑﺨﺶﭘﺬﻳﺮ ﺑﺎﺷﺪ؟
x 15 + y 25
ﻛﺪام اﺳﺖ؟
-12 (2ﻳﺎ -4
(2
x6 +y9
(4
54 (3
1 (2
.22ﺑﺎﻗﻲﻣﺎﻧﺪه ﺗﻘﺴﻴﻢ
saati.com
-12 (4ﻳﺎ 4
ﻛﺪام اﺳﺖ؟
1 (4
9 (3
ﺗﺎﺑﻊ ﻧﻤﺎﻳﻲ
.1ﻫﺮ ﺗﺎﺑﻌﻲ ﻛﻪ ﺑﻪ ﺻﻮرت
y =ax
ﻛﻪ a > 0و a ≠ 1اﺳﺖ ﻳﻚ ﺗﺎﺑﻊ ﻧﻤﺎﻳﻲ ﻧﺎﻣﻴﺪه ﻣﻲﺷﻮد.
y
y =ax
0 <a < 1
y =ax
a >1
y
1
x
0
1
x
0
ﻓﺼﻞ :2رﻳﺎﺿﻲ ﭘﺎﻳﻪ 103
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.2اﮔﺮ در ﻳﻚ دﺳﺘﮕﺎه ﻣﺨﺘﺼﺎت ،ﻧﻤﻮدار ﺗﻮاﺑﻊ ﻧﻤﺎﻳﻲ را ﺑﻪ ﻓﺮم
ﻣﻲﻛﻨﻨﺪ.
T
( )x ( 1 ) x
3
1
4
( )x
2
y
4x 2x
5x
y
1
1
x
.3اﮔﺮ ﻧﻘﺎط ﺗﺎﺑﻊ ﻧﻤﺎﻳﻲ ﺑﻪ ﺻﻮرت
y = a xرا رﺳﻢ ﻛﻨﻴﻢ ﻫﻤﻪي ﻧﻤﻮدارﻫـﺎ در ﻳـﻚ ﻧﻘﻄـﻪ ﻳﻌﻨـﻲ ) (0 , 1
x
0
f (x ) = ab x
ﻫﻤـﺪﻳﮕﺮ را ﻗﻄـﻊ
0
را در ﻳﻚ ﺟﺪول ﺗﻨﻈﻴﻢ ﻛﻨﻴﻢ در ﺻﻮرﺗﻲ ﻛﻪ ﻃﻮل ﻧﻘﺎط ﺑﻪ ﺻـﻮرت ﺗـﺼﺎﻋﺪ ﺣـﺴﺎﺑﻲ اﻧﺘﺨـﺎب ﻛﻨـﻴﻢ
ﻋﺮض ﻧﻘﺎط ﺑﻪ ﺻﻮرت ﺗﺼﺎﻋﺪ ﻫﻨﺪﺳﻲ ﺗﻐﻴﻴﺮ ﺧﻮاﻫﺪ ﻛﺮد .ﻣﺜﻼً ﺑﻪ ازاي xﻫﺎي ﻃﺒﻴﻌﻲ
4
ab 4
.4ﺗﺎﺑﻊ ﻧﻤﺎﻳﻲ
f (x ) = a x
3
3
2
ab
2
1
و a > 0و .a ≠ 1
x
ab ab
ab
f (x ) = logax
ﺑﺮاي a ≠ 1و a > 0ﻳﻚ ﺗﺎﺑﻊ ﻳﻚ ﺑﻪ ﻳﻚ و ﻣﻌﻜﻮسﭘﺬﻳﺮ اﺳﺖ و ﻣﻌﻜﻮس آن ﺗﺎﺑﻊ ﻧﻤﺎﻳﻲ ﺑﻪ ﺻـﻮرت
y
x
y
log x
a >1
x
1
0
0 1
x
logax
0 <a < 1
.5ﺑﺮاي ﻫﺮ ﻋﺪد ﻧﻤﺎﻳﻲ b y = xﺑﺎ ﺷﺮط b > 0و b ≠ 1دارﻳﻢ:
b >0 , b ≠1
b y = x ⇔ y = logbx
ﻛﻪ bرا ﺑﻪ ﻋﻨﻮان ﭘﺎﻳﻪ ﻳﺎ ﻣﻨﺒﺎي ﻟﮕﺎرﻳﺘﻢ ﺗﻌﺮﻳﻒ ﻣﻲﺷﻮد.
.6ﺑﺎ اﺳﺘﻔﺎده از ﺗﻌﺮﻳﻒ رﻳﺎﺿﻲ ﻟﮕﺎرﻳﺘﻢ دارﻳﻢ) :رواﺑﻂ اوﻟﻴﻪ(
loga1 = 0
)ب
logaa = 1
log AB = log A + log B
)د
logx10 = log x
)ج
log An = n log A
)و
log
)ه
= log n A
)ز
A
= log A − log B
B
1
log A
n
)اﻟﻒ
.7ﺑﺎ اﺳﺘﻔﺎده از رواﺑﻂ اوﻟﻴﻪ دارﻳﻢ:
1
logba
n
logba
m
= logab
)اﻟﻒ
= logam
)ب
n
b
104ﺟﺰوه ﻛﺎر در ﻛﻼس ﺣﺴﺎب و دﻳﻔﺮاﻧﺴﻴﻞ و اﻧﺘﮕﺮال و رﻳﺎﺿﻲ ﭘﺎﻳﻪ
saati.com
logca
c >0 , c ≠1
logbc
= logab
=x
a
= b logc
)ج
x
) a logaد
b
) a logcه
.8
→ a ≥c
) b > 1 , logab ≥ logcbاﻟﻒ
) b > 1 , logab ≥ c → a ≥ bcب
0 < b < 1 , logab ≥ logcb → a ≤ c
)ج
0 < b < 1 , logab ≥ c → a ≤ b c
)د
log 34 > 0
و
.9اﮔﺮ در ﻳﻚ ﻟﮕﺎرﻳﺘﻢ ،ﻋﺪد و ﻣﺒﻨﺎ ﻫﺮ دو ﺑﺰرﮔﺘﺮ از 1و ﻳﺎ ﻫـﺮ دو ﺑـﻴﻦ ﺻـﻔﺮ و ﻳـﻚ ﺑﺎﺷـﻨﺪ ﺣﺎﺻـﻞ ﻟﮕـﺎرﻳﺘﻢ ﻣﺜﺒـﺖ اﺳـﺖ .ﺑـﺮاي ﻣﺜـﺎل
1
log 21 > 0
و اﮔﺮ ﻋﺪد و ﻣﺒﻨﺎ ﻳﻜﻲ ﺑﺰرﮔﺘﺮ از ﻳﻚ و دﻳﮕﺮي ﺑﻴﻦ ﺻﻔﺮ و ﻳﻚ ﺑﺎﺷﺪ ﺣﺎﺻﻞ ﻟﮕﺎرﻳﺘﻢ ﻣﻨﻔﻲ اﺳﺖ ﺑﺮاي ﻣﺜﺎل
log 30/5 < 0
و
1
log52 < 0
3
n ≤ logab
logab
و ، n ∈Zﻳﻌﻨﻲ ﻟﮕﺎرﻳﺘﻢ ﻫﺮ ﻋﺪد ﺑﻴﻦ ﺻﺤﻴﺢ ﻣﺘـﻮاﻟﻲ ﻗـﺮار
ﻋﺪد ﮔﻨﮓ و ﮔﺎﻫﻲ ﻋﺪدي ﮔﻮﻳﺎ اﺳﺖ ﻳﻌﻨﻲ < n + 1
.10در اﻏﻠﺐ ﻣﻮارد
دارد و ﺑﺮاي ﭘﻴﺪا ﻛﺮدن ﻣﺤﺪودهي ، logabﺑﺎﻳﺴﺘﻲ aرا ﺑﻴﻦ دو ﺗﻮان ﻣﺘﻮاﻟﻲ از bﻗﺮار دﻫﻴﻢ ﻳﻌﻨﻲ b n < a < b n +1و از آﻧﺠﺎ ﻧﺘﻴﺠﻪ ﻣﻲﺷـﻮد ﻛـﻪ
. n ≤ logab < n + 1
.11اﮔﺮ در a ، logaxرا ﻋﺪد ﻧﭙﺮ ﻳﻌﻨـﻲ ) (e 2 / 7182818200در ﻧﻈـﺮ ﺑﮕﻴـﺮﻳﻢ آنﮔـﺎه logex = Lnxو آن را ﺗـﺎﺑﻊ ﻟﮕـﺎرﻳﺘﻢ ﻃﺒﻴﻌـﻲ )ﻟﮕـﺎرﻳﺘﻢ ﻧﭙـﺮﻳﻦ(
ﻣﻲﻧﺎﻣﻨﺪ و دارﻳﻢ:
) ln 1 = 0اﻟﻒ
) ln e = 1ب
) ln xy = ln x + ln yج
ln x n = n ln x
)ه
) e ln x = xو
y = ln x ⇒ x = e y
)ح
x
= ln x − ln y
y
.12
y
1
x
1
x
y
0
x
1
y = Lnx
Ln
y
1
y =e x
y = ex
y
1
x
0 1
0
1
x
0
ln
)د
ln x
= ln xy
ln y
)ز
ﻓﺼﻞ :2رﻳﺎﺿﻲ ﭘﺎﻳﻪ 105
saati.com
.13در ﺣﻞ ﻣﻌﺎدﻻت ﻟﮕﺎرﻳﺘﻤﻲ و ﻧﻤﺎﻳﻲ ﻣﻮارد زﻳﺮ را ﺑﺮرﺳﻲ ﻣﻲﻛﻨﻴﻢ:
اﻟﻒ :داﻣﻨﻪ ﺗﻌﺮﻳﻒ ﺗﺎﺑﻊ را ﺗﻌﻴﻴﻦ ﻣﻲﻛﻨﻴﻢ.
ب :ﭼﻨﺎﻧﭽﻪ از وﻳﮋﮔﻲﻫﺎي ﻟﮕﺎرﻳﺘﻢ اﺳﺘﻔﺎده ﻣﻲﻛﻨﻴﻢ ﻣﻤﻜﻦ اﺳﺖ ﺟﻮاب ﺧﺎرﺟﻲ ﻇﺎﻫﺮ ﺷﻮد ﻛﻪ ﺑﺎﻳﺪ از ﻣﺠﻤﻮﻋﻪ ﺟﻮاب ﺑﺎ ﺗﻮﺟﻪ ﺑـﻪ داﻣﻨـﻪي ﺗﻌﺮﻳـﻒ ﻛـﺴﺮ
ﮔﺮدد.
ﻣﺠﻤﻮﻋﻪ ﺗﺴﺖﻫﺎي ﺗﺎﺑﻊ ﻟﮕﺎرﻳﺘﻢ و ﺗﺎﺑﻊ ﻧﻤﺎﻳﻲ
.1در ﺗﺎﺑﻊ ﺑﺎ ﺿﺎﺑﻄﻪي
f (x ) = ab x = b > 0
6 (1
.2اﮔﺮ
(1
f (x ) = 3 3x
(2
)∞ (0 , +
3
4
12 (3
)(−∞ , 0
)(−∞ , 0
f (x ) = a (b )x − 1
- 1 (1
(1
f (5 ) = −6
و
1
2
(2
ﺑﺎﺷﺪ
log 9A2
(4
)∞ (1 , +
f (25 ) = −96
ﺑﺎﺷﺪ،
1 1
) ,
2 2
A(−
و
(3
logx2
1
2
24 (4
) B (1 , 1
1
4
) (−∞ , 1
? = ) f (15
51 (3
−
.5از ﺗﺴﺎوي ، log(x 2 + 4 ) = 1 + logx5ﻣﻘﺪار
3 +a
اﺳﺖ اﮔﺮ
از دو ﻧﻘﻄﻪ ي
(2
−
3a = A
(3
-36 (2
.4اﮔﺮ ﻧﻤﻮدار ﺗﺎﺑﻊ
24 (4
و ، g (x ) = 5 2xﺑﺰرﮔﺘﺮﻳﻦ ﺑﺎزهاي ﻛﻪ در آن ﻧﻤﻮدار fﭘﺎﻳﻴﻦ ﻧﻤﻮدار gﻗﺮار ﻣﻲﮔﻴﺮد ﻛﺪام اﺳﺖ؟
-24 (1
.6اﮔﺮ
2
8 (2
f .3ﺗﺎﺑﻌﻲ ﻧﻤﺎﻳﻲ ﺑﺎ داﻣﻨﻪ Rو ﺑﺮد
(1
دارﻳﻢ
3
2
= )f (0
و ، f (−2 ) = 3ﻣﻘﺪار
3
) (f
2
ﻛﺪام اﺳﺖ؟
ﺑﮕﺬرد .آنﮔﺎه
−
? = ) f (−1
)ﺗﺠﺮﺑﻲ (93
(4
3
4
−
ﻛﺪام اﺳﺖ؟ )ﺳﺮاﺳﺮي ﺗﺠﺮﺑﻲ (93
(3
3
2
2 (4
ﻛﺪام اﺳﺖ؟
(2
3 + 2a
(3
2 +a 2
(4
2 + 2a
106ﺟﺰوه ﻛﺎر در ﻛﻼس ﺣﺴﺎب و دﻳﻔﺮاﻧﺴﻴﻞ و اﻧﺘﮕﺮال و رﻳﺎﺿﻲ ﭘﺎﻳﻪ
x = 8 log 24 2
.7اﮔﺮ
(1
ﺑﺎﺷﺪ ،ﻟﮕﺎرﻳﺘﻢ ﻋﺪد
4
3
(2
.8اﮔﺮ ﻟﮕﺎرﻳﺘﻢ ﻋﺪد
2 3 0 / 25
- 3 (1
.9ﺿﺎﺑﻄﻪ ﺗﺎﺑﻊ وارون
1 −x
1 +x
(1
1 + 10x
=y
(2
y = log
1053
) (30 + x − x 2
(1ﺑﻲﺷﻤﺎر
1
4
.13اﮔﺮ
(1
x +1
x −1
y = log
(2
log 2 =k
2k
1
3
ﺑﺎﺷﺪ ﺣﺎﺻﻞ
1
)− 1
A
(
در ﭘﺎﻳﻪ 4ﻛﺪام اﺳﺖ؟
2
3
(3
x −1
x +1
y = log
(4
3
2
1047
) (x 2 −3
1014
7 (3
ﺧﻂ
y =2
5 (4
را در ﻧﻘﻄﻪ ﺑﺎ ﻛﺪام ﻃﻮل ﻗﻄﻊ ﻣﻲﻛﻨﺪ؟
5
4
4k
(4
109
ﺷﺎﻣﻞ ﭼﻨﺪ ﻋﺪد ﺻﺤﻴﺢ اﺳﺖ؟
) log(6 − 2 5 ) + 2 log(1 + 5
(2
y = log
(4
10 − x
10 + x
ﻛﺪام اﺳﺖ؟
(3
y = log
1 +x
1 −x
(3
9 (2
.12ﻧﻤﻮدار ﺗﺎﺑﻊ ﺑﺎ ﺿﺎﺑﻄﻪي
(1
2 (3
log x − 9 1ogx + 14 = 0
(2
.11داﻣﻨﻪي ﺗﺎﺑﻊ ﺑﺎ ﺿﺎﺑﻄﻪي
3
2
3 (4
ﻛﺪام اﺳﺖ؟
.10ﺣﺎﺻﻞ ﺿﺮب ﺟﻮابﻫﺎي ﻣﻌﺎدﻟﻪي
(1
در ﭘﺎﻳﻪ xﻛﺪام اﺳﺖ؟
در ﻣﺒﻨﺎي 8ﺑﺮاﺑﺮ Aﺑﺎﺷﺪ آنﮔﺎه logﻋﺪد
(2
1 − 10x
) 4 (x + 3
saati.com
3
4
(3
1 (4
ﻛﺪام اﺳﺖ؟
(3
1 +k
(4
2 + 4k
y = log
ﻓﺼﻞ :2رﻳﺎﺿﻲ ﭘﺎﻳﻪ 107
saati.com
.14ﻣﺠﻤﻮﻋﻪ ﺟﻮاب ﻣﻌﺎدﻟﻪﻫﺎي
1
=0
x
log x + log
(2
R (1
.15ﻣﻌﺎدﻟﻪ
x
x = 33
)∞ (0 , +
2 (3
1 (2
) log(6x − 1 ) log(1 − x
) log(1 − x ) log(6x − 1
.16ﭼﻨﺪ ﻣﻘﺪار ﻣﻮرد ﻗﺒﻮل ،xﺣﺎﺻﻞ دﺗﺮﻣﻴﻨﺎن
(1ﺻﻔﺮ
(1
1
2
1 (2
logx2 = log 3
، log(2x − 1) + 1ﻣﻘﺪار
2
(2
−
1
4
2
3
.19اﮔﺮ
(1
(2
log 2 + log 3 + log 4 =a
a2
a +2
.20اﮔﺮ
4 (1
log xy 2 = 2
−
) log 4 (x + 2
4
3
3log 6 + 2 log 8
log 2400
2 (2
3
4
(4
3
2
ﻛﺪام اﺳﺖ؟
3a
a +2
و ، log x 2y = 4ﺣﺎﺻﻞ
1
4
(4
1
3
ﻛﺪام اﺳﺖ؟
(3
logxy 4
3 (4
ﻛﺪام اﺳﺖ؟
(3
ﺑﺎﺷﺪ ﺣﺎﺻﻞ
(2
x
3
log 4
(4ﺑﻲﺷﻤﺎر
را ﺻﻔﺮ ﻣﻲﻛﻨﺪ؟
2 (3
.18اﮔﺮ ) ، 2 log(x − 2 ) = log(x + 10آﻧﮕﺎه
(1
(3
)∞ [1 , +
(4
}{
1
ﭼﻨﺪ ﺟﻮاب دارد؟
(1ﺻﻔﺮ
.17از ﺗﺴﺎوي
ﻛﺪام اﺳﺖ؟
(3
a +3
a +2
(4
a
3a + 6
ﭼﻘﺪر اﺳﺖ؟
8 (3
6 (4
108ﺟﺰوه ﻛﺎر در ﻛﻼس ﺣﺴﺎب و دﻳﻔﺮاﻧﺴﻴﻞ و اﻧﺘﮕﺮال و رﻳﺎﺿﻲ ﭘﺎﻳﻪ
.21از ﺗﺴﺎوي ، log5 (2x − 1) + log5 (3x − 5 ) = 1ﻣﻘﺪار
2 (1
.22اﮔﺮ
(1
=A
، log 5 e 2ﺣﺎﺻﻞ
و
(2
k1 + k 2 + k 3
log 32
ﻛﺪام اﺳﺖ؟
e
A
4
.24ﺑﻪ ازاي ﻛﺪام ﻣﻘﺪار mﺗﺎﺑﻊ ﺑﺎ ﺿﺎﺑﻄﻪ
.25اﮔﺮ
t +1
t
A=x
و
1
B = x t +1
(3
x + x2 +m
2
f (x ) = log
5 (4
ﻛﺪام اﺳﺖ؟
2k1 + k3
2
A
(4
(4
2k3 + k1
4
A
ﻳﻚ ﺗﺎﺑﻊ ﻓﺮد اﺳﺖ؟ )ﺟﺎﻣﻊ ﺳﻨﺠﺶ (93
-4 (3
(4ﻫﻴﭻ ﻣﻘﺪار m
ﺑﺎﺷﻨﺪ ﻟﮕﺎرﻳﺘﻢ Aدر ﭘﺎﻳﻪ Bﻛﺪام اﺳﺖ؟ )ﺳﻨﺠﺶ (93
(2
A = 4 log x + 1 − 3 log 3 x
5/25 (1
loga 3b 2c
(3
1 (2
(t + 1 )2
t
.26اﮔﺮ
ﺑﺎﺷﺪ ﺣﺎﺻﻞ
A
2
(2
4 (1
4 (3
log ac = k3
2k1 + k2
.23اﮔﺮ
(1
log ab = k1
و
ﻛﺪام اﺳﺖ؟
) log 2 (6x + 3
3 (2
log bc = k2
saati.com
t
(3
(t + 1 )2
ﺑﺎﺷﺪ ﺣﺎﺻﻞ
6/5 (2
10A
ﺑﻪ ازاي
x =4
1
t
t (4
ﭼﻘﺪر اﺳﺖ؟
6/25 (3
5/5 (4
ﻓﺼﻞ :2رﻳﺎﺿﻲ ﭘﺎﻳﻪ 109
saati.com
.27اﮔﺮ
(1
p = loga2
و q = logb4آنﮔﺎه
(2
log 2 ab
log 147
3
.28ﻣﻘﺪار
ﭼﻨﺪ واﺣﺪ از ﻣﻘﺪار
2 (1
log 27
16
4a − 8
3
.30ﺑﺎ ﻓﺮض
= 1 + loga8
)
2
4 (3
4
3a − 6
loga3
] [log 0 / 1 ] + [log 0 / 02 ] + [log 0 / 003
(2
.32اﮔﺮ ! a = 100ﺑﺎﺷﺪ ،ﺣﺎﺻﻞ
(3
(3
.33اﮔﺮ
1
6a
log 312 =a
(3
−5
log 23 × log 43 × ...× log 26
27
(2
1 −a
6a
1
2
(4
1
2
−
ﺑﺮاﺑﺮ اﺳﺖ ﺑﺎ:
5 (2
ﺑﺎﺷﺪ ،ﺣﺎﺻﻞ
3
4a − 8
(4
3a − 6
4
ﻛﺪام اﺳﺖ؟
1
1
1
1
+
+
+ .... +
a
a
a
log 2 log 3 log 4
loga100
1 (1
7 (4
ﺑﺮﺣﺴﺐ aﻛﺪام اﺳﺖ؟
، loga(1 −9aﺣﺎﺻﻞ
−3
log 2 a b
(4
log 2 ab 2
ﺑﻴﺶﺗﺮ اﺳﺖ؟
2 (2
.31ﺣﺎﺻﻞ
(1
2 log
(2
- 2 (1
(1
7
3
log 2 a 2b
(3
3 (2
.29ﺑﺎ ﻓﺮض ، log183 =aﺣﺎﺻﻞ
(1
1
1
+
p q
ﺑﺮاﺑﺮ اﺳﺖ ﺑﺎ:
−6
(4
−9
ﻛﺪام اﺳﺖ؟
10 (3
100 (4
ﺑﺮاﺑﺮ اﺳﺖ ﺑﺎ:
(3
1 −a
3a
(4
1
3a
110ﺟﺰوه ﻛﺎر در ﻛﻼس ﺣﺴﺎب و دﻳﻔﺮاﻧﺴﻴﻞ و اﻧﺘﮕﺮال و رﻳﺎﺿﻲ ﭘﺎﻳﻪ
.34ﺣﺎﺻﻞ
) A = log 1 (x 3 − 3x 2 + 3x + 1
ﺑﻪ ازاي
x =1 /1
saati.com
در ﻛﺪام ﮔﺰﻳﻨﻪ ﺻﺪق ﻣﻲﻛﻨﺪ؟
x −1
(1
A = −3
.35اﮔﺮ
(1
(2
=A
1
500
log 3
(3
−4 < A < −3
−3 < A < −2
(4
0<A<1
ﺑﺎﺷﺪ آنﮔﺎه:
(2
−5 < A < −4
(3
4 <A<5
−6 < A < −5
(4
5 <A<6
.36اﮔﺮ a = log26ﺑﺎﺷﺪ آﻧﮕﺎه:
(1
1
1
< <a
3
2
.37داﻣﻨﻪ ﺗﺎﺑﻊ
98 (1
(2
) 2 − log(x + 1
1 + logx2
=y
(1
ﺷﺎﻣﻞ ﭼﻨﺪ ﻋﺪد ﺻﺤﻴﺢ اﺳﺖ؟
99 (2
.38ﺑﻪ ازاي ﭼﻪ ﻣﻘﺎدﻳﺮ از ،aﻣﻌﺎدﻟﻪي
1 < a < 100
1
1
< <a
4
3
(3
1
1
< <a
5
4
(4
1
1
< <a
6
5
100 (3
x 2 + x log a 2 + log a = 0
(2
10 <a
ﻳﺎ
0 <a < 1
101 (4
دو ﺟﻮاب ﻣﺘﻤﺎﻳﺰ دارد؟
(3
0 < a < 10
(4
1 <a
.39ﺣﺎﺻﻞ ﻛﺪام ﻟﮕﺎرﻳﺘﻢ ﺑﺰرگﺗﺮ اﺳﺖ؟
(1
log 215
(2
log 143
(3
log 152
(4
log 160/5
.40ﻣﻌﺎدﻟﻪي e x + x 2 = 4ﭼﻨﺪ رﻳﺸﻪ دارد؟
(1ﺻﻔﺮ
1 (2
2 (3
4 (4
ﻓﺼﻞ :2رﻳﺎﺿﻲ ﭘﺎﻳﻪ 111
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.41ﻣﺠﻤﻮع رﻳﺸﻪﻫﺎي ﻣﻌﺎدﻟﻪي e 2x − 4e x + 3 = 0ﻛﺪام اﺳﺖ؟
(1
(2
ln 4
ln(logx2 ) = −1
.42رﻳﺸﻪي ﻣﻌﺎدﻟﻪ
(1
2
e
.43ﺣﺎﺻﻞ
(1
(2
2
6
log 6 + log 2
y = (log53 ) cos x
.46ﻓﺎﺻﻠﻪ ﻧﻘﻄﻪ ﺑﺮﺧﻮرد ﺗﺎﺑﻊ ﻧﻤﺎﻳﻲ
1 (1
4
1
≤x ≤ −
3
2
.48از ﻧﺎﻣﻌﺎدﻟﻪي
(1
−
2 (3
1
2
y = 2x
(2
.47ﻣﺠﻤﻮع ﺟﻮاب ﻧﺎﻣﻌﺎدﻟﻪي
e
2
e (4
3 (4
ﭼﻴﺴﺖ؟
(2
x ≥4
e
4
(3
1 (2
1 (1
(1
e
ﻛﺪام اﺳﺖ؟
(1ﺻﻔﺮ
.45ﻣﺎﻛﺰﻳﻤﻢ
(3
e
3
(4
2
3
ﻛﺪام اﺳﺖ؟
e
8
.44ﺣﺎﺻﻞ
4
ﻛﺪام اﺳﺖ؟
(2
( e )2 −3 ln 4
(3
ln 3
3 (4
ﺑﺎ ﻣﺤﻮر yﻫﺎ و ﻧﻘﻄﻪ ﺑﺮﺧﻮرد ﻣﻌﻜﻮس اﻳﻦ ﺗﺎﺑﻊ ﻧﻤﺎﻳﻲ ﺑﺎ ﻣﺤﻮر xﻫﺎ ﻛﺪام اﺳﺖ؟
2 (3
2
3x + 4
≤ −1
5
(2
(3
1
2
log 2
(4
2 2
ﻛﺪام اﺳﺖ؟
x ≤−
) log 0/1 (6 − x ) ≥ log 0/1 (x − 2
(2
log53
(4
log53
(3
4
3
x >−
(4
4
<x <1
3
ﺣﺪود xﻛﺪام اﺳﺖ؟
4 ≤x ≤6
(3
4 ≤x <6
(4
2 <x <6
−
112ﺟﺰوه ﻛﺎر در ﻛﻼس ﺣﺴﺎب و دﻳﻔﺮاﻧﺴﻴﻞ و اﻧﺘﮕﺮال و رﻳﺎﺿﻲ ﭘﺎﻳﻪ
.49ﺟﻮاب ﻣﻌﺎدﻟﻪي
x log 3 + 3 log x = 162
100 (1
log 2
.50اﮔﺮ
log 2 log 5
(2
.51از دﺳﺘﮕﺎه
13
4
(2
.52ﻣﻌﺎدﻟﻪي
x log3 = 81
(3
ﻛﺪام اﺳﺖ؟
13
6
2 (2
.53اﮔﺮ
log 3 = 0 / 477آﻧﮕﺎه ﻋﺪد (81 )25
45 (1
.54از ﻣﻌﺎدﻻت
2
5
4
3
(4
3
2
(3
6
7
(4
7
6
ﭼﻨﺪ ﺟﻮاب دارد؟
1 (1
(1
5
2
x + y = 25
6
، ﻣﻘﺪار | | x − y
log x + log y = 2 log 2
x
30.000 (3
10.000 (4
= ) ، log(3x − 2آنﮔﺎه ﻣﻘﺪار xﻛﺪام اﺳﺖ؟
1 (1
(1
ﻛﺪام اﺳﺖ؟
30 (2
log 5
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ﭼﻨﺪ رﻗﻤﻲ اﺳﺖ؟
46 (2
2x × 8y = 4
3 (3
(4ﻫﻴﭻ
47 (3
48 (4
و ، log x = log 2 + log yﻣﻘﺪار xﻛﺪام اﺳﺖ؟
(2
3
4
(3
3
5
(4
4
5
.55اﮔﺮ ﺑﻪ ﻋﺪد 15 ،Aواﺣﺪ اﺿﺎﻓﻪ ﺷﻮد ﺑﻪ ﻟﮕﺎرﻳﺘﻢ آن در ﻣﺒﻨﺎي 4ﻳﻚ واﺣﺪ اﺿﺎﻓﻪ ﻣﻲﺷﻮد؟ = A
7 / 5 (1
15 (2
3 (3
5 (4
ﻓﺼﻞ :2رﻳﺎﺿﻲ ﭘﺎﻳﻪ 113
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.56ﻟﮕﺎرﻳﺘﻢ ﻋﺪدي از ﻟﮕﺎرﻳﺘﻢ ﻋﻜﺲ ﻣﺠﺬور آن ﻋﺪد در ﭘﺎﻳﻪ 9ﺑﻪ اﻧﺪازه 6واﺣﺪ ﺑﻴﺸﺘﺮ اﺳﺖ آن ﻋﺪد ﻛﺪام اﺳﺖ؟
81 (2
27 (1
.57ﺣﺎﺻﻞ
(1
(log515 )2 + log 315 .log 1575
log 315
48 (4
54 (3
ﻛﺪام اﺳﺖ؟
(2
log515
8 (4
6 (3
.58اﮔﺮ ﻣﺠﻤﻮع ﻟﮕﺎرﻳﺘﻢ دو ﻋﺪد ﻣﺜﺒﺖ aو bدر ﭘﺎﻳﻪ 2ﺑﺮاﺑﺮ 4ﺑﺎﺷﺪ آن ﮔﺎه ﺣﺪاﻗﻞ a + bﻛﺪام اﺳﺖ؟
2 (1
.59اﮔﺮ
(1
4 (2
!
log 27
3 =k
آﻧﮕﺎه
!
log 26
3
6 (3
ﺑﺮﺣﺴﺐ kﻛﺪام اﺳﺖ؟
(2
3 −k
8 (4
(3
k −3
(4
k +3
k
3
.60ﺑﻪ ﻋﺪد 301ﭼﻨﺪ واﺣﺪ اﺿﺎﻓﻪ ﻛﻨﻴﻢ ﺗﺎ ﻟﮕﺎرﻳﺘﻢ ﻋﺪد ﺣﺎﺻﻞ در ﻣﺒﻨﺎي 8ﺑﺮاﺑﺮ 3ﮔﺮدد؟
112 (2
103 (1
301 (4
211 (3
ﺗﺎﺑﻊ و ﻣﻌﺎدﻟﻪ درﺟﻪ 2؛ ﻣﻌﺎدﻻت ﮔﻮﻳﺎ و اﺻﻢ
(1در ﺳﻬﻤﻲ
y = ax 2 + bx + c
ﻫﻤﻮاره دارﻳﻢ:
→ ﺳـــﻬﻤﻲ ﻣـــﻲ ﻧﻴﻤـــﻢ دارد → a > 0
: اﻟﻒ
→ ﺳـــــﻬﻤﻲ ﻣـــــﺎﻛﺰﻳﻤﻢ دارد → a < 0
b
2a
x =−
S
ﻃــﻮل راس ﺳــﻬﻤﻲ
b
→
ﻣﺤــــﻮر ﺗﻘــــﺎرن ﺳــــﻬﻤﻲ : x = − = ب
2a
ﻃــــﻮل ﻣﺎﻛﺴــــﻴﻤﻢ ﻳــــﺎ ﻣــــﻲ ﻧﻴﻤــــﻢ ﺳــــﻬﻴﻢ
b
∆−
=,y
) , ∆ = b 2 − 4ac
2a
4a
⇒ S (x = −ﻣﺨﺘﺼﺎت راس ﺳﻬﻤﻲ :ج
114ﺟﺰوه ﻛﺎر در ﻛﻼس ﺣﺴﺎب و دﻳﻔﺮاﻧﺴﻴﻞ و اﻧﺘﮕﺮال و رﻳﺎﺿﻲ ﭘﺎﻳﻪ
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y
∆−
∆
a > 0 ⇒ y
Rf = [−ﺑــــــﺮد ﺗــــــﺎﺑﻊ →
)∞ , +
= min
4a
4a
∆−
∆
a < 0 ⇒ y
] Rf = (−∞ , −ﺑــــــﺮد ﺗــــــﺎﺑﻊ →
= max
4a
4a
S
0
−b
2a
:د
=x
(2ﺑﺮاي اﻳﻨﻜﻪ از ﻋﺒﺎرت ax 2 + bx + cﻳﻚ ﻋﺒﺎرت ﻣﺮﺑﻊ ﻛﺎﻣﻞ در ﺑﻴﺎورﻳﻢ اﺑﺘﺪا از ﺿﺮﻳﺐ
x2
ﻓﺎﻛﺘﻮر ﮔﺮﻓﺘﻪ و ﺳﭙﺲ در ﻋﺒـﺎرت داﺧـﻞ ﭘﺮاﻧﺘـﺰ،
ﻣﺮﺑﻊ ﻧﺼﻒ ﺿﺮﻳﺐ xرا ﻛﻢ و زﻳﺎد ﻣﻲﻛﻨﻴﻢ.
y = ax 2 + bx + c
(3در ﺳﻬﻤﻲ
ﺣﺎﻟﺖﻫﺎي زﻳﺮ را دارﻳﻢ:
y
y
ﻳﺎ
0
x
→ ﺳﻬﻤﻲ در 2ﻧﻘﻄﻪ ﻣﺘﻤﺎﻳﺰ ﻣﺤﻮر xرا ﻗﻄﻊ ﻣﻲﻛﻨﺪ
0
x
y
ﻃﻮل ﻧﻘﻄﻪ ﻣﻤﺎس
−b
2a
= ⇒x
y
ﻳﺎ
0
x
y
x
0
→ ∆ >0
x
ﺳﻬﻤﻲ ﺑﺮ ﻣﺤﻮر xﻫﺎ ﻣﻤﺎس اﺳﺖ.
→
0
→ ∆ =0
ﻳﺎ
x
ﺳﻬﻤﻲ ﺑﺮ ﻣﺤﻮر xﻫﺎ را ﻗﻄﻊ ﻧﻤﻲﻛﻨﺪ.
→
0
∆ −b ±
2a
→ ∆ <0
ﻣﻌﺎدﻟﻪ دو رﻳﺸﻪ ﺣﻘﻴﻘﻲ ﻣﺘﻤﺎﻳﺰ دارد∆ > 0 → .
:اﻟﻒ
→ ∆ =0
:اﻟﻒ
2a
ﻣﻌﺎدﻟﻪ رﻳﺸﻪي ﺣﻘﻴﻘﻲ ﻧﺪارد.
.5اﮔﺮ
α
:اﻟﻒ
= x1 ,x2
→ x 1 = x 2 = −bﻣﻌﺎدﻟﻪ 2رﻳﺸﻪ ﺣﻘﻴﻘﻲ ﻣﺴﺎوي ﻳﺎ ﻳﻚ رﻳﺸﻪي ﻣﻀﺎﻋﻒ دارد
و
:اﻟﻒ
y
(4رﻳﺸﻪﻫﺎي ﻣﻌﺎدﻟﻪ درﺟﻪ دوم ax 2 + bx + c = 0در ﺻﻮرت وﺟﻮد ﻋﺒﺎرﺗﻨﺪ از:
β
:اﻟﻒ
→ ∆ <0
:ج
رﻳﺸﻪﻫﺎي ﻣﻌﺎدﻟﻪ درﺟﻪ دوم ax 2 + bx + c = 0ﺑﺎﺷﻨﺪ:
= S = α + β = −bﻣﺠﻤﻮع رﻳﺸﻪﻫﺎ :اﻟﻒ
a
= p = αβ = cﺿﺮب رﻳﺸﻪﻫﺎ :ب
a
∆
| |a
.6در ﻣﻌﺎدﻟﻪ درﺟﻪ ،2اﮔﺮ ﻳﻜﻲ از رﻳﺸﻪﻫﺎ kﺑﺮاﺑﺮ دﻳﮕﺮي ﺑﺎﺷﺪ دارﻳﻢ:
2
) b 2 (k + 1
=
ac
k
=| d =| α − bﻗﺪرﻣﻄﻠﻖ ﺗﻘﺎﺿﻞ رﻳﺸﻪﻫﺎ :ج
ﻓﺼﻞ :2رﻳﺎﺿﻲ ﭘﺎﻳﻪ 115
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.7اﮔﺮ رﻳﺸﻪﻫﺎي ﻣﻌﺎدﻟﻪ درﺟﻪ دوم ﻳﻌﻨﻲ
α
و
β
را داﺷﺘﻪ ﺑﺎﺷﻴﻢ ،ﺑﺎ ﺗﺸﻜﻴﻞ sو ،pﻣﻌﺎدﻟﻪي ﺷﺎﻣﻞ اﻳﻦ رﻳﺸﻪ ﻫـﺎ ﺑـﻪ ﺻـﻮرت
x 2 − Sx + p = 0
ﺧﻮاﻫﺪ ﺑﻮد.
.8در ﻣﻌﺎدﻟﻪي ، ax 2 + bx + c = 0اﮔﺮ ﻣﻌﺎدﻟﻪ 2رﻳﺸﻪ داﺷﺘﻪ ﺑﺎﺷﺪآﻧﮕﺎه:
α = 1
: a + b + c = 0 → اﻟﻒ
c
= β
a
α = −1
b =a +c →
c
β = −
a
.9ﺑﺮاي ﻧﻮﺷﺘﻦ ﻣﻌﺎدﻟﻪاي ﻛﻪ رﻳﺸﻪﻫﺎﻳﺶ ﺑﺮﺣﺴﺐ رﻳﺸﻪﻫﺎي ﻣﻌﺎدﻟﻪ ﻣﻔﺮوض اﺳﺖ ،ﻓﺮض ﻣﻲﻛﻨﻴﻢ ﻣﻌﺎدﻟـﻪي ﺟﺪﻳـﺪ رﻳـﺸﻪﻫـﺎﻳﺶ
ﺳﭙﺲ ﺑﺎ ﺗﺸﻜﻴﻞ
S′
و
p′
و ﺑﺎ اﺳﺘﻔﺎده از Sو pﻣﻌﺎدﻟﻪي ﻣﻔﺮوض ،ﻣﻌﺎدﻟﻪي ﺟﺪﻳﺪ ﺧﻮاﺳﺘﻪ ﺷﺪه ﺑﻪ ﺻﻮرت
α′
x 2 − S ′x + p ′ = 0
و
β′
:ب
ﺑﺎﺷـﺪ،
ﺧﻮاﻫﺪ ﺑﻮد.
.10ﺑﺮاي ﻧﻮﺷﺘﻦ ﻣﻌﺎدﻟﻪاي ﻛﻪ رﻳﺸﻪﻫﺎﻳﺶ ﺗﻐﻴﻴﺮ ﻳﺎﻓﺘﻪاي از رﻳﺸﻪﻫﺎي ﻣﻌﺎدﻟﻪي ax 2 + bx + c = 0اﺳﺖ ﺟﺪول زﻳﺮ را دارﻳﻢ:
وﺿﻌﻴﺖ رﻳﺸﻪﻫﺎي ﻣﻌﺎدﻟﻪي ﺟﺪﻳﺪ ﻧﺴﺒﺖ ﺑﻪ ﻣﻌﺎدﻟﻪي ﻗﺒﻞ
اﻟﻒ :ﻗﺮﻳﻨﻪ
ب :ﻋﻜﺲ
ج :ﻗﺮﻳﻨﻪ و ﻋﻜﺲ
د k :ﺑﺮاﺑﺮ
ه k :واﺣﺪ ﺑﻴﺸﺘﺮ
و k :واﺣﺪ ﻛﻤﺘﺮ
ز :ﻣﺮﺑﻊ
ح :ﻣﻜﻌﺐ
.11در ﻣﻌﺎدﻟﻪ درﺟﻪ دوم:
ﻛﺎري ﻛﻪ ﺑﺎﻳﺪ اﻧﺠﺎم دﻫﻴﻢ.
bرا ﺑﻪ -bﺗﺒﺪﻳﻞ ﻣﻲﻛﻨﻴﻢ.
ﺟﺎي aو cرا ﻋﻮض ﻣﻲﻛﻨﻴﻢ.
ﻫﻢ bرا ﺑﻪ -bﺗﺒﺪﻳﻞ ﻣﻲﻛﻨﻴﻢ و ﻫﻢ ﺟﺎي aو cرا ﻋﻮض ﻣﻲﻛﻨﻴﻢ.
bرا در kو cرا در k 2ﺿﺮب ﻣﻲﻛﻨﻴﻢ.
xرا ﺑﻪ x − kﺗﺒﺪﻳﻞ ﻣﻲﻛﻨﻴﻢ.
xرا ﺑﻪ x + kﺗﺒﺪﻳﻞ ﻣﻲﻛﻨﻴﻢ.
xرا ﺑﻪ xﺗﺒﺪﻳﻞ ﻣﻲﻛﻨﻴﻢ.
xرا ﺑﻪ 3 xﺗﺒﺪﻳﻞ ﻣﻲﻛﻨﻴﻢ.
ax 2 + bc + c = 0
x = 0
c =0 ⇒
ﻣﻌﺎدﻟﻪ ﻳﻚ رﻳﺸﻪي ﺻﻔﺮ و ﻳﻚ رﻳﺸﻪي ﻏﻴﺮﺻﻔﺮ داردb ⇒ .
x = −
a
اﮔﺮ
c
<0
a
آﻧﮕﺎه دو رﻳﺸﻪي ﻣﺘﻤﺎﻳﺰ ﻗﺮﻳﻨﻪ دارد
c
⇒
a
−
:اﻟﻒ
: b = 0 ⇒ x = ±ب
ﻣﻌﺎدﻟﻪ ﻳﻚ رﻳﺸﻪي ﻣﻀﺎﻋﻒ ﺻﻔﺮ دارد: c = b = 0 ⇒ x = 0 ⇒ .ج
(12اﮔﺮ دو ﻣﻌﺎدﻟﻪي ax 2 + bx + c = 0و a ′x 2 + bx + c = 0ﻓﻘﻂ ﻳﻚ رﻳﺸﻪي ﻣﺸﺘﺮك داﺷﺘﻪ ﺑﺎﺷﻨﺪ ،ﺑﺮاي ﻳـﺎﻓﺘﻦ رﻳـﺸﻪي ﻣـﺸﺘﺮك ﻛـﺎﻓﻲ اﺳـﺖ
x2
را از ﻫﺮ دو ﻣﻌﺎدﻟﻪ ﺣﺬف ﻛﻨﻴﻢ.
(13ﻣﻌﺎدﻟﻪﻫﺎي ax 2 + bx + c = 0و a ′x 2 + bx + c ′ = 0ﻛﻪ داراي 2رﻳﺸﻪ ﻫﺴﺘﻨﺪ زﻣﺎﻧﻲ رﻳﺸﻪﻫﺎي ﻣﺸﺘﺮك دارﻧﺪ ﻛﻪ
(14اﮔـــــــﺮ rرﻳـــــــﺸﻪاي از ﭼﻨﺪﺟﻤﻠـــــــﻪاي
p (x ) = an x n + an −1x n −1 + ... + a 1x + a 0
Q (x ) = a0x n + a1x n −1 + ... + an −1x + anﺧﻮاﻫﺪ ﺑﻮد.
)(a 0 ≠ 0
ﺑﺎﺷـــــــﺪ آﻧﮕـــــــﺎه
1
r
a b c
= =
a ′ b′ c′
رﻳـــــــﺸﻪي ﭼﻨﺪﺟﻤﻠـــــــﻪاي
116ﺟﺰوه ﻛﺎر در ﻛﻼس ﺣﺴﺎب و دﻳﻔﺮاﻧﺴﻴﻞ و اﻧﺘﮕﺮال و رﻳﺎﺿﻲ ﭘﺎﻳﻪ
.15اﮔﺮ
α
و
β
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رﻳﺸﻪﻫﺎي ﻣﻌﺎدﻟﻪي ax 2 + bx + c = 0ﺑﺎﺷﻨﺪ دارﻳﻢ:
:α 2 + β 2 = (α + β )2 − 2αβ = S 2 − 2pاﻟﻒ
: α 3 + β 3 = (α + β )3 − 3αβ (α + β ) = S 3 − 3pSب
: α 4 + β 4 = (α 2 + β 2 )2 − 2α 2 β 2 = (S 2 − 2 p )2 − 2p 2ج
.16در ﻣﻌﺎدﻻﺗﻲ ﻛﻪ ﺗﻮان ﻳﻜﻲ از ﻣﺘﻐﻴﺮﻫﺎ دو ﺑﺮاﺑﺮ ﺗﻮان ﻣﺘﻐﻴﺮ دﮔﻴﺮ ﺑﺎﺷﺪ ﻣﻲﺗﻮاﻧﻴﻢ ﻣﺘﻐﻴﺮ ﺑﺎ ﺗﻮان ﻛﻮﭼﻜﺘﺮ را tﻓﺮض ﻛﺮده و ﻣﻌﺎدﻟﻪ را ﺑﻪ ﻳﻚ ﻣﻌﺎدﻟﻪي
درﺟﻪ 2ﺗﺒﺪﻳﻞ ﻛﻨﻴﻢ.
.17در ﻫﺮ ﻣﻌﺎدﻟﻪ درﺟﻪ ax 2 + bx + c = 0 ،2اﮔﺮ
.18ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﺳﻬﻤﻲ
f (x ) = ax 2 + bx + c
c
<0
a
ﺑﺎﺷﺪ ،ﻫﻤﻮاره
∆ >0
ﺧﻮاﻫﺪ ﺑﻮد و ﻣﻌﺎدﻟﻪ دو رﻳﺸﻪي ﻣﺨﺘﻠﻒاﻟﻌﻼﻣﻪ دارد.
اﮔﺮ ﻣﻌﺎدﻟﻪي ax 2 + bx + c = 0دو رﻳﺸﻪي
α
و
β
داﺷﺘﻪ ﺑﺎﺷﺪ و ﻋﺪد tﺑﻴﻦ اﻳﻦ دو رﻳﺸﻪ ﺑﺎﺷﺪ
ﻫﻤﻮاره af (t ) < 0ﺧﻮاﻫﺪ ﺑﻮد.
y
) ﺗـــــﺎﺑﻊ ﻣـــــﻲ ﻧﻴﻤـــــﻢ دارد( a > 0
af (t ) < 0 ⇒
f (t ) < 0
.19
x
t
0
α
β
) f (t
اﮔﺮ ﻣﺠﻤﻮع ﭼﻨﺪ ﻣﺘﻐﻴﺮ ﻣﺜﺒﺖ ﻣﻘﺪار ﺛﺎﺑﺘﻲ ﺑﺎﺷﺪ ﺣﺎﺻﻠﻀﺮب آﻧﻬﺎ زﻣﺎﻧﻲ ﺑﻴﺸﺘﺮﻳﻦ ﻣﻘﺪار را دارد ﻛﻪ ﻣﺘﻐﻴﺮﻫﺎ ﺑﺎ ﻫﻢ ﺑﺮاﺑﺮ ﺑﺎﺷﻨﺪ.
اﮔﺮ ﺣﺎﺻﻠﻀﺮب ﭼﻨﺪ ﻣﺘﻐﻴﺮ ﻣﺜﺒﺖ ﻣﻘﺪار ﺛﺎﺑﺘﻲ ﺑﺎﺷﺪ ﺣﺎﺻﻠﺠﻤﻊ آﻧﻬﺎ زﻣﺎﻧﻲ ﻛﻤﺘﺮﻳﻦ ﻣﻘﺪار را دارد ﻛﻪ ﻣﺘﻐﻴﺮﻫﺎ ﺑﺎ ﻫﻢ ﺑﺮاﺑﺮ ﺑﺎﺷﻨﺪ.
.20اﮔﺮ ﺧﻄﻮط اﻓﻘﻲ ﺳﻬﻤﻲ را ﻗﻄﻊ ﻛﻨﻨﺪ ،ﭘﺎرهﺧﻄﺎﻳﻲ اﻳﺠﺎد ﻣﻲﺷﻮد ﻛﻪ ﻣﺤﻮر ﺗﻘﺎرن ﺳﻬﻤﻲ اﻳﻦ ﭘﺎرهﺧﻂﻫﺎ را ﺑﻪ دو ﻗﺴﻤﺖ ﻣﺴﺎوي ﺗﻘﺴﻴﻢ ﻣﻲﻛﻨـﺪ ﻳﻌﻨـﻲ
ﻣﻜﺎن ﻫﻨﺪﺳﻲ وﺳﻂ ﭘﺎرهﺧﻂﻫﺎي اﻳﺠﺎد ﺷﺪه ،ﻫﻤﺎن ﻣﺤﻮر ﺗﻘﺎرن ﺳﻬﻤﻲ اﺳﺖ.
A
B
B′
b
2a
.21اﮔﺮ ﺑﺨﻮاﻫﻴﻢ ﻣﻌﺎدﻟﻪي درﺟﻪ دوﻣﻲ ﺑﺎ ﺿﺮاﻳﺐ ﮔﻮﻳﺎ ﺑﻨﻮﻳـﺴﻴﻢ ﻛـﻪ ﻳﻜـﻲ از ﺟـﻮابﻫـﺎي آن
b′
a ′ −ﺑﺎﺷﻨﺪ:
b′
0
0′
x =−
a ′ +ﺑﺎﺷـﺪ ﻣـﻲﺑﺎﻳـﺴﺖ ﺟـﻮاب دﻳﮕـﺮ ﺑـﻪ ﺻـﻮرت
a ′ , b ′ ∈Q
ﻣﺠﻤﻮﻋﻪ ﺗﺴﺖﻫﺎي ﻣﻌﺎدﻟﻪ درﺟﻪ و ﻣﻌﺎدﻻت ﮔﻮﻳﺎ و اﺻﻢ
.1ﻣﻌﺎدﻟﻪي
(1
) y = (x − 1 )(x 2 − ax + a
−4 < a < 0
ﻣﺤﻮر xﻫﺎ را ﻓﻘﻂ در ﻳﻚ ﻧﻘﻄﻪ ﻗﻄﻊ ﻣﻲﻛﻨﺪ ،ﻣﺠﻤﻮﻋﻪي ﻣﻘﺎدﻳﺮ aﺑﻪ ﻛﺪام ﺻﻮرت اﺳﺖ؟
(2
0 <a < 2
(3
0 <a < 4
A′
(4
a >4
ﻓﺼﻞ :2رﻳﺎﺿﻲ ﭘﺎﻳﻪ 117
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.2ﺣﺪود kﺑﺮاي آﻧﻜﻪ ﻣﻌﺎدﻟﻪي
(1
x 2 + (k 2 + 1 )x − k 2 − 2 = 0
(2
| k |≤ 1
.3اﮔﺮ ﻳﻜﻲ از رﻳﺸﻪﻫﺎي ﻣﻌﺎدﻟﻪي
داراي دو رﻳﺸﻪي ﺣﻘﻴﻘﻲ ﻣﺘﻤﺎﻳﺰ ﺑﺎﺷﺪ.
(3
k ∈R
(4
k ≥1
k ≤ −1
، x 2 − 5x + 2m = 0رﺑﻊ رﻳﺸﻪ دﻳﮕﺮ ﺑﺎﺷﺪ ? = m
2 (1
(3
4 (2
.4ﺑﻪ ازاي ﻛﺪام ﻣﻘﺪار ،mﻣﺠﻤﻮع ﻣﺠﺬورات دو رﻳﺸﻪ ﺣﻘﻴﻘﻲ ﻣﻌﺎدﻟﻪ
- 2 (2
- 6 (1
1
2
(4
2x 2 − mx + m − 1 = 0
1
4
ﺑﺮاﺑﺮ 4ﺑﺎﺷﺪ.
6 (4
2 (3
.5ﺑﻪ ازاي ﭼﻪ ﻣﻘﺪار ،kﻳﻜﻲ از رﻳﺸﻪﻫﺎي ﻣﻌﺎدﻟﻪي ، x 2 + kx + 8 = 0ﻣﺮﺑﻊ رﻳﺸﻪ دﻳﮕﺮي اﺳﺖ؟
- 6 (2
6 (1
-4 (4
4 (3
.6در ﻣﻌﺎدﻟﻪي ، 4x 2 − 16x + m = 0ﻳﻜﻲ از رﻳﺸﻪﻫﺎ 3واﺣﺪ از رﻳﺸﻪي دﻳﮕﺮ ﺑﻴﺸﺘﺮ اﺳﺖ در آن ﺻﻮرت ﻣﺠﻤﻮع ﻣﺮﺑﻌﺎت رﻳﺸﻪﻫﺎ ﻛﺪام اﺳﺖ؟
13/5 (2
12/5 (1
.7در ﻣﻌﺎدﻟﻪي
(1
3x 2 − 15x + m = 0
59
5
.8اﮔﺮ
4 (1
α
و
β
14/5 (3
اﮔﺮ ﻳﻜﻲ از رﻳﺸﻪﻫﺎ 2واﺣﺪ ﺑﻴﺸﺘﺮ از رﻳﺸﻪي دﻳﮕﺮ ﺑﺎﺷﺪ.
(2
رﻳﺸﻪﻫﺎي ﻣﻌﺎدﻟﻪ
15/5 (4
63
5
x 2 − 14x + 1 = 0
8 (2
(3
ﺑﺎﺷﺪ ﺣﺎﺻﻞ
?= m
59
4
α β +β α
12 (3
(4
63
4
ﻛﺪام اﺳﺖ؟
16 (4
118ﺟﺰوه ﻛﺎر در ﻛﻼس ﺣﺴﺎب و دﻳﻔﺮاﻧﺴﻴﻞ و اﻧﺘﮕﺮال و رﻳﺎﺿﻲ ﭘﺎﻳﻪ
.9اﮔﺮ
α
و
رﻳﺸﻪﻫﺎي ﻣﻌﺎدﻟﻪ
β
x 2 − 14x + 1 = 0
(2
4 (1
.10اﮔﺮ راﺑﻄﻪي
3x 1 + x 2 = 12
3x 2 − 2x + m − 1 = 0
- 6 (1
α
و
β
رﻳﺸﻪﻫﺎي ﻣﻌﺎدﻟﻪي
و
β
رﻳﺸﻪﻫﺎي ﻣﻌﺎدﻟﻪي
α
و
β
رﻳﺸﻪﻫﺎي ﻣﻌﺎدﻟﻪي
1 (1
ﺑﺎﺷﺪ ﺣﺎﺻﻞ
.15در ﻣﻌﺎدﻟﻪي ، x 2 − 4x + 1 = 0اﮔﺮ
ﺑﺎﺷﺪ ﺣﺎﺻﻞ
3
2 (2
4 (2
-3 (4
1 2
1
) + (α + )2
α
β
(β +
ﻛﺪام اﺳﺖ؟
) α 2 (3 β − 1
40 (4
ﻛﺪام اﺳﺖ؟
1 (3
x 2 − 4x − 2 = 0
α
?= m
32 (3
x 2 − 3x + 1 = 0
و
13 (4
-5 (3
x 2 − 3x + 1 = 0
(2
2
3 (1
= 2x 1 − x 2
ﺑﺎﺷﺪ ﻣﻘﺪار
36 (2
α
.14اﮔﺮ
12 (3
16
3
اﮔﺮ
(4
ﺑﺎﺷﺪ ﺣﺎﺻﻞ
2 (4
α 2 − 5α − β
-2 (3
β
2
x 2 − 8x + m − 1 = 0ﺑﺮﻗﺮار ﺑﺎﺷﺪ ? = m
- 7 (2
28 (1
(1
1
4
6 (2
.11در ﻣﻌﺎدﻟﻪي
.13اﮔﺮ
α
+
β
ﻛﺪام اﺳﺖ؟
2 (3
ﺑﻴﻦ رﻳﺸﻪﻫﺎي ﻣﻌﺎدﻟﻪي
2 (1
.12اﮔﺮ
ﺑﺎﺷﺪ ﺣﺎﺻﻞ
β
α
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رﻳﺸﻪﻫﺎي ﻣﻌﺎدﻟﻪ ﺑﺎﺷﻨﺪ ﺣﺎﺻﻞ
6 (3
ﻛﺪام اﺳﺖ؟
-1 (4
) (α 2 − 4α + 4 )( β 2 − 4 β + 2
ﭼﻘﺪر اﺳﺖ؟
8 (4
ﻓﺼﻞ :2رﻳﺎﺿﻲ ﭘﺎﻳﻪ 119
saati.com
.16ﻧﻤﻮدار ﺗﺎﺑﻊ
(1
) y = (x − 3 )(x 2 − ax + a
(2
}{4
.17اﮔﺮ
α
و
β
رﻳﺸﻪﻫﺎي ﻣﻌﺎدﻟﻪي
2 (1
در ﻧﻘﻄﻪاي ﺑﺮ ﻣﺤﻮر xﻫﺎ ﻣﻤﺎس اﺳﺖ ﻣﺠﻤﻮﻋﻪ ﻣﻘﺎدﻳﺮي ﻛﺪام اﺳﺖ؟ )ﺳﻨﺠﺶ ﺟﺎﻣﻊ (93
(3
}{0 , 4
x 2 − 6x + 1 = 0
ﺑﺎﺷﺪ ﺣﺎﺻﻞ
(1
.21اﮔﺮ
x1
و
رﻳﺸﻪﻫﺎي ﻣﻌﺎدﻟﻪ
2
α
و
β
و
x 2 − 2x + m − 1 = 0
رﻳﺸﻪﻫﺎي ﻣﻌﺎدﻟﻪي
27 (1
.22اﮔﺮ a > 0و دو ﻣﻌﺎدﻟﻪي
- 2 (1
ﺑﺮﻗﺮار اﺳﺖ
?= a
1 (4
2 (3
-2 (4
ﺑﺎﺷﺪ ﺑﻪ ازاي ﭼﻪ ﻣﻘﺪار mراﺑﻄﻪي
1 (2
γ
رﻳﺸﻪﻫﺎي ﻣﻌﺎدﻟﻪ راﺑﻄﻪاي
2α 2 + 3αβ + β 2 = 12
-1 (3
- 1 (2
x2
6 (4
، (x + 1 )(x 2 − x − 6m ) = 0ﺣﺎﺻﻞ ﺿﺮب 3رﻳﺸﻪي ﺣﻘﻴﻘﻲ ﻣﻌﺎدﻟﻪ ﺑﺮاﺑﺮ ) (-6اﺳﺖ ? = m
1 (1
.20اﮔﺮ
α
و
β
- 2 (2
.19در ﻣﻌﺎدﻟﻪي درﺟﻪ ﺳﻮم
ﭼﻘﺪر اﺳﺖ؟
5 (3
.18در ﻣﻌﺎدﻟﻪي درﺟﻪ دوم ، ax 2 + 3ax + a − 2 = 0ﺑﻴﻦ
- 3 (1
}{4 / 5
| |α | + | β
3 (2
(4
}{0 , 4 , 4 / 5
x1 − x2 = 2 3
0 (3
x 3 + 5x 2 − x = 0
ﺑﺎﺷﺪ ﺣﺎﺻﻞ
29 (2
x 2 + 2x + a = 0
- 1 (2
α2 + β 2 +γ 2
30 (3
و
x 2 − x − 2a = 0
ﺑﺮﻗﺮار اﺳﺖ؟
-1 (4
ﻛﺪام اﺳﺖ؟
31 (4
داراي ﻳﻚ رﻳﺸﻪي ﻣﺸﺘﺮك ﺑﺎﺷﻨﺪ اﻳﻦ رﻳﺸﻪي ﻣﺸﺘﺮك ﻛﺪام اﺳﺖ؟
1 (3
2 (4
120ﺟﺰوه ﻛﺎر در ﻛﻼس ﺣﺴﺎب و دﻳﻔﺮاﻧﺴﻴﻞ و اﻧﺘﮕﺮال و رﻳﺎﺿﻲ ﭘﺎﻳﻪ
.23ﺑﻪ ازاي ﻛﺪام ﻣﻘﺪار ،mﻣﺠﻤﻮع و ﺣﺎﺻﻠﻀﺮب و ﺗﻔﺎﺿﻞ رﻳﺸﻪﻫﺎي
- 1 (2
1 (1
(1
9
5
x 2 − (1 − 3m )x + 3 = 0
ﺗﺸﻜﻴﻞ ﺗﺼﺎﻋﺪ ﺣﺴﺎﺑﻲ ﻣﻲدﻫﻨﺪ؟
(3ﺻﻔﺮ
.24ﺑﻪ ازاي ﻛﺪام ﻣﻘﺪار mﻣﺠﻤﻮع ﻣﺮﺑﻌﺎت رﻳﺸﻪﻫﺎي ﺣﻘﻴﻘﻲ ﻣﻌﺎدﻟﻪي
−
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1 (2
2 (4
mx 2 − (m + 3 )x + 5 = 0
1 (3و
9
5
ﺑﺮاﺑﺮ 6ﻣﻲﺑﺎﺷﺪ )ﺳﺮاﺳﺮي ﺗﺠﺮﺑﻲ (93
−
(4
9
5
و -1
.25ﺣﺎﺻﻠﻀﺮب دو ﻋﺪد ﻃﺒﻴﻌﻲ ﻣﺘﻮاﻟﻲ از 5ﺑﺮاﺑﺮ ﻋﺪد ﻛﻮﭼﻜﺘﺮ 32واﺣﺪ ﺑﻴﺸﺘﺮ اﺳﺖ ،ﻣﺠﻤﻮع آن دو ﻋﺪد ﻛﺪام اﺳﺖ؟
21 (1
19 (2
.26در ﻣﻌﺎدﻟﻪي درﺟﻪ دوم ، 7x 2 + 6x + 1 = 0ﺣﺎﺻﻞ
- 1 (1
(2ﺻﻔﺮ
.27ﻣﻌﺎدﻟﻪي درﺟﻪ دوﻣﻲ ﻛﻪ رﻳﺸﻪﻫﺎﻳﺶ
(1
x 2 − 10 2 x − 1 = 0
17 (3
] [x 1 + x 2 ] + [x 1 ] + [x 2
(2
( 2 + 1 )3
(1
x 2 − 6x − 4 = 0
ﭼﻘﺪر ات؟
-3 (3
و
( 2 − 1 )3
x 2 − 10 2 x + 1 = 0
.28ﻣﻌﺎدﻟﻪي درﺟﻪ دوﻣﻲ ﺑﺎ ﺿﺮاﻳﺐ ﮔﻮﻳﺎ ﻛﻪ ﻳﻜﻲ از رﻳﺸﻪﻫﺎي آن
(2
x 2 − 6x + 5 = 0
(1
a = 6 , b = −4
a = −4 , b = 2
-2 (4
ﺑﺎﺷﻨﺪ ﻛﺪام اﺳﺖ؟
(3
3− 5
x 2 − 198x + 1 = 0
(4
x 2 + 10 2 x + 1 = 0
ﺑﺎﺷﺪ ﻛﺪام اﺳﺖ؟
(3
x 2 − 6x + 4 = 0
.29ﻫﺮ ﮔﺎه رﻳﺸﻪﻫﺎي ﻣﻌﺎدﻟﻪي ، (a + b )x 2 − 6x + b = 0ﻋﻜﺲ رﻳﺸﻪﻫﺎي ﻣﻌﺎدﻟﻪي
(2
15 (4
(3
2x 2 + 3x − 1 = 0
a =4 , b =2
(4
x 2 − 3x + 1 = 0
ﺑﺎﺷﻨﺪ دارﻳﻢ؟
(4
a = −4 , b = −2
ﻓﺼﻞ :2رﻳﺎﺿﻲ ﭘﺎﻳﻪ 121
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.30در ﻣﻌﺎدﻟﻪ
2 + 1 )x + 2 = 0
( ، x 2 −ﺣﺎﺻﻞ
5 (1
x 16 + x 26
ﭼﻘﺪر اﺳﺖ؟
65 (2
.31ﺑﻪ ازاي ﻛﺪام ﻣﻘﺪار mﻋﺪد
1
8
17 (3
واﺳﻄﻪ ﻋﺪدي ﺑﻴﻦ دو رﻳﺸﻪي ﻣﻌﺎدﻟﻪي
- 3 (2
3 (1
(m 2 − 4 )x 2 − 3x + m = 0
4 (3
9 (4
اﺳﺖ؟
-4 (4
.32ﻣﻨﺤﻨﻲ ﺑﻪ ﻣﻌﺎدﻟﻪي ) ، y = (x − 1)(x 2 − ax + aﻣﺤﻮر xﻫﺎ را ﻓﻘﻂ در ﻳﻚ ﻧﻘﻄﻪ ﻗﻄﻊ ﻣﻲﻛﻨﺪ ،ﻣﺠﻤﻮﻋﻪي ﻣﻘﺎدﻳﺮ aﺑﻪ ﻛﺪام ﺻﻮرت اﺳﺖ؟
(1
(2
−4 < a < 0
0 <a < 2
.33ﺑﻪ ازاي ﻛﺪام ﻣﻘﺪار mﻣﻌﺎدﻟﻪ درﺟﻪ دوم
- 3 (1
13 (1
2x + 1
α
و
27 (1
β
رﻳﺸﻪﻫﺎي ﻣﻌﺎدﻟﻪي
و
2x − 1
x (5x + 3 ) = 2
و xاﺳﺖ
) (x > 1
+3 (4
ﻃﻮل ﺿﻠﻊ ﻣﺘﻮﺳﻂ ﻛﺪام اﺳﺖ؟
17 (3
19 (4
ﺑﺎﺷﻨﺪ ﺑﻪ ازاي ﻛﺪام ﻣﻘﺪار kﻣﺠﻤﻮﻋﻪ ﺟﻮابﻫﺎي ﻣﻌﺎدﻟﻪي
4x 2 − kx + 25 = 0
اﺳﺖ؟
28 (2
.36ﺑﺎ ﻛﺪام ﻣﻘﺎدﻳﺮ ،mﻣﻨﺤﻨﻲ ﺑﻪ ﻣﻌﺎدﻟﻪي
m < −2
دو رﻳﺸﻪ ﺣﻘﻴﻘﻲ و ﻣﻌﻜﻮس ﻫﻢ دارد؟
2 (3
15 (2
1 , 1
2
2
α β
(1
mx 2 + 5x + m 2 − 6 = 0
- 2 (2
.34ﻃﻮل اﺿﻼع ﻣﺜﻠﺚ ﻗﺎﺋﻢاﻟﺰاوﻳﻪاي
.35اﮔﺮ
(3
0 <a < 4
(4
a >4
(2
y = (m + 2 )x 2 − 2x + 1
m < −1
29 (3
31 (4
از ﻫﺮ 4ﻧﺎﺣﻴﻪي ﻣﺤﻮرﻫﺎي ﻣﺨﺘﺼﺎت ﻣﻲﮔﺬرد؟
(3
−2 < m < −1
(4
−4 < m < −2
ﺑﻪ ﺻﻮرت
122ﺟﺰوه ﻛﺎر در ﻛﻼس ﺣﺴﺎب و دﻳﻔﺮاﻧﺴﻴﻞ و اﻧﺘﮕﺮال و رﻳﺎﺿﻲ ﭘﺎﻳﻪ
saati.com
.37اﮔﺮ ﻣﻨﺤﻨﻲ ﺑﻪ ﻣﻌﺎدﻟﻪي ، y = 2x 2 − 4x + m − 3ﻣﺤﻮر xﻫﺎ را در دو ﻧﻘﻄﻪ ﺑﻪ ﻃﻮلﻫـﺎي ﻣﺜﺒـﺖ ﻗﻄـﻊ ﻛﻨـﺪ آﻧﮕـﺎه ﻣﺠﻤﻮﻋـﻪ ﻣﻘـﺎدﻳﺮ mﺑـﻪ ﻛـﺪام
ﺻﻮرت اﺳﺖ.
(1
(2
m >3
.38اﮔﺮ ﻳﻜﻲ از رﻳﺸﻪﻫﺎي ﻣﻌﺎدﻟﻪي
(1
(2
(2
9 < m < 25
3x 2 − 2x
x2 +4
4 (1
.41ﻧﻤﻮدار ﺗﺎﺑﻊ ﺑﺎ ﺿﺎﺑﻄﻪي
−
y = mx
در ﺑﺎزهي
(3
) (a , b
f (x ) = x 3 − 4x 2 − x + 4 : x > −1
m > −2
.43ﺑﻪ ازاي ﻛﺪام ﻣﻘﺎدﻳﺮ ،mﻋﺒﺎرت
در ﺑﺎزهي
−2 < m < −1
(m − 1 )x 2 + 6x + 2m + 1
(2
y =2
اﺳﺖ؟
8 (3
) (a , b
(4
زﻳﺮﻣﺤﻮر xﻫﺎﺳﺖ
5 < m < 13
? = ) Max (b − a
∞
? = ) Max (b − a
4 (3
y = (m + 2 )x 2 − 2mx + 1
m > 2 /5
(4
7 < m < 15
ﭘﺎﻳﻴﻦﺗﺮ از ﺧﻂ ﺑﻪ ﻣﻌﺎدﻟﻪ
3 (2
(2
1
2
(4
3
2
ﻧﻘﻄﻪ ﻣﺸﺘﺮك ﻧﺪارد ،ﻣﺠﻤﻮﻋﻪ ﻣﻘﺎدﻳﺮ mﻛﺪام اﺳﺖ؟
15 < m < 23
= ) f (x
.42ﺑﻪ ازاي ﻛﺪام ﻣﻘﺪار ،mﻧﻤﻮدار ﺗﺎﺑﻊ
(1
(3
ﺑﺎ ﺧﻄﻮط
3 <m <5
ﺑﺮاﺑﺮ 2ﺑﺎﺷﺪ ،ﻣﺠﻤﻮع دو رﻳﺸﻪي دﻳﮕﺮ آن ﻛﺪام اﺳﺖ؟
6 (2
2 (1
m < −2
3
2
) y = (2x + 1 )(x + 8
.40ﻧﻤﻮدار ﺗﺎﺑﻊ ﺑﺎ ﺿﺎﺑﻄﻪي
(1
3 <m <4
x (ax 2 − x − 5 ) = 2
- 2 (1
.39ﻣﻨﺤﻨﻲ ﺑﻪ ﻣﻌﺎدﻟﻪي
(3
(4
4 <m <5
5 (4
ﻫﻤﻮاره در ﺑﺎﻻي ﻣﺤﻮر xﻫﺎﺳﺖ؟
(3
| m |< 2
(4
−1 < m < 2
ﺑﺮاي ﻫﺮ ﻣﻘﺪار دﻟﺨﻮاه xﻣﺜﺒﺖ اﺳﺖ؟
(3
1 <m <2
(4
1 <m < 2 /5
ﻓﺼﻞ :2رﻳﺎﺿﻲ ﭘﺎﻳﻪ 123
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.44ﺗﻌﺪاد رﻳﺸﻪﻫﺎي ﻣﻌﺎدﻟﻪي
2 (x − 3 )6 + (x − 3 )2 + 5 = 0
6 (1رﻳﺸﻪ
3 (2رﻳﺸﻪ
2
x3 x3
+
= 30
1 +x3 1 +x3
.45ﻣﻌﺎدﻟﻪي
2 (1
.47ﻣﻌﺎدﻟﻪ
1 2
1
) + 3(x + ) = 4
x
x
1 (1
11
5
y = 3 (a − 1 )x 2 − 2x − 1
<a
(2
3 (3
a < −2
y − 2x − 4 = 0
>a
.50ﺑﻪ ازاي ﻛﺪام ﻣﻘﺎدﻳﺮ aﻣﻨﺤﻨﻲ ﺑﻪ ﻣﻌﺎدﻟﻪي
(2
(3
)(−∞ , −3
11
5
y = ax 2 − (a + 2 )x
a > −2
4 (4
ﻓﻘﻂ از ﻧﺎﺣﻴﻪ اول ﻋﺒﻮر ﻧﻜﻨﺪ ،ﻣﺤﺪودهي aﻛﺪام اﺳﺖ؟
ﭘﺎﻳﻴﻦﺗﺮ از ﺧﻂ
(2
2 (4
ﭼﻨﺪ رﻳﺸﻪ دارد؟
y = (a − 2 )x 2 + (a + 3 )x
)∞ (−3 , +
.49اﮔﺮ ﺗﺎﺑﻊ
(1
1 (3
2 (2
.48اﮔﺮ ﻧﻤﻮدار ﺗﺎﺑﻊ
6 (4
ﻛﺪام اﺳﺖ؟
(2ﺻﻔﺮ
(x +
(4رﻳﺸﻪ ﺣﻘﻴﻘﻲ ﻧﺪارد.
ﭼﻨﺪ رﻳﺸﻪي ﮔﻨﮓ دارد؟
5 (3
x 4 + 2x 2 − 4 = 0
- 1 (1
(1
2 (3رﻳﺸﻪ
3 (2
.46ﺟﻤﻊ رﻳﺸﻪﻫﺎي ﻣﻌﺎدﻟﻪي
(1
ﻛﺪام اﺳﺖ؟
] [−1 , 2
(4
] [−3 , 2
ﻗﺮار ﮔﻴﺮد .ﺣﺪود aﻛﺪام اﺳﺖ؟
(3
a <1
(4
a >1
از ﻧﺎﺣﻴﻪي دوم ﻣﺤﻮرﻫﺎي ﻣﺨﺘﺼﺎت ﻧﻤﻲﮔﺬرد؟
(3
a >0
(4
−2 ≤ a < 0
124ﺟﺰوه ﻛﺎر در ﻛﻼس ﺣﺴﺎب و دﻳﻔﺮاﻧﺴﻴﻞ و اﻧﺘﮕﺮال و رﻳﺎﺿﻲ ﭘﺎﻳﻪ
.51ﻣﺠﻤﻮع ارﻗﺎم رﻳﺸﻪي ﻣﻌﺎدﻟﻪي
10 (1
3 − 3p = 3 − 3p + 2
(1ﺻﻔﺮ
1 (3
1 − x 2 + x 2 − 3x + 2 = 0
(1ﺻﻔﺮ
3 (2
.54اﮔﺮ ﻣﻌﺎدﻟﻪي
-20 (2
.55ﻣﻌﺎدﻟﻪي
x −1 −2 = 0
(1ﺻﻔﺮ
x = 1 −x
(4ﺳﻪ
ﭼﻨﺪ ﺟﻮاب دارد؟
2 (3
1 (4
داراي ﺟﻮاب ﺑﺎﺷﺪ ﻣﻘﺪار mﻛﺪام اﺳﺖ؟
22 (3
-24 (4
ﭼﻨﺪ رﻳﺸﻪ دارد؟
(2ﻳﻚ
(3دو
(4ﺳﻪ
ﭼﻨﺪ رﻳﺸﻪ دارد؟
2 (1
.57ﻣﻌﺎدﻟﻪي
(3دو
2x 2 + 3x − 14 + x 2 + x − 6 + x 3 + x 4 + m = 0
24 (1
11 (4
ﭼﻨﺪ ﺟﻮاب دارد؟
(2ﻳﻚ
.53ﻣﻌﺎدﻟﻪي
(1ﺻﻔﺮ
ﻛﺪام اﺳﺖ؟
2 (2
.52ﻣﻌﺎدﻟﻪي
.56ﻣﻌﺎدﻟﻪ
15 + 2x + 80 = 5
saati.com
(2ﺻﻔﺮ
x x =x +1
3 (3
1 (4
ﭼﻨﺪ ﺟﻮاب دارد؟
1 (2
2 (3
3 (4
ﻓﺼﻞ :2رﻳﺎﺿﻲ ﭘﺎﻳﻪ 125
saati.com
.58ﻣﻌﺎدﻟﻪي
3
3 −x 2
= 3 1 −x
ﭼﻨﺪ ﺟﻮاب دارد؟
1 (2
(1ﺻﻔﺮ
.59ﻣﻌﺎدﻟﻪي
x 2 − cos x + 1 = 0
(1ﺻﻔﺮ
.60ﻣﻌﺎدﻟﻪ
.61ﻣﻌﺎدﻟﻪي
1 (2
x
− sin x = 0
9
3 (2
(x + 1 )3 = 3x + 2
(1ﺻﻔﺮ
.63ﻣﻌﺎدﻟﻪي
(1ﺻﻔﺮ
.64ﻣﻌﺎدﻟﻪي
2 (3
3 (4
ﭼﻨﺪ رﻳﺸﻪ دارد؟
1 (2
= 2x
5 (3
7 (4
ﭼﻨﺪ ﺟﻮاب دارد؟
1 (2
2x = x 2
2 (3
3 (4
ﭼﻨﺪ ﺟﻮاب دارد؟
1 (1
.62ﻣﻌﺎدﻟﻪي
2 (3
3 (4
ﭼﻨﺪ ﺟﻮاب دارد؟
(1ﺻﻔﺮ
(1ﺻﻔﺮ
ﭼﻨﺪ ﺟﻮاب دارد؟
1 (2
x
1
=
x
sin x
2 (3
3 (4
4x + 1
2 −x 2
2 (3
3 (4
ﭼﻨﺪ رﻳﺸﻪ دارد؟
1 (2
2 (3
3 (4
126ﺟﺰوه ﻛﺎر در ﻛﻼس ﺣﺴﺎب و دﻳﻔﺮاﻧﺴﻴﻞ و اﻧﺘﮕﺮال و رﻳﺎﺿﻲ ﭘﺎﻳﻪ
.65ﻛﺪام ﻋﺪد ﻣﻲﺗﻮاﻧﺪ رﻳﺸﻪي ﻣﻌﺎدﻟﻪي
- 1 / 5 (1
x 2 −a x = 1
ﺑﺎﺷﺪ؟
saati.com
) (0 < a < 1
1 / 5 (2
0/5 (4
-0/3 (3
ﺗﺎﺑﻊ
(1ﺗﺎﺑﻊ راﺑﻄﻪاي اﺳﺖ از ﻣﺠﻤﻮﻋﻪي Aﺑﻪ Bﻛﻪ در آن ﺑﻪ ﻫﺮ ﻋﻀﻮ از Aدﻗﻴﻘﺎً ﻳﻚ ﻋﻀﻮ از Bﻧﻈﻴﺮ ﻣﻲﺷﻮد.
F :A → B
(2ﻫﺮ ﺗﺎﺑﻊ را ﻣﻲﺗﻮان ﺑﺎ ﻧﻤﻮدار ون ﻧﻤﺎﻳﺶ داد ﺑﻪ اﻳﻦ ﺻﻮرت ﻛﻪ ﻣﺠﻤﻮﻋﻪي ، Aﻣﺠﻤﻮﻋﻪ ﻣﺒﺪاء ﺑﺎﺷﺪ و ﻣﺠﻤﻮﻋﻪ Bﻣﻘﺼﺪ ﺑﺎﺷﺪ و در ﻧﻤـﻮدار ون
ﻳﻚ ﺗﺎﺑﻊ ﺑﺎﻳﺪ دو اﺗﻔﺎق زﻳﺮ ﺑﻴﻔﺘﺪ.
1
c
e
اﻟﻒ :از ﻫﺮ ﻋﻀﻮ Aدﻗﻴﻘﺎً ﻳﻚ ﻓﻠﺶ ﺧﺎرج ﺷﻮد.
a
b
d
ب .ﻻزم ﻧﻴﺴﺖ ﺑﻪ ﻫﺮ ﻋﻀﻮ Bدﻗﻴﻘﺎً ﻳﻚ ﻓﻠﺶ وارد ﺷﻮد ،ﻣﻤﻜﻦ اﺳﺖ ﺑﻪ ﻫﺮ ﻋﻀﻮ Bﺑﻴﺶ از ﻳﻚ ﻓﻠﺶ وارد ﺷﻮد ﻳﺎ اﺻﻼً ﻓﻠﺸﻲ وارد ﻧﺸﻮد.
(3اﮔﺮ :fﺗﺎﺑﻌﻲ از Aﺑﻪ Bﺑﺎﺷﺪ آن ﮔﺎه ﻣﺠﻤﻮﻋﻪ ،Aداﻣﻨﻪ fاﺳﺖ ،اﻣﺎ ﻟﺰوﻣﻲ ﻧﺪارد ﻛﻪ ﻣﺠﻤﻮﻋﻪ
B
ﺑﺮد ﺗﺎﺑﻊ ﺑﺎﺷﺪ و ﻣﺠﻤﻮﻋﻪ Bرا ﻫﻢ داﻣﻨﻪ ﻳـﺎ
ﻣﻘﺼﺪ ﻣﻲﻧﺎﻣﻴﻢ.
(4ﺗﺎﺑﻊ )زوج ﻣﺮﺗﺐ( :دﺳﺘﻪاي از زوج ﻣﺮﺗﺐﻫﺎ را ﺗﺎﺑﻊ ﻣﻲﻧﺎﻣﻴﻢ ﻛﻪ ﻣﻮﻟﻔﻪ اول آن ﺗﻜﺮار ﻧﺪاﺷﺘﻪ ﺑﺎﺷﺪ وﻟﻲ ﻣﺆﻟﻔﻪﻫﺎي دوم آﻧﻬﺎ ﻣﻲﺗﻮاﻧﻨـﺪ ﺗﻜـﺮار
داﺷﺘﻪ ﺑﺎﺷﻨﺪ.
ﺑﺮاي ﻣﺜﺎل راﺑﻄﻪﻫﺎ
}) f2 = {(1, 2 ),(3, 3 )(4, 5 )}, f1 = {(1, 2 ),(2, 3 ),(3, 4
ﺗﺎﺑﻊ ﻫﺴﺘﻨﺪ وﻟﻲ راﺑﻄﻪ
}) f3 = {(1, 2 ),(1, 3 ),(4, 5
ﺗﺎﺑﻊ ﻧﻴﺴﺖ.
(5ﺗﺎﺑﻊ را ﺑﻪ ﻋﻨﻮان ﻣﺎﺷﻴﻦ ﻣﻲﺗﻮان در ﻧﻈﺮ ﮔﺮﻓﺖ ﻛﻪ وروديﻫﺎي ﻣﺎﺷﻴﻦ ) (xاز ﻳﻚ ﻃﺮف وارد ﻣﻲﺷﻮد .و ﺧﺮوﺟﻲﻫﺎ) yﻫﺎ( از ﻃﺮف دﻳﮕـﺮ ﺧـﺎرج
ﻣﻲﺷﻮﻧﺪ.
ﺧﺮوﺟﻲ )(y
ورودي x
ﻣﺎﺷﻴﻦ f
(6ﺗﺎﺑﻊ را ﻣﻲﺗﻮان ﮔﺎﻫﻲ اوﻗﺎت ﺑﻪ ﺻﻮرت ﺟﺪوﻟﻲ ﺑﺮاﺳﺎس xو yﻧﻮﺷﺖ.
5
7
4
3
x
3
2
6
1
y
(7ﮔﺎﻫﻲ اوﻗﺎت ﺑﻴﻦ وروديﻫﺎ ) xﻫﺎ( و ﺧﺮوﺟﻲﻫﺎ )(yﻫﺎ ﺿﺎﺑﻄﻪي ﻗﺎﻧﻮنﻣﻨﺪ وﺟﻮد دارد )y=f(x
(8اﮔﺮ ﻧﻤﻮدار fرا داﺷﺘﻪ ﺑﺎﺷﻴﻢ آن ﮔﺎه ﺑﻪ ﺷﺮﻃﻲ fﺗﺎﺑﻊ اﺳﺖ ﻛﻪ ﺑﻪ ازاي ﻫﺮ ﺧﻂ ﻗﺎﺋﻢ ﻛﻪ ﻣﻮازي ﻣﺤﻮر yﻫﺎ رﺳﻢ ﺷﻮد ،ﻧﻤﻮدار ﺣﺪاﻛﺜﺮ در ﻳﻚ ﻧﻘﻄـﻪ
ﻗﻄﻊ ﺷﻮد.
y
x
0
ﺗﺎﺑﻊ ﻧﻴﺴﺖ
y
x
0
ﺗﺎﺑﻊ اﺳﺖ
ﻓﺼﻞ :2رﻳﺎﺿﻲ ﭘﺎﻳﻪ 127
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(9ﻫﺮ ﺗﺎﺑﻊ ﺧﻄﻲ درﺟﻪ اول را ﺑﻪ ﺻﻮرت y=ax+bﻧﻤﺎﻳﺶ ﻣﻲدﻫﻴﻢ ﻛﻪ aﺷﻴﺐ ﺧﻂ و ) bﻋﺮض از ﻣﺒﺪاء آن اﺳﺖ(
(10ﺗﺎﺑﻊ ﮔﻮﻳﺎ :اﮔﺮ دو ﭼﻨﺪ ﺟﻤﻠﻪاي را ﺑﻪ ﻫﻢ ﺗﻘﺴﻴﻢ ﻛﻨﻴﻢ ،اﻳﻦ ﺗﺎﺑﻊ را ﺗﺎﺑﻊ ﮔﻮﻳﺎ ﮔﻮﻳﻨﺪ.
ax n + bx n −1 + ...
a ′x m + b ′m −1 + ...
(11ﺗﺎﺑﻊ ﺛﺎﺑﺖ :ﻳﻌﻨﻲ ﺑﻪ ازاي ﻫﺮ ،xﻣﻘﺪار yﺛﺎﺑﺖ اﺳﺖ.
B
(12ﺗﺎﺑﻊ ﻫﻤﺎﻧﻲ :اﮔﺮ در ﺗﺎﺑﻊ ﺧﻄﻲ a = 1 . b = 0 ، y = ax + bﺑﺎﺷﺪ آن ﮔﺎه
y =x
= ) f (x
A
را ﺗﺎﺑﻊ ﻫﻤﺎﻧﻲ ﮔﻮﻳﻨﺪ.
A
B
(13اﮔﺮ ﺗﺎﺑﻊ fاز ﭼﻨﺪ ﺿﺎﺑﻄﻪ ﻣﺘﻤﺎﻳﺰ ﺗﺸﻜﻴﻞ ﺷﺪه ﺑﺎﺷﺪ آن را ﺗﺎﺑﻊ ﭼﻨﺪ ﺿﺎﺑﻄﻪاي ﮔﻮﻳﻨﺪ.
f1 : x ∈D1
f2 : x ∈D2
f (x ) =
.
fn : x ∈ Dn
ﺑﻪ ﻃﻮري ﻛﻪ
Df = D1 ∪ D2 ∪ ... ∪ Dn
(14اﮔﺮ در ﻳﻚ راﺑﻄﻪ ﺑﻪ ازاي ﻫﺮ xﻓﻘﻂ ﻳﻚ yداﺷﺘﻪ ﺑﺎﺷﻴﻢ آن راﺑﻄﻪ را ﻳﻚ ﺗﺎﺑﻊ ﮔﻮﻳﻨﺪ ﺑﻪ ﻋﺒﺎرﺗﻲ :
∀x 1 , x 2 ∈ Df : x 1 = x 2 ⇒ y 1 = y 2
(15در ﺑﺮﺧﻲ ﻣﺴﺎﺋﻞ ﻻزم اﺳﺖ ﻛﻪ راﺑﻄﻪاي ﺑﻴﻦ ﻣﺘﻐﻴﺮﻫﺎ ﺑﺮﻗﺮار ﻛﻨﻴﻢ ﺑـﻪ ﻋﻨـﻮان ﻣﺜـﺎل اﮔـﺮ دو ﻣﺜﻠـﺚ ﻗـﺎﺋﻢ اﻟﺰاوﻳـﻪ ﺑـﻪ اﺿـﻼع ﻗﺎﺋﻤـﻪ aو bاﮔـﻪ
a = b2ﺑﺎﺷﺪ در اﻳﻦ ﺻﻮرت ﻣﺴﺎﺣﺖ ﻣﺜﻠﺚ ﺑﺮاﺑﺮ ﺑﺎ S = 21 ab = 41 b 2ﺧﻮاﻫﺪ ﺑﻮد ،ﻳﻌﻨﻲ ﻣﺴﺎﺣﺖ ﻣﺜﻠﺚ ﺗﺎﺑﻌﻲ از aاﺳﺖ.
(16اﮔﺮ ﺑﻪ ازاي ) x=aورودي ﺗﺎﺑﻊ (fﺧﺮوﺟﻲ ﺗﺎﺑﻌﻲ ﺑﺮاﺑﺮ ﺑﺎ bﺑﺎﺷﺪ آن ﮔﺎه دارﻳﻢ y=f(a)=bﻳﻌﻨﻲ ﻣﻘﺪار ﺗﺎﺑﻊ fدر aﺑﺮاﺑﺮ ﺑﺎ bاﺳﺖ.
f (3 ) = 2
f (5 ) = 2
f (6 ) = π
(17
} Df = {x :(x , y ) ∈ f
و ﺑﻪ ﻋﺒﺎرﺗﻲ اﮔﺮ
f :A → B
(19اﮔﺮ
B
2
2
π
3
5
6
ﺗﻌﺮﻳﻒ ﺷﻮد A .داﻣﻨﻪ ﺗﺎﺑﻊ fﺧﻮاﻫﺪ ﺑﻮد.
(18اﮔﺮ fﻳﻚ ﺗﺎﺑﻊ ﮔﻮﻳﺎ ﺑﺎﺷﺪ ﻳﻌﻨﻲ )) f (x ) = qp((xxدر اﻳﻦ ﺻﻮرت
) f = 2n p (x
A
}D f = R − {x :q (x ) = 0
و ) p (xﻳﻚ ﭼﻨﺪ ﺟﻤﻠﻪاي ﺑﺎﺷﺪ.
}Df = R − {x : p (x ) ≥ 0
128ﺟﺰوه ﻛﺎر در ﻛﻼس ﺣﺴﺎب و دﻳﻔﺮاﻧﺴﻴﻞ و اﻧﺘﮕﺮال و رﻳﺎﺿﻲ ﭘﺎﻳﻪ
(20اﻟﻒ :ﺑﺮد ﺗﺎﺑﻊ ﻋﺒﺎرﺗﺴﺖ از ﺗﻐﻴﻴﺮات yﺑﻪ ازاي ﺗﻐﻴﻴﺮات ، xﻳﻌﻨﻲ
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} Rf = {y :(x , y ) ∈ f
ب :ﺑﺮد ﺗﺎﺑﻊ fﻋﺒﺎرﺗﺴﺖ از ﺗﺼﻮﻳﺮ fروي ﻣﺤﻮر yﻫﺎ
ج :در ﻧﻤﻮدار ون :ﻣﺠﻤﻮﻋﻪ اﻋﺪادي ﻛﻪ ﭘﻴﻜﺎن ﺑﻪ آنﻫﺎ ﻧﻈﻴﺮ ﺷﺪه ﺑﺮد ﺗﺎﺑﻊ اﺳﺖ.
(21اﻟﻒ :اﻧﺘﻘﺎل اﻓﻘﻲ :اﮔﺮ
a >0
ﻧﻤﻮدار
ﻫﻤﺎن ﻧﻤﻮدار fاﺳﺖ ﻛﻪ aواﺣﺪ ﺑﻪ راﺳﺖ اﻧﺘﻘـﺎل داده ﺷـﺪه و ) y=f(x+aﻫﻤـﺎن
) y = f (x − a
ﻧﻤﻮدار fاﺳﺖ ﻛﻪ aواﺣﺪ ﺑﻪ ﭼﭗ اﻧﺘﻘﺎل داده ﺷﺪه اﺳﺖ.
ب :اﻧﺘﻘﺎل ﻋﻤﻮدي :ﻧﻤﻮدار y = f (x ) + kﻫﻤﺎن ﻧﻤﻮدار fاﺳﺖ ﻛﻪ kواﺣﺪ در راﺳﺘﺎي ﻣﺤﻮر yﻫﺎ ﺟﺎﺑﺠﺎ ﺷـﺪه اﺳـﺖ .اﮔـﺮ
ﺑﺎﻻ و اﮔﺮ
k <0
k >0
ﺑﺎﺷـﺪ ﺑـﻪ ﻃـﺮف
ﺑﻪ ﻃﺮف ﭘﺎﻳﻴﻦ اﻧﺘﻘﺎل داده ﻣﻲﺷﻮد.
ج .اﻧﺒــﺴﺎط و اﻧﻘﺒــﺎض اﻓﻘــﻲ :در ﻧﻤــﻮدار ) ، y = f (kxﻧــﺴﺒﺖ ﺑــﻪ ﻧﻤــﻮدار fﻓــﺸﺮدهﺗــﺮ ﻳــﺎ ﻛــﺸﻴﺪهﺗــﺮ اﺳــﺖ ،اﮔــﺮ
k >1
ﻧﻤــﻮدار ﻓــﺸﺮدهﺗــﺮ و اﮔــﺮ
0 < k < 1ﻧﻤﻮدار ﻛﺸﻴﺪهﺗﺮ اﺳﺖ.
د( اﻧﺒﺴﺎط و اﻧﻘﺒﺎض ﻋﻤﻮدي :ﻧﻤﻮدار ) ، y = kf (xﺑﻪ ﻧﻤﻮدار ،fﻓﺸﺮدهﺗﺮ اﺳﺖ اﮔﺮ 0 < k < 1و ﻛﺸﻴﺪهﺗﺮ اﺳﺖ اﮔﺮ
1 <l =k
ه( ) ، y = − f (xﻗﺮﻳﻨﻪ fﻧﺴﺒﺖ ﺑﻪ ﻣﺤﻮر xﻫﺎﺳﺖ و ) ، y = f (−xﻗﺮﻳﻨﻪي ﻧﻤﻮدار fﻧﺴﺒﺖ ﺑﻪ ﻣﺤﻮر yﻫﺎﺳﺖ.
(22ﺷﺮط اﻳﻨﻜﻪ دو ﺗﺎﺑﻊ ) y = f (xو
) y = g (x
ﺑﺎ ﻫﻢ ﺑﺮاﺑﺮ ﺑﺎﺷﻨﺪ اﻳﻦ اﺳﺖ ﻛﻪ اوﻻً داﻣﻨﻪﻫﺎ ﺑﺎ ﻫﻢ ﺑﺮاﺑﺮ ﺑﺎﺷﻨﺪ و ﺛﺎﻧﻴﺎً ﺑﻪ ازاي ﻫﺮ xاز داﻣﻨﻪﻫـﺎ
) f (x ) = g (x
(23اﮔﺮ ) f(xو ) g(xداﻣﻨﻪ ﻣﺸﺘﺮﻛﻲ داﺷﺘﻪ ﺑﺎﺷﻨﺪ آن ﮔﺎه:
( f + g )(x ) = f (x ) + g (x ), Df +g = Df ∩ Dg
( f − g )(x ) = f (x ) − g (x ), Df −g = Df ∩ Dg
( f × g )(x ) = f (x ) × g (x ) = Df ×g = Df ∩ Dg
) f (x
f
f
= ) ( )(x
}, D = Df ∩ Dg − {x : g (x ) = 0
g
) g (x
g
(24اﻟﻒ :ﺗﺮﻛﻴﺐ دو ﺗﺎﺑﻊ fو gرا ﺑﺎ fogﻧﻤﺎﻳﺶ ﻣﻲدﻫﻨﺪ و )) ، fog(x)=f(g(xﻳﻌﻨﻲ اﺑﺘـﺪا x ،وارد gو ﺧﺮوﺟـﻲ آن ) g(xاﺳـﺖ ،ﺳـﭙﺲ
) g(xوارد fﻣﻲﺷﻮد و ﺧﺮوﺟﻲ آن )) f(g(xاﺳﺖ.
ب:
)) x → g → g (x ) → f → f (g (x
ج:
} Dfog = {x : x ∈ Dg , g (x ) ∈ D f
(25ﻓﺮض ﻛﻨﻴﺪ fﺗﺎﺑﻌﻲ ﺑﺎ داﻣﻨﻪ ﻣﺘﻘﺎرن اﺳﺖ ﻳﻌﻨﻲ اﮔﺮ
اﻟﻒ :اﮔﺮ ﺑﺮاي ﻫﺮ
x ∈ Df
x ∈Df
آن ﮔﺎه
−X ∈ Df
در اﻳﻦ ﺻﻮرت:
و ) f (−x ) = f (xآن ﮔﺎه fزوج اﺳﺖ.
ب :اﮔﺮ ﺑﺮاي ﻫﺮ f (−x ) = − f (x ) ، x ∈Dfآن ﮔﺎه fﻓﺮد اﺳﺖ.
(26ﺗﻮاﺑﻊ زﻳﺮ ،ﺗﻮاﺑﻊ ﻓﺮدﻧﺪ
ﭼﻨﺪ ﺟﻤﻠﻪاي از درﺟﻪ ﻓﺮد
2n −1
) f (x ) = a1x + a3 + a3x 3 + ... + a2xn −1اﻟﻒ
) ) y = tan 2n −1 (axب
ﻓﺼﻞ :2رﻳﺎﺿﻲ ﭘﺎﻳﻪ 129
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) ) y = sin 2n −1 (axج
n −1
) y = cot 2(axد
)
) y = a x − a − xه
| ) y =| x −a | − | x + aو
) ) y = sgn(xز
bx
) y = log aa +− bxح
(27ﺗﻮاﺑﻊ زﻳﺮ زوجاﻧﺪ:
) y = a0 + a 2x 2 + ... + a2n x 2n + ...اﻟﻒ
ﭼﻨﺪ ﺟﻤﻠﻪاي از درﺟﻪ زوج
) ) y = sin 2n (axب
) ) y = cosn (axج
) ) y = tg 2n (axد
) ) y = cot g 2n (axه
) y = a x + a −xو
| ) y = y =| x − a | + | x + aز
) y = x sgn xح
| ) y =| xط
(28اﻟﻒ :ﻧﻤﻮدار ﺗﻮاﺑﻊ زوج ﻧﺴﺒﺖ ﺑﻪ ﻣﺤﻮر yﻫﺎ ﻣﺘﻘﺎرناﻧﺪ )ﻣﺤﻮر yﻫﺎ ﻣﺤﻮر ﺗﻘﺎرن ﺗﺎﺑﻊ ﻓﺮد اﺳﺖ(
ب :ﻧﻤﻮدار ﺗﺎﺑﻊ ﻓﺮد ﻧﺴﺒﺖ ﺑﻪ ﻣﺒﺪاء ﻣﺨﺘﺼﺎت ﻣﺘﻘﺎرن اﺳﺖ )ﻣﺒﺪاء ﻣﺨﺘﺼﺎت ،ﻣﺮﻛﺰ ﺗﻘﺎرن ﺗﺎرﺑﻊ ﻓﺮد اﺳﺖ.
ج :ﺗﻨﻬﺎ ﺗﺎﺑﻊ ﻫﻢ زوج و ﻫﻢ ﻓﺮد ،ﺗﺎﺑﻊ
f (x ) = 0
(29اﮔﺮ fو gدو ﺗﺎﺑﻊ ﻏﻴﺮﺻﻔﺮ ﺑﺎﺷﻨﺪ آن ﮔﺎه
اﻟﻒ :اﮔﺮ fو gﻫﺮ دو ﻓﺮد ﺑﺎﺷﻨﺪ آن ﮔﺎه f ± gﻓﺮد و
ب :اﮔﺮ fو gﻫﺮ دو زوج ﺑﺎﺷﻨﺪ آن ﮔﺎه
f +g
و
fg
f −g
و gfزوجاﻧﺪ و fogﻓﺮد اﺳﺖ.
و fgو
f
g
ج( اﮔﺮ fزوج و gﻓﺮد ﺑﺎﺷﻨﺪ آن ﮔﺎه f ± gﻧﻪ زوج و ﻧﻪ ﻓﺮد Fg .و
د( اﮔﺮ ) f(xﺗﺎﺑﻊ زوج ﺑﺎﺷﻨﺪ آن ﮔﺎه
) f n (x
ﻧﻴﺰ زوج اﺳﺖ.
ﻫـ( اﮔﺮ ) f(xﺗﺎﺑﻊ ﻓﺮد ﺑﺎﺷﺪ آن ﮔﺎه
) f 2n (x
f(2xn)−1
زوج و
زوجاﻧﺪ و fogزوج اﺳﺖ.
f
g
ﻓﺮد اﺳﺖ و fogزوج اﺳﺖ.
ﻓﺮد اﺳﺖ.
(30ﻫﺮ ﺗﺎﺑﻊ ﻣﺎﻧﻨﺪ ) f(xﻛﻪ داﻣﻨﻪ ﻣﺘﻘﺎرن دارد را ﻣﻲﺗﻮان ﺑﻪ ﺻﻮرت ﻣﺠﻤﻮع دو ﺗﺎﺑﻊ زوج و ﻓﺮد ﻧﻤﺎﻳﺶ داد.
1
1
) f (x ) = ( f (x ) + f (−x )) + ( f (x ) − f (−x
2
2
ﺗـــﺎﺑﻊ ﻓـــﺮد
ﺗـــــــــﺎﺑﻊ زوج
130ﺟﺰوه ﻛﺎر در ﻛﻼس ﺣﺴﺎب و دﻳﻔﺮاﻧﺴﻴﻞ و اﻧﺘﮕﺮال و رﻳﺎﺿﻲ ﭘﺎﻳﻪ
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(31اﻟﻒ :اﮔﺮ fﻳﻚ ﺗﺎﺑﻊ ﺑﺎﺷﺪ ﺑﻪ ﺷﺮﻃﻲ ﺗﺎﺑﻊ fﻳﻚ ﺑﻪ ﻳﻚ اﺳﺖ ﻛﻪ از ﻫﺮ ﻋﻀﻮ ﻣﺠﻤﻮﻋﻪي ﻣﺒﺪاء ﺑﻪ ﻫﺮ ﻋﻀﻮ ﻣﺠﻤﻮﻋﻪي ﻣﻘﺼﺪ ﻓﻘﻂ ﻳﻚ ﭘﻴﻜﺎن ﻧﻈﻴﺮ
ﺑﺎﺷﺪ.
ب f :ﻳﻚ ﺑﻪ ﻳﻚ اﺳﺖ ﻫﺮ ﮔﺎه در زوجﻫﺎي ﻣﺮﺗﺐ ﺑﻪ ﺷﺮﻃﻲ ﻛﻪ ﻣﺆﻟﻔﻪﻫﺎي دوم ﺗﻜﺮاري ﻧﺒﺎﺷﻨﺪ ﺗﺎﺑﻊ ﻳﻚ ﺑﻪ ﻳﻚ اﺳﺖ.
ج( در ﻧﻤﻮدار :ﺑﻪ ازاي ﻫﺮ ﺧﻂ رﺳﻢ ﺷﺪه ﻣﻮازي ﻣﺤﻮر xﻫﺎ ،ﺗﺎﺑﻊ ﺣﺪاﻛﺜﺮ در ﻳﻚ ﻧﻘﻄﻪ ﻗﻄﻊ ﺷﻮد.
(32اﮔﺮ ) y = f (xﻳﻚ ﺗﺎﺑﻊ ﺑﺎﺷﺪ آن ﮔﺎه ﺷﺮط اﻳﻨﻜﻪ ﻳﻚ ﺑﻪ ﻳﻚ ﺑﺎﺷﻨﺪ اﻳﻦ اﺳﺖ ﻛﻪ
f (x 1 ) = f (x 2 ) ⇒ x 1 = x 2
(33ﺗﺎﺑﻊ زوج روي داﻣﻨﻪي ﺧﻮد ﺗﺎﺑﻌﻲ اﺳﺖ ﻏﻴﺮ ﻳﻚ ﺑﻪ ﻳﻚ ﻣﮕﺮ آن ﻛﻪ }) f = {(0,aﺑﺎﺷﺪ.
(34اﮔﺮ
x =a
ﻣﺤﻮر ﺗﻘﺎرن ﺗﺎﺑﻊ
) y = f (x
ﺑﺎﺷﺪ آن ﮔﺎه ﺗﺎﺑﻊ در داﻣﻨﻪي ﺧﻮد ﻳﻚ ﺑﻪ ﻳﻚ ﻧﺴﺒﺖ ﺑﻪ ﺟﺰء
}) f = {(0,a
(35اﮔﺮ fﻳﻚ ﺑﻪ ﻳﻚ ﻧﺒﺎﺷﺪ ﻣﻌﻜﻮس ﻧﺎﭘﺬﻳﺮ اﺳﺖ ،ﭘﺲ در ﻧﻤﻮدار ﺗﻮاﺑﻊ :ﺗﺎﺑﻌﻲ ﻛﻪ ﻳﻚ ﺑﻪ ﻳﻚ ﻧﺒﺎﺷﺪ ﻣﻌﻜﻮس ﻧﺎﭘﺬﻳﺮ اﺳـﺖ) ،آزﻣـﻮن ﺧـﻂ اﻓﻘـﻲ را
ﺑﺰﻧﻴﺪ(
(36اﻟﻒ :ﺷﺮط ﻻزم و ﻛﺎﻓﻲ ﺑﺮاي آن ﻛﻪ fﻣﻌﻜﻮسﭘﺬﻳﺮ ﺑﺎﺷﺪ آن اﺳﺖ ﻛﻪ ﻳﻚ ﺑﻪ ﻳﻚ ﺑﺎﺷﺪ.
ب :اﮔﺮ fﺗﺎﺑﻌﻲ ﭘﻴﻮﺳﺘﻪ و اﻛﻴﺪاً ﻳﻜﻨﻮا ﺑﺎﺷﺪ ،ﻣﻌﻜﻮسﭘﺬﻳﺮ اﺳﺖ.
ج :ﺗﺎﺑﻌﻲ ﻛﻪ داراي ﻣﺤﻮر ﺗﻘﺎرن
د(
x =a
ﺑﺎﺷﺪ در داﻣﻨﻪ ﺧﻮد ﻣﻌﻤﻮﻻً ﻣﻌﻜﻮسﭘﺬﻳﺮ ﻧﻴﺴﺖ.
f f−1 = Rf−1 , Df = Rf−1
ه( ﺗﻮاﺑﻊ ﭼﻨﺪ ﺟﻤﻠﻪاي در ﺻﻮرﺗﻲ ﻛﻪ داراي اﻛﺴﺘﺮﻣﻢ ﺑﺎﺷﻨﺪ در Rﻣﻌﻜﻮسﭘﺬﻳﺮ ﻧﻴﺴﺘﻨﺪ.
(37در ﺗﻮاﺑﻊ ﭼﻨﺪ ﺿﺎﺑﻄﻪاي ﻣﺎﻧﻨﺪ
g 1 (x ): x ∈ D1
f (x ) =
g 2 (x ): x ∈D2
در ﺻﻮرﺗﻲ وارون ﭘﺬﻳﺮﻧﺪ ﻛﻪ
اﻟﻒ :ﻫﺮ ﻳﻚ از ﺿﺎﺑﻄﻪﻫﺎ در داﻣﻨﻪ ﺗﻌﺮﻳﻒ ﺧﻮد وارونﭘﺬﻳﺮ ﺑﺎﺷﺪ.
ب:
Rf 1 ∩ R f 2 = φ
ﺗﺬﻛﺮ :اﮔﺮ ﺷﺮاﻳﻂ اﻟﻒ و ب ﺑﺮﻗﺮار ﺑﺎﺷﺪ ،ﺿﺎﺑﻄﻪي ﺗﺎﺑﻊ ﻣﻌﻜﻮس ﺑﻪ ﺻﻮرت
g 1−1 (x ): x ∈Rg 1
f −1 (x ) =
g 2−1 (x ): x ∈Rg 2
(38اﮔﺮ ﺗﺎﺑﻊ fﻣﻌﻜﻮسﭘﺬﻳﺮ و ﻓﺮد ﺑﺎﺷﺪ آن ﮔﺎه f −1ﻧﻴﺰ ﻓﺮد اﺳﺖ.
(39ﺗﺎﺑﻊ | f (x ) =| ax + bﻳﻚ ﺑﻪ ﻳﻚ ﻧﻴﺴﺖ ﻣﺜﺎل f (x ) =| x − 1 | :ﻳﻚ ﺑﻪ ﻳﻚ ﻧﻴﺴﺖ.
(40ﻧﻤﻮدار ﺗﺎﺑﻊ fو f −1از ﻧﻈﺮ ﻫﻨﺪﺳﻲ ﻧﺴﺒﺖ ﺑﻪ ﺧﻂ
(41اﮔﺮ
اﻟﻒ:
f −1
y =x
ﻣﺘﻘﺎرناﻧﺪ.
وارون ﺗﺎﺑﻊ ﺑﺎﺷﺪ آن ﮔﺎه
A = (a ,b ) ∈ f
آن ﮔﺎه
A′(b ,a ) ∈ f −1
ب :اﮔﺮ fﺻﻌﻮدي اﻛﻴﺪ و fو f −1ﻣﺘﻘﺎﻃﻊ ﺑﺎﺷﻨﺪ آن ﮔﺎه ﻧﻘﻄﻪ ﺗﻘﺎﻃﻊ روي ﺧﻂ
y =x
اﺳﺖ و ﺑﺮاي ﭘﻴﺪا ﻛﺮدن اﻳﻦ ﻧﻘﻄـﻪ ﻛـﺎﻓﻲ اﺳـﺖ ﻛـﻪ ﻣﻌﺎدﻟـﻪ
f (x ) = xﻧﺒﺎﺷﺪ.
ج :ﻗﺎﺑﻞ ﺗﺬﻛﺮ اﺳﺖ ﻛﻪ اﮔﺮ fﻧﺰوﻟﻲ اﻛﻴﺪ ﺑﺎﺷﺪ ،ﻣﻤﻜﻦ اﺳﺖ ﻧﻘﻄﻪ ﺑﺮﺧﻮرد fﺑﺎ f −1روي y=xﻧﺒﺎﺷﺪ
ﻓﺼﻞ :2رﻳﺎﺿﻲ ﭘﺎﻳﻪ 131
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د:
fof(−x1) = x : x ∈Df , fof(−x1) = x : x ∈Rf
(42اﮔﺮ دو ﺗﺎﺑﻊ fو gﻣﻌﻜﻮسﭘﺬﻳﺮ ﺑﺎﺷﻨﺪ آن ﮔﺎه
( fog ) −1 = g −1of −1
ax + b
، f (x ) = cxاﮔﺮ a + d = 0آن ﮔﺎه
(43در ﺗﻮاﺑﻊ ﻫﻤﻮﮔﺮاﻓﻴﻚ
+d
و ﻗﺎﺑﻞ ﺗﻌﻤﻴﻢ ﻧﻴﺰ اﺳﺖ.
) f(−x1) = f(x
ﻣﺠﻤﻮﻋﻪ ﺗﺴﺖﻫﺎي ﺗﺎﺑﻊ
.1در ﺻﻮرﺗﻲ ﻛﻪ دو زوج ﻣﺮﺗﺐ
1 (1
) (x − 2 , y + 3
و ) (6 − x , 5 − yﺑﺎ ﻫﻢ ﺑﺮاﺑﺮ ﺑﺎﺷﻨﺪ x + y ،ﻛﺪام اﺳﺖ؟
3 (2
.2اﮔﺮ ﻣﺠﻤﻮﻋﻪ }) {(−1, 2 ),(a ,b + 1),(c ,d − 1ﻓﻘﻂ ﻳﻚ ﻋﻀﻮ داﺷﺘﻪ ﺑﺎﺷﺪ
- 2 (1
2 (2
4 (3
? = a +b +c +d
4 (3
.3در ﺻﻮرﺗﻲ ﻛﻪ راﺑﻄﻪي }) {(a , 1),(5, m ),(3, 4 ),(3,a − 1),(2, m + 1ﻳﻚ ﺗﺎﺑﻊ ﺑﺎﺷﺪ
1 (1
- 1 (2
5 (4
(3ﺻﻔﺮ
6 (4
?= m
(4ﺑﺮاي ﻫﺮ ﻣﻘﺪار
.4راﺑﻄﻪي }) R = {(3, m 2 ),(2, 1),(−3, m ),(−2, m ),(3, m + 2 ),(m , 4ﺑﻪ ازاي ﻛﺪام ﻣﻘﺪار ،mﻳﻚ ﺗﺎﺑﻊ اﺳﺖ؟
- 2 (1
- 1 (2
2 (3
(4ﻫﻴﭻ ﻣﻘﺪار m
132ﺟﺰوه ﻛﺎر در ﻛﻼس ﺣﺴﺎب و دﻳﻔﺮاﻧﺴﻴﻞ و اﻧﺘﮕﺮال و رﻳﺎﺿﻲ ﭘﺎﻳﻪ
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.5ﻛﺪام ﻳﻚ ﺗﺎﺑﻊ اﺳﺖ؟
y
(1
x
0
3
1
1
(3
(2
32
}), 3
2
({(3, −1 )(4, 8 ),
1
a0
4
(4
4
-1
2
3
-1
X
-1
5
2
y
.6ﻛﺪام ﺗﺎﺑﻊ زﻳﺮ ﻫﻤﺎﻧﻲ اﺳﺖ؟
(1
y =x
(2
(3
y =x +1
(4
y = −x
y = 1 −x
.7ﻛﺪام ﺗﺎﺑﻊ زﻳﺮ ﻳﻚ ﺗﺎﺑﻊ ﺧﻄﻲ اﺳﺖ؟
(1
1
x
=y
(2
4x 3 + 4x + x 2 + 1
x2 +1
=y
(3
2x 2 − 1
x −1
=y
(4
y= x
.8ﻛﺪام ﺗﺎﺑﻊ زﻳﺮ ﻳﻚ ﺗﺎﺑﻊ ﮔﻮﻳﺎ ﻧﻴﺴﺖ؟
(1
x −1
x +1
=y
(2
y= x
(3
y =x2
(4
y =x
.9در ﻛﺪام راﺑﻄﻪ y ،ﺗﺎﺑﻌﻲ از xاﺳﺖ؟
(1
x 2 − x − xy = 0
(2
x + y +1 =y
(3
x y
+ +2 =0
y x
(4
| x | + | y |= 1
.10درﻛﺪام ﮔﺰﻳﻨﻪ ، yﺗﺎﺑﻊ xاﺳﺖ؟
(1
x + y +2 =y
(2
x = y 3 − 4y + 1
(3
x =| 2y + 1 | +y
(4
| x = y3 +y + |y
ﻓﺼﻞ :2رﻳﺎﺿﻲ ﭘﺎﻳﻪ 133
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.11در ﻛﺪام ﮔﺰﻳﻨﻪ xﺗﺎﺑﻊ yﻧﻴﺴﺖ:
(1
(2
xy = 2
(x − 1 )2 + y 2 = 0
(3
x 2 + 4xy + 4y 2 = 1
(4
x = 1 − sin y
.12ﻃﻮل ) (xﻳﻚ ﻣﺴﺘﻄﻴﻞ 3واﺣﺪ ﺑﻴﺸﺘﺮ از ﻋﺮض ) (yآن اﺳﺖ اﮔﺮ ﻣﺤﻴﻂ ﻣﺴﺘﻄﻴﻞ را ﺑـﺎ Pﻧﻤـﺎﻳﺶ دﻫـﻴﻢ ﻛـﺪام راﺑﻄـﻪ زﻳـﺮ ﻣﺤـﻴﻂ را ﺑﺮﺣـﺴﺐ
ﻋﺮض ﺑﻴﺎن ﻣﻲﻛﻨﺪ؟
(1
P (y ) = 2y + 3
(2
P (y ) = 2y − 3
(3
P (y ) = 4y + 6
(4
P (y ) = 4y − 6
.13داﻳﺮهاي ﺑﻪ ﺷﻌﺎع rﻣﻔﺮوض اﺳﺖ اﮔﺮ ﻣﺤﻴﻂ ﻣﺮﺑﻌﻲ ﺑﻪ ﺿﻠﻊ xﺑﺎ ﻣﺤﻴﻂ داﻳﺮه ﺑﺮاﺑﺮ ﺑﺎﺷﺪ ﻣﺴﺎﺣﺖ ﻣﺮﺑﻊ ﺑﺮﺣﺴﺐ rﻛﺪام اﺳﺖ؟
(1
π2 2
r
4
(2
π2 2
r
2
(3
π 2r 2
(4
4π r 2
.14ﺧﻄﻲ از ﻧﻘﻄﻪي ) A = (3, 2ﻋﺒﻮر ﻣﻲﻛﻨﺪ و ﺷﻴﺐ آن ﻣﻨﻔﻲ اﺳﺖ و ﻣﺤﻮر xﻫﺎ و ﻣﺤﻮر yﻫﺎ را ﺑﻪ ﺗﺮﺗﻴﺐ در ﻧﻘﺎط ) B = (a , 0و ) C = (0, Pﻗﻄﻊ
ﻣﻲﻛﻨﺪ و ﺑﺎ ﻣﺤﻮرﻫﺎي ﻣﺨﺘﺼﺎت ﻣﺜﻠﺚ ﻗﺎﺋﻢاﻟﺰاوﻳﻪاي ﻣﻲﺳﺎزد ،ﻣﺴﺎﺣﺖ ﻣﺜﻠﺚ ﺑﺮﺣﺴﺐ Pﻛﺪام اﺳﺖ؟
(1
3P 2
P −2
.15در ﺻﻮرﺗﻲ ﻛﻪ
- 1 (1
(2
3P 2
2P − 4
}) f = {(1, −1 ),(−1, 5 ),(3, 1
4 (2
ﻣﻘﺪار ﻋﺪدي
(3
3P 2
2P + 4
)) f ( f (3 )) + f ( f (1
5 (3
(4
3P 2
P −2
ﻛﺪام اﺳﺖ؟
(4ﺻﻔﺮ
134ﺟﺰوه ﻛﺎر در ﻛﻼس ﺣﺴﺎب و دﻳﻔﺮاﻧﺴﻴﻞ و اﻧﺘﮕﺮال و رﻳﺎﺿﻲ ﭘﺎﻳﻪ
ax + 3 : x ≤ 1
f (x ) =
b[−x ] + a : x ≥ 1
.16اﮔﺮ
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ﻧﻤﺎﻳﺶ ﻳﻚ ﺗﺎﺑﻊ ﺑﺎﺷﺪ ﻣﻘﺪار bﻛﺪام اﺳﺖ؟
- 2 (2
2 (1
.17ﻧﻤﻮدار ) f(xﺑﻪ ﺻﻮرت ﻣﻘﺎﺑﻞ اﺳﺖ ،ﺣﺎﺻﻞ
(1ﺻﻔﺮ
2 (2
4 (3
1 (4
-1 (4
1 (3
? = ) f (−1 ) + f (1
y
2
x
.18داﻣﻨﻪ ﺗﺎﺑﻊ
y = −x 2 (x 2 − 4 )2
(1ﺻﻔﺮ
.19داﻣﻨﻪ ﺗﺎﺑﻊ
17 (1
.20داﻣﻨﻪ ﺗﺎﺑﻊ
(1
(1
3 (3
1− | x |
1 + | x |
5 (3
4 (4
= f = (x ,y ): yﻛﺪام ﻣﺠﻤﻮﻋﻪ اﺳﺖ؟
(2
x ≤1
y = | x + 1 | + | x − 3 | −6
) R − (−2, 4
(4ﺑﻲﺷﻤﺎر
ﺷﺎﻣﻞ ﭼﻨﺪ ﻋﺪد ﺻﺤﻴﺢ اﺳﺖ؟
16 (2
IR
.21ﺗﻤﺎم داﻣﻨﻪ
ﭼﻨﺪ ﻋﻀﻮ دارد؟
1 (2
y = 4 − x +1
0
(2
(3
x ≥1
(4
| x |≤ 1
ﻛﺪام اﺳﺖ؟
] R − [−2, 4
(3
] [−2, 4
(4
) (−2, 4
−6
ﻓﺼﻞ :2رﻳﺎﺿﻲ ﭘﺎﻳﻪ 135
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.22داﻣﻨﻪي ﺗﺎﺑﻊ
) (x 2 − 1 )2 + 1
( y = Sin −1ﺷﺎﻣﻞ ﭼﻨﺪ ﻋﺪد ﺻﺤﻴﺢ ﻣﻲﺑﺎﺷﺪ؟
1 (2
(1ﺻﻔﺮ
.23داﻣﻨﻪي ﺗﺎﺑﻊ
2 (1
(3ﺻﻔﺮ
)] f (x ) = ([x ] − 2 )(3 − [x
(2
][ 2 , 3
−45
4
.26اﮔﺮ
(2
f (x ) = x 3 + 2x 2 + ax + b
35 (1
.27داﻣﻨﻪ ﺗﺎﺑﻊ
(1
و
(3
15
4
(3
و
f (1 ) = 0
32 (2
f (x ) = 4 2 sin x − 1
π , 5π
6 6
) [2, 4
(4
] [1, 3
ﭼﻘﺪر اﺳﺖ؟
) f (2
f (2 ) = 1
(4ﺑﻲﺷﻤﺎر
ﻛﺪام اﺳﺖ؟
) [1, 3
.25اﮔﺮ ، f ( x1 ) − 2 f (x ) = x 2 + x1ﺣﺎﺻﻞ
(1
ﺷﺎﻣﻞ ﭼﻨﺪ ﻋﻀﻮ اﺳﺖ؟
1 (2
.24داﻣﻨﻪي ﺗﺎﺑﻊ
(1
2 (3
) y = sin −1 (x 2 + x − 3 ) + sin −1 (x 2 + x − 5
4 (4
،ﻣﻘﺪار
−15
4
) f (−2
(4
45
4
ﻛﺪام اﺳﺖ؟
31 (3
33 (4
در ﻳﻚ ﺳﻴﻜﻞ ﻣﺜﻠﺜﺎﺗﻲ ﻛﺪام اﺳﺖ؟
(2
π , 2π
3 3
(3
π , 2π
6
(4
π , 2π
3
136ﺟﺰوه ﻛﺎر در ﻛﻼس ﺣﺴﺎب و دﻳﻔﺮاﻧﺴﻴﻞ و اﻧﺘﮕﺮال و رﻳﺎﺿﻲ ﭘﺎﻳﻪ
.28اﮔﺮ ﺑﺎ ﺷﺮط x < 6داﻣﻨﻪي ﺗﺎﺑﻊ
f (x ) = x 3 − 6x 2 − 2x + 12
(2
2 (1
.29اﮔﺮ }) ، f = {(1, 2 ),(3, −1ﺑﺮد ﺗﺎﺑﻊ
(1
.30اﮔﺮ ﺑﺮد ﺗﺎﺑﻊ f (x ) = x 2 + 1را
(1
(1
(2
] [0 , 3
.31ﺑﺮد ﺗﺎﺑﻊ
f (x ) = x + 4
(1ﺻﻔﺮ
1
2x − 1
= ) f (x
(4
2 2
{ − 32 ,3 } (3
(4
}{3
در ﻧﻈﺮ ﺑﮕﻴﺮﻳﻢ آن ﮔﺎه ﺑﺰرﮔﺘﺮﻳﻦ داﻣﻨﻪي آن ﻛﺪام اﺳﺖ؟
] [−3, 3
(3
) (−3, 3
(4
) [0 , 3
ﻛﺪام اﺳﺖ؟
(2
)∞[0 , +
.32ﺑﺮد ﺗﺎﺑﻊ
) [1, 10
) (b − a
ﻛﺪام اﺳﺖ؟
ﻛﺪام اﺳﺖ؟
{ − 32 , − 21 } (2
}{2, −1
ﺑﻪ ﺻﻮرت
] [a ,b
ﺑﺎﺷﺪ ،ﺣﺪاﻛﺜﺮ ﻣﻘﺪار
4 (3
2
−3
f
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] (−∞, 4
(3
)∞[−4, +
(4
) (−∞, −4
ﭼﻨﺪ ﻋﺪد ﺻﺤﻴﺢ را ﺷﺎﻣﻞ ﻧﻤﻲﺷﻮد؟
1 (2
2 (3
(4ﺑﻲﺷﻤﺎر
ﻓﺼﻞ :2رﻳﺎﺿﻲ ﭘﺎﻳﻪ 137
saati.com
1
x +1
.33ﺑﺮد ﺗﺎﺑﻊ ﺑﺎ ﺿﺎﺑﻄﻪي
1 (1
2 (2
(2
] [0, 4
.35ﺑﺮد ﺗﺎﺑﻊ ﺑﺎ ﺿﺎﺑﻄﻪ
(1
x
1 +x2
1 1
] [− ,
2 2
.36اﮔﺮ ﻧﻤﻮدار
5 (3
f (x ) = 4 − x 2
.34ﺑﺮد ﺗﺎﺑﻊ ﺑﺎ ﺿﺎﺑﻄﻪاي
(1
y =x +
ﺷﺎﻣﻞ ﭼﻨﺪ ﻋﺪد ﺻﺤﻴﺢ ﻧﻴﺴﺖ؟
ﻛﺪام اﺳﺖ؟
(3
] [0 , 2
(4
] [0 , 1
= ) f (x
(3
] [−1, 1
ﺑﻪ ﺻﻮرت ﺷﻜﻞ ﻣﻘﺎﺑﻞ ﺑﺎﺷﺪ ،ﻧﻤﻮدار
(4
)∞[2, +
) y = −2 f (1 − 2x
(1
x
(3
x
0
y
(2
x
0
y
y
.37اﮔﺮ داﻣﻨﻪي ﺗﺎﺑﻊ fﺑﺮاﺑﺮ
4 (1
.38اﮔﺮ ﺑﺮد ﺗﺎﺑﻊ fﺑﺮاﺑﺮ
5 (1
R
ﺷﺒﻴﻪ ﻛﺪام اﺳﺖ؟
y
0
] [1, 2
ﻛﺪام اﺳﺖ؟
(2
) y = f (x
(4ﺑﻲﺷﻤﺎر
(4
] = [−2, 6
x
0
Dfﺑﺎﺷﺪ ،داﻣﻨﻪي ﻧﻤﻮدار ) 3 f (2x + 1ﺷﺎﻣﻞ ﭼﻨﺪ ﻋﺪد ﺻﺤﻴﺢ اﺳﺖ؟
17 (2
] Rf = [ − 3 , 2
3 (2
9 (3
ﺑﺎﺷﺪ ،ﺑﺮد ﺗﺎﺑﻊ
2 f (x − 1 ) + 1
25 (4
ﺷﺎﻣﻞ ﭼﻨﺪ ﻋﺪد ﺻﺤﻴﺢ اﺳﺖ؟
2 (3
4 (4
138ﺟﺰوه ﻛﺎر در ﻛﻼس ﺣﺴﺎب و دﻳﻔﺮاﻧﺴﻴﻞ و اﻧﺘﮕﺮال و رﻳﺎﺿﻲ ﭘﺎﻳﻪ
x
.39ﻧﻤﻮدار | −2
2
|= y
saati.com
را 4واﺣﺪ ﺑﻪ ﻃﺮف xﻫﺎي ﻣﻨﻔﻲ و ﻳﻚ واﺣﺪ ﺑﻪ ﻃﺮف yﻫﺎي ﻣﺜﺒﺖ اﻧﺘﻘﺎل ﻣﻲدﻫﻴﻢ ،ﻧﻤﻮدار ﺟﺪﻳﺪ ﻧﻤـﻮدار اوﻟﻴـﻪ ﺑـﺎ ﻛـﺪام
ﻃﻮل ﻣﺘﻘﺎﻃﻊاﻧﺪ) .ﺗﺠﺮﺑﻲ (93
- 3 (2
- 3 / 5 (1
-2 (4
-2/5 (3
.40دو ﺗﺎﺑﻊ fو gﻣﻔﺮوضاﻧﺪ ،در ﻛﺪام ﮔﺰﻳﻨﻪ دو ﺗﺎﺑﻊ ﻣﺴﺎوياﻧﺪ؟
(1
f (x ) = 2 log x , g (x ) = log x 2
(2
x2
, g (x ) = 1
| |x
(3
f (x ) = ( x )2 , g (x ) = x
(4
| |x
x
= ) , g (x
| |x
x
.41ﻛﺪام ﻳﻚ از ﺗﻮاﺑﻊ زﻳﺮ ﺑﺎ ﺗﺎﺑﻊ
(1
x2 −4
x +2
f (x ) = x + 2
(2
= ) g (x
= ) f (x
= ) f (x
ﺑﺮاﺑﺮ اﺳﺖ؟
k (x ) = 2 + x 2
(3
h (x ) = x 2 + 4x + 4
(4
x 3 + 2x 2 + x + 2
x2 +1
.42دو ﺗﺎﺑﻊ fو gﺑﻪ ﺻﻮرت ﻣﺠﻤﻮﻋﻪي زوجﻫﺎي ﻣﺮﺗﺐ ﺑﻴﺎن ﺷﺪهاﻧﺪ در ﺣﺎﻟﺖ ﻛﻠﻲ ﻛﺪام راﺑﻄﻪ ﻣﻤﻜﻦ اﺳﺖ ﺗﺎﺑﻊ ﻧﺒﺎﺷﺪ؟
(1
f ∪g
.43اﮔﺮ ﺗﻮاﺑﻊ fو gﺑﻪ ﺻﻮرت
(1
f (x − 3 ) = x 2 − 4x + 5
x2 +1
f ∩g
}) f = {(4, 2 ),(1, 2
}) {(4, 2 ),(1, 0
.44اﮔﺮ
(1
(2
(3
(2
) f (1 − x
x2 +3
f −g
}) g = {(1, 0 ),(4, 1 ),(2, 1
}) {(4, 1 ),(2, 1
ﺑﺎﺷﺪ آن ﮔﺎه
(2
و
(3
(4
ﺑﺎﺷﺪ ﺗﺎﺑﻊ
}) {(1, 2
f
g
fog
ﻛﺪام اﺳﺖ؟
(4
}) {(4, 2
ﻛﺪام اﺳﺖ؟ )ﺳﺮاﺳﺮي (90
(3
x 2 + 4x + 5
(4
x 2 − 4x + 5
= ) t (x
ﻓﺼﻞ :2رﻳﺎﺿﻲ ﭘﺎﻳﻪ 139
saati.com
.45اﮔﺮ
(1
f (x ) = x 2 + 3x
x
2 −x
ﺑﺎﺷﺪ داﻣﻨﻪي
(2
= ) f (x
f (x ) = cos x
73 − 40 3
) xf (x
= yﻛﺪام ﺑﺎزه اﺳﺖ؟ )ﺳﺮاﺳﺮي (93
(3
) (−∞, 0
)∞(−∞, +
(4
)∞(0, +
(−2, 1 ) (3
) (−3, 2
(4
) (4, −1
و ، gof (x ) = 21 xﺿﺎﺑﻄﻪي ﺗﺎﺑﻊ gﺑﺮاﺑﺮ ﻛﺪام اﺳﺖ؟
(2
x −1
x
(3
و ، (gof )(x ) = 1 + tg 2xآن ﮔﺎه
(2
.50اﮔﺮ ، f (x ) =| x | −xﺿﺎﺑﻄﻪي ﺗﺎﺑﻊ
x (1
x 2 − 4x + 5
(3
x 2 − 2x + 5
(4
x 2 − 2x + 3
و ، g (x ) = − 21 x + 2ﻣﺠﻤﻮﻋﻪي ﻃﻮل ﻧﻘﺎط در ﻣﻨﺤﻨﻲ ﺗﺎﺑﻊ gofﻛﻪ در ﺑﺎﻻي ﻣﺤﻮر xﻫﺎ ﻗﺮار ﮔﻴﺮﻧﺪ ﺑﺮاﺑﺮ ﻛﺪام ﺑﺎزه اﺳﺖ؟
(2
x
x +1
.49اﮔﺮ
(1
1
f (x ) = 1 − ( )x
2
) (−4, 1
.48اﮔﺮ
(1
(2
] [−1, 1
.47اﮔﺮ
(1
g (x ) = 2x − 3
x 2 − 4x + 2
.46اﮔﺮ
(1
و
) fog (x ) = 4 (x 2 − 4x + 5
ﺑﺎﺷﻨﺪ ﺗﺎﺑﻊ
) f (x
ﻛﺪام اﺳﺖ؟ )ﺳﺮاﺳﺮي (93
(2
) (gog )( 3 − 2
(3
5 −4 3
) ( fof )(x
| |x
x
x −1
(4
x +1
x
ﻛﺪام اﺳﺖ؟
69 − 30 3
(4
25 + 48 3
ﺑﺮاﺑﺮ ﻛﺪام اﺳﺖ؟
(3
| x + |x
(4ﺻﻔﺮ
140ﺟﺰوه ﻛﺎر در ﻛﻼس ﺣﺴﺎب و دﻳﻔﺮاﻧﺴﻴﻞ و اﻧﺘﮕﺮال و رﻳﺎﺿﻲ ﭘﺎﻳﻪ
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.51اﮔﺮ ﺧﺮوﺟﻲ ﻣﺎﺷﻴﻦ ﺷﻜﻞ ﻣﻘﺎﺑﻞ ﺑﺮاي ورودي ،2ﺑﺮاﺑﺮ -5ﺑﺎﺷﺪ Aﻛﺪام اﺳﺖ؟
ﺧﺮوﺟﻲ → 2x + A → x − 2x − 4
(1
−15
4
−3 (2
4
3
.52اﮔﺮ ﺧﻮرﺟﻲ از ﻣﺎﺷﻴﻦ ﺷﻜﻞ ﻣﻘﺎﺑﻞ
(4
3 (3
15
4
ﺑﺎﺷﺪ ﻣﻘﺪار ورودي ﻛﺪام اﺳﺖ؟
x
ﺧﺮوﺟﻲ →
x +1
(1
11
9
.53اﮔﺮ
(2
g (x ) = 2x − 1
و
x
x −3
- 4 (1
.54
7
2
= ) ( fog ) (x
3 (3
ﻣﻘﺪار
2 (2
ﺗﻮاﺑﻊ }) f = {(2, 1 ),(3, 2 ),(4, 5 ),(1, 7
)f (3
ﻛﺪام اﺳﺖ؟
-2 (3
و }) g = {(1, 2 ),(3, 1),(a , 3),(b , 1ﻣﻔﺮوضاﻧـﺪ اﮔـﺮ
.55اﮔﺮ
f (x 2 − 6x ) = x + 1
(1ﺻﻔﺮ
3) (2و(4.
ﺑﺎﺷﺪ
4 (4
(4, 2 ) ∈ fog
و
(4, 1 ) ∈gof
.56اﮔﺮ f (x 2 − x ) = (2x − 1)2آن ﮔﺎه
19 (1
5) (3و (4
4) (4و (5
? = ) f (−9
8 (2
10 (3
4 (4
? = ) f (5
20 (2
→ 2x − 2
→ ورودي
4 (4
ﻛﺪام اﺳﺖ؟
4 ) ( 1و (3
→ ورودي
21 (3
22 (4
ﺑﺎﺷـﺪ دو
ﺗـﺎﺋﻲ ) (a ,b
ﻓﺼﻞ :2رﻳﺎﺿﻲ ﭘﺎﻳﻪ 141
saati.com
1
1
) =x6 − 6
2
x
x
.57اﮔﺮ
f (x 2 −
x 3 − 3x
(1
ﺑﺎﺷﺪ آن ﮔﺎه
(2
.58اﮔﺮ | ، f (x 2 + 2x ) =| x + 1آن ﮔﺎه
11 (1
x 3 + 3x
(4
}A = {1, 2, 3, 4, 5
13 (3
و } ، f = {(x , 2x − 1), x ∈Aﺗﺎﺑﻊ
1 (1
)) f ( f (x
2 (2
f ( x + 1) = x + 2 x + 2
3 (1
x 3 − 3x + 1
(4
x 3 + 3x − 1
? = ) f (n ) + f (15 ) + f (35
12 (2
.59اﮔﺮ
.60اﮔﺮ
) f (x
ﺑﺮاﺑﺮ اﺳﺖ ﺑﺎ
x ≠0
آن ﮔﺎه
14 (4
ﭼﻨﺪ ﻋﻀﻮ دوﺗﺎﺋﻲ دارد؟
3 (3
)f( 2
4 (4
ﻛﺪام اﺳﺖ؟
5 (2
26 (4
10 (3
.61ﻧﻤﻮدار ﺗﺎﺑﻊ زوج ﻧﺴﺒﺖ ﺑﻪ ............................ﺗﻘﺎرن دارد؟
(1ﻣﺤﻮر xﻫﺎ
.62اﮔﺮ
3x 2 + 1
x +1
(2ﻣﺤﻮر yﻫﺎ
= ) g = {(7, 3 ),(2, 5 ),(−19 , 10 )}, f (x
(3ﻧﻴﻤﺴﺎز ﻧﺎﺣﻴﻪ اول و ﺳﻮم
ﺑﺎﺷﺪ ،ﻣﺠﻤﻮع اﻋﻀﺎء داﻣﻨﻪ
gof
(4ﻧﻴﻤﺴﺎز ﻧﺎﺣﻴﻪ دوم و ﭼﻬﺎرم
ﻛﺪام اﺳﺖ )اﻋﻀﺎء داﻣﻨﻪ را ﻳﻚ ﺑﺎر ﻟﺤﺎظ ﻛﻨﻴـﺪ( )ﺳـﻨﺠﺶ
ﺟﺎﻣﻊ (93
- 1 (1
(2
7
3
−
(3
10
3
−
3 (4
142ﺟﺰوه ﻛﺎر در ﻛﻼس ﺣﺴﺎب و دﻳﻔﺮاﻧﺴﻴﻞ و اﻧﺘﮕﺮال و رﻳﺎﺿﻲ ﭘﺎﻳﻪ
.63ﻧﻤﻮدار ﺗﺎﺑﻊ
| y =| | x | − 1
ﻛﺪام اﺳﺖ؟
(1
(2
(2
(4
.64اﮔﺮ
x2 −1
= ) g = {(1, 2),(2, 3),(3, −1),(−1, 0 ),(−2, 1),(−3, 2)}, f (xﺑﺎﺷﺪ آن ﮔﺎه ﺗﺎﺑﻊ
3 (1
.65اﮔﺮ
(1
saati.com
4 (2
1 +x2
1 −x 2
= ) f (x
و
) g (x ) = x (1 − x
(2
) [0 , 1
.66اﮔﺮ
}) f = {(0 , 1 ),(1, 2 ),(2, 3
(1ﺻﻔﺮ
.67ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﻣﺎﺷﻴﻦ
3 (1
6 (3
(3
1 (2
(4
) (−1, 1
و }) ، g = {(1, 0 ),(2, 1),(−1, 2آن ﮔﺎه ﻣﻌﺎدﻟﻪي
5 (2
(4ﺑﻲﺷﻤﺎر
ﺑﺎﺷﺪ آن ﮔﺎه Dfog ∩ Dgofﻛﺪام اﺳﺖ؟
] (−1, 0
x → f → g → 6x + a
g −3
f− 3
=h
ﭼﻨﺪ ﻋﻀﻮ دارد؟
) f 2 (x ) = 1 + 2 (gof ) (x
2 (3
اﮔﺮ
−19
f (x ) = 3x 2 − 2و ) = 1
9
7 (3
φ
ﭼﻨﺪ رﻳﺸﻪ دارد؟
3 (4
(g
ﺣﺎﺻﻞ
) g (1
ﻛﺪام اﺳﺖ؟
9 (4
ﻓﺼﻞ :2رﻳﺎﺿﻲ ﭘﺎﻳﻪ 143
saati.com
.68اﮔﺮ
) f (2x + 1 ) + f (0 ) = 2x − f (1
19
3
(1
(1
(2
−
1
1
) =x −
x
x
.69اﮔﺮ
ﺑﺎﺷﺪ ﺣﺎﺻﻞ
x+
7
3
(3
−
( ، fآن ﮔﺎه ﺣﺎﺻﻞ
2
- 4 (1
(1
= ) f (x
.72ﺑﺮاي ﻛﺪام ﻣﻘﺪار ، aﺗﺎﺑﻊ
(1
ﺑﺎﺷﺪ ﺣﺎﺻﻞ
(2
(2
±3
5
3 (3
π
π
) f ( ) + f (−
4
4
4 2
π 2 −4
3x − a
f (x ) = x
3 +a
3 (4
ﻛﺪام اﺳﺖ؟
(2ﺻﻔﺮ
4 2
π2
±1
x
)
x −1
(f
19
3
(4
7
3
) (x > 1
(3
.70اﮔﺮ f ( x +x 1 ) = x x+ 1آن ﮔﺎه ﺣﺪاﻗﻞ ﻣﻘﺪار
.71اﮔﺮ
)f( 5
ﻛﺪام اﺳﺖ؟
1 (2
3 5
sin 3 x + x cos x
x2 −1
) f (3
ﻛﺪام اﺳﺖ؟
-2 (4
ﭼﻘﺪر اﺳﺖ.
(3
π +4 2
π2
(4ﺻﻔﺮ
ﺗﺎﺑﻌﻲ ﻓﺮد اﺳﺖ؟
(3
1
3
±
(4ﻫﻴﭻ ﻣﻘﺪار a
144ﺟﺰوه ﻛﺎر در ﻛﻼس ﺣﺴﺎب و دﻳﻔﺮاﻧﺴﻴﻞ و اﻧﺘﮕﺮال و رﻳﺎﺿﻲ ﭘﺎﻳﻪ
.73ﺑﺮاي ﻛﺪام ﻣﻘﺪار aﺗﺎﺑﻊ
(1ﻓﻘﻂ
(2
a = ±10
.74ﺑﺮاي ﻛﺪام ﻣﻘﺪار aﺗﺎﺑﻊ
(1
a −x
a +x
f (x ) = log
(3
±1
.75اﮔﺮ fﺗﺎﺑﻌﻲ زوج و gﺗﺎﺑﻌﻲ ﻓﺮد ﺑﺎﺷﺪ از دﺳﺘﮕﺎه
(1
(3ﻫﺮ ﻣﻘﺪار
a ≠0
(4ﻫﻴﭻ ﻣﻘﺪار a
f (x ) = log(ax +ﺗﺎﺑﻌﻲ ﻓﺮد اﺳﺖ؟
(2
10
3
ﺗﺎﺑﻌﻲ ﻓﺮد اﺳﺖ؟
a = ±1
) 4x 2 + 1
±2
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(2
±16
(2 f + g ) (1 ) = 4
، ﻣﻘﺪار
( f − 2g )(−1 ) = 2
−10
3
(4
) f (1
±4
ﻛﺪام اﺳﺖ؟
2 (4
-2 (3
.76ﻣﺠﻤﻮع رﻳﺸﻪﻫﺎي ﻣﻌﺎدﻟﻪي 2x 6 + kx 2 − 3x 4 − 4 = 0ﻛﺪام اﺳﺖ؟
(1ﺻﻔﺮ
.77اﮔﺮ ﺗﺎﺑﻊ fﺑﺎ ﺿﺎﺑﻄﻪي
(1
−x − 3 x − 1
1 (2
-1 (3
x − 3 x − 1 :x ≥ 2
f (x ) = ﺗﺎﺑﻌﻲ ﻓﺮد ﺑﺎﺷﺪ،
g (x 0
x ≤ −2
(2
−x + 3 x − 1
(3
) g (x
(4ﺑﻪ
k
ﺑﺴﺘﮕﻲ دارد.
ﻛﺪام اﺳﺖ؟
x + 3 x +1
(4
x −3 x +1
.78اﮔﺮ ﺑﺮاي ﻫﺮ b ∈Rو aداﺷﺘﻪ ﺑﺎﺷﻴﻢ ) ، f (a + b ) = f (a ) + f (bآن ﮔﺎه fﭼﻪ ﺗﺎﺑﻌﻲ اﺳﺖ؟
(1ﻓﺮد
(2زوج
(3ﻫﻢ زوج و ﻫﻢ ﻓﺮد
(4ﻧﻪ زوج و ﻧﻪ ﻓﺮد
ﻓﺼﻞ :2رﻳﺎﺿﻲ ﭘﺎﻳﻪ 145
saati.com
.79اﮔﺮ ﺗﺎﺑﻊ fﺑﺎ ﺿﺎﺑﻄﻪي
- 1 (1
| f (x ) = a | x | +b | x + 3 | +2 | x − 3
2 (2
.80اﮔﺮ ﺗﺎﺑﻊ | f (x ) =| x + a | + | x + b | + | x + c +زوج ﺑﺎﺷﺪ
- 1 (1
ﺗﺎﺑﻌﻲ ﻓﺮد ﺑﺎﺷﺪ
-2 (3
1 (3
(2ﺻﻔﺮ
.81اﮔﺮ ﺗﺎﺑﻊ
- 9 (1
7 (2
- 3 (1
1 (3
2 (4
-3 (4
x 2 + x + 1 :x > 0
f (x ) = ﻓﺮد ﺑﺎﺷﺪ ? = a + b + c
2
a x + bx + c : x < 0
1 (2
.83اﮔﺮ ﺗﺎﺑﻊ f (x ) = x 4 + 3x 3 + A(x + 1)2 + Bxزوج ﺑﺎﺷﺪ
- 6 (1
1 (4
? = a +b +c
3x
: x >0
x 2 + 5x − 1
، f (x ) = ﻓﺮد ﺑﺎﺷﺪ ? a + b + c
ax
: x <0
x 2 + bx + c
.82اﮔﺮ ﺗﺎﺑﻊ
) (a + b
ﻛﺪام اﺳﺖ؟
−3 (2
3 (3
-1 (4
? = A+B
(3ﺻﻔﺮ
6 (4
146ﺟﺰوه ﻛﺎر در ﻛﻼس ﺣﺴﺎب و دﻳﻔﺮاﻧﺴﻴﻞ و اﻧﺘﮕﺮال و رﻳﺎﺿﻲ ﭘﺎﻳﻪ
.84اﮔﺮ ﺗﺎﺑﻊ
x3 +4
x2
= ) f (x
در ﺑﺎزهاي ﺻﻌﻮدي ﺑﺎﺷﺪ آن ﮔﺎه
saati.com
3x 3 − x 2 + 12
x2
= ) g (x
در ﻫﻤﺎن ﺑﺎزه ﭼﮕﻮﻧﻪ اﺳﺖ؟
(1ﺻﻌﻮدي
(2ﻧﺰوﻟﻲ
(3اﺑﺘﺪا ﺻﻌﻮدي ،ﺳﭙﺲ ﻧﺰوﻟﻲ
(4اﺑﺘﺪا ﻧﺰوﻟﻲ و ﺳﭙﺲ ﺻﻌﻮدي
.85ﺑﻪ ازاي ﭼﻪ ﻣﻘﺎدﻳﺮي از aﺗﺎﺑﻊ
(1ﻓﻘﻂ
a ≤1
x + 1 :x ≥ 0
f (x ) =
2x + a : x < 0
ﺻﻌﻮدي ﺧﻮاﻫﺪ ﺑﻮد؟
(2ﻓﻘﻂ a > 1
.86اﮔﺮ ﺗﻮاﺑﻊ fو gﺑﻪ ﻋﻨﻮان ﻣﺎﺷﻴﻦﻫﺎﻳﻲ در ﻧﻈﺮ ﮔﺮﻓﺘﻪ ﺷﻮﻧﺪ ﻛﻪ
(4ﻫﻴﭻ ﻣﻘﺪار a
(3ﻫﺮ ﻣﻘﺪار a
= (2x 3 + 1 ) → g (x ) → x
x → fدر اﻳﻦ ﺻﻮرت ﺗﺎﺑﻊ gﻋﺪد 55را ﺑﻪ ﭼـﻪ
ﻋﺪدي ﻣﻲﺑﺮد؟
55 (2
2 (1
.87دو ﺗﺎﺑﻊ ﺑﺎ ﺿـﺎﺑﻄﻪﻫـﺎي
)ﺳﺮاﺳﺮي (93
1 (1
}) g = {(2, 5 ),(3, 4 ),(1, 6 ),(4, 7 ),(8, 1
و
2 (2
.88ﻛﺪام ﮔﺰﻳﻨﻪ ﻧﻘﻄﻪاي از ﻧﻤﻮدار ﻣﻌﻜﻮس ﺗﺎﺑﻊ
5) (1و (1
3 (3
f (x ) = 2x − 5
6 (4
ﻣﻔـﺮوضاﻧـﺪ اﮔـﺮ
3 (3
f (x ) = x 5 + 8x − 4
-2) (2و (-52
.89اﮔﺮ ﺗﺎﺑﻊ
}) f = {(m 2 + 2m , 2 ),(m + 3, 4 ),(4 − m , 2 ),(2, −2
(1ﺻﻔﺮ
1 (2
( fog −1 ) (a ) = 6
ﺑﺎﺷـﺪ aﻛـﺪام اﺳـﺖ؟
4 (4
ﻣﻲﺑﺎﺷﺪ؟
-5) (3و (-1
ﻣﻌﻜﻮسﭘﺬﻳﺮ ﺑﺎﺷﺪ،
2 (3
2) (4و (52
m
ﭼﻨﺪ ﻣﻘﺪار ﻣﺨﺘﻠﻒ ﻣﻲﺗﻮاﻧﺪ داﺷﺘﻪ ﺑﺎﺷﺪ؟
(4ﺑﻲﺷﻤﺎر
ﻓﺼﻞ :2رﻳﺎﺿﻲ ﭘﺎﻳﻪ 147
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.90در ﺗﺎﺑﻊ ﻣﻌﻜﻮسﭘﺬﻳﺮ
f (x ) = 2x 5 + (k − 1 )x 4 + 2k + 5
- 9 (1
(1
و
− 3
f (x ) = x 3 + x x
.92اﮔﺮ
(1
-57 (2
f (x ) = 2x 4 − 4x 2
.91اﮔﺮ
ﺣﺎﺻﻞ )) f (− f −1 (9ﻛﺪام اﺳﺖ؟
2
3
( )3
2
) Df = (−∞, −1
(2
ﺑﺎﺷﺪ آن ﮔﺎه
ﻛﺪام اﺳﺖ؟
(3
3
15
) ( f −1
4
ﺑﺎﺷﺪ آن ﮔﺎه
(2
5 (3
) f −1 (6
3
2
(3
4
3
( )3
4
.93اﮔﺮ fو gدو ﺗﺎﺑﻊ ﻣﻌﻜﻮسﭘﺬﻳﺮ ﺑﺎﺷﻨﺪ ﻛﺪام ﻳﻚ از ﺗﻮاﺑﻊ زﻳﺮ اﻟﺰاﻣﺎً ﻣﻌﻜﻮسﭘﺬﻳﺮ ﻧﻴﺴﺖ
(1
gof
(2
.94درﺟﻪ ﺻﻮرﺗﻲ ﻣﻌﻜﻮس ﺗﺎﺑﻊ fﺑﺎ ﺿﺎﺑﻄﻪي
(1
c =1
(2
(3
5
x −a
bx + c
c = −1
(4
− 2
ﻛﺪام اﺳﺖ؟
2
( )2
3
1
f
33 (4
= ) f (x
(4
4 3
) (
3 4
) ( f (x ) ≠ 0
−3g + 2
(4
f ×g
ﺧﻮدش ﻣﻲﺷﻮد؟
(3
ab ≠ −1 ,c = −1
(4
ab ≠ 1 ,c = 1
148ﺟﺰوه ﻛﺎر در ﻛﻼس ﺣﺴﺎب و دﻳﻔﺮاﻧﺴﻴﻞ و اﻧﺘﮕﺮال و رﻳﺎﺿﻲ ﭘﺎﻳﻪ
saati.com
.95ﭼﻪ ﺗﻌﺪاد از ﺗﻮاﺑﻊ زﻳﺮ ﻳﻚ ﺑﻪ ﻳﻚاﻧﺪ؟
f (x ) = x 2 , x ≤ 0
اﻟﻒ(
ب(
(1ﺻﻔﺮ
g (x ) = (1 − 2x )3
1 (2
2 (3
.96اﮔﺮ fﺗﺎﺑﻌﻲ ﻣﻌﻜﻮسﭘﺬﻳﺮ ﺑﺎﺷﺪ ﻣﻌﻜﻮس ﺗﺎﺑﻊ fﺑﺎ ﺿﺎﺑﻄﻪي
(1
2x + 2
)
x −1
(3
= ) g −1 (x
.97ﺑﻪ ازاي ﻛﺪام ﻣﻘﺪار mﺗﺎﺑﻊ ﺑﺎ ﺿﺎﺑﻄﻪي
4 (1
.98اﮔﺮ
x + x2 +m
2
x ≥0
) x + 4 = f (x
2 (2
.99ﻣﻌﻜﻮس ﺗﺎﺑﻊ
x >0
و
2
logx2 , x > 0
f (x ) = x 3 + 3x + 2
.100اﮔﺮ
−a (1
3 (4
ﻛﺪام اﺳﺖ؟
(2
2x + 2
)
1 −x
(4
1 −1 1 − x
( f
)
3
2x + 2
( g −1 (x ) = 3 f −1
= ) g −1 (x
ﻳﻚ ﺗﺎﺑﻊ ﻓﺮد اﺳﺖ )ﺳﻨﺠﺶ ﺟﺎﻣﻊ (93
-4 (3
و ، f (x ) = x 2 − 4آن ﮔﺎه ﻣﻌﺎدﻟﻪي
f (x ) = 2x
= ) g (x
f (x ) = log
1 (2
1 (1
(1
) 1 − 2 f (3x
) 1 + 2 f (3x
( g −1 (x ) = 3 f −1
1 −1 x − 1
( f
)
3
2x + 2
ج(
1 − 2x
1 + 2x
= ) h (x
(4ﻫﻴﭻ ﻣﻘﺪار m
ﭼﻨﺪ رﻳﺸﻪ دارد؟
(3ﻫﻴﭻ
3 (4
ﻛﺪام اﺳﺖ؟
(2
1
logx2 x > 1
2
و f (a ) = 16آن ﮔﺎه
a (2
(3
− logx2 , x > 0
(4
logx2 x > 1
? = ) f −1 (−12
(3
2a
(4
−2a
ﻓﺼﻞ :2رﻳﺎﺿﻲ ﭘﺎﻳﻪ 149
saati.com
.101ﻧﻤﻮدار ﺗﺎﺑﻊ
y = −3 x
و ﻣﻌﻜﻮس آن در ﭼﻨﺪ ﻧﻘﻄﻪ ﻣﺘﻘﺎﻃﻊاﻧﺪ؟
(1ﻳﻚ ﻧﻘﻄﻪ
.102اﮔﺮ
f (x ) = x x + 1
(x − 1 )2 , x ≥ 1
(1
.103
(2دو ﻧﻘﻄﻪ
(2
3
f (x ) = x 2 − 4x + 4
1 (1
.104ﻣﻌﻜﻮس
(1ﺻﻔﺮ
ﺑﺎﺷﺪ ﺿﺎﺑﻄﻪ
f −1
و
x ≥2
2x − 1
x −2
(x − 1 )2 : x ∈ R
3
(3
(x − 1 )2 , x ≤ 1
3
(4
(x − 1 )2 , x ≥ 0
در اﻳﻦ ﺻﻮرت ﻣﻌﺎدﻟﻪ ) f(x ) = f(−x1ﭼﻨﺪ رﻳﺸﻪ دارد؟)ﺳﺮاﺳﺮي (92
(3ﻫﻴﭻ
(4ﺑﻲﺷﻤﺎر
و ﺧﻂ ، y = xﻫﻤﺪﻳﮕﺮ را در ﭼﻨﺪ ﻧﻘﻄﻪ ﻗﻄﻊ ﻣﻲﻛﻨﻨﺪ؟
1 (2
.105ﺗﺎﺑﻊ ﻣﻌﻜﻮسﭘﺬﻳﺮ fﺑﺮ روي Rﺗﻌﺮﻳﻒ ﺷﺪه ،ﺗﺎﺑﻊ
(1ﻣﺘﻨﺎوب
(3ﺳﻪ ﻧﻘﻄﻪ
ﺑﺮاﺑﺮ ﻛﺪام اﺳﺖ؟
2 (2
=y
(4ﻳﻜﺪﻳﮕﺮ را ﻗﻄﻊ ﻧﻤﻲﻛﻨﻨﺪ.
(2ﻳﻜﻨﻮا
2 (3
) g (x ) = f (−x ) + f (x
4 (4
ﭼﮕﻮﻧﻪ اﺳﺖ؟
(3ﻣﻌﻜﻮسﭘﺬﻳﺮ
(4ﻳﻚ ﺑﻪ ﻳﻚ
3
150ﺟﺰوه ﻛﺎر در ﻛﻼس ﺣﺴﺎب و دﻳﻔﺮاﻧﺴﻴﻞ و اﻧﺘﮕﺮال و رﻳﺎﺿﻲ ﭘﺎﻳﻪ
.106ﻧﻤﺎﻳﺶ ﻫﻨﺪﺳﻲ ﺗﺎﺑﻊ ﻣﻌﻜﻮس
(1
2
), 2
2
.107اﮔﺮ ﺗﺎﺑﻊ
x
=y
1 +x2
(2
(
f (x ) = x 3 + ax 2 + a − 3
5) (1و (2
saati.com
از ﻛﺪام ﻧﻘﻄﻪ ﻣﻲﮔﺬرد؟
(3
) (1, 0
) (0, 1
(4
3
), 3
2
(
ﻣﻌﻜﻮسﭘﺬﻳﺮ ﺑﺎﺷﺪ ،ﻣﻨﺤﻨﻲ ﻣﻌﻜﻮس آن از ﻛﺪام ﻧﻘﻄﻪ ﻣﻲﮔﺬرد؟
1) (3و (0
0) (2و (1
2) (4و (5
.108ﺗﺎﺑﻊ ﻓﺮد fﻣﻌﻜﻮس ﭘﺬﻳﺮ اﺳﺖ ،ﻧﻤﻮدار f −1ﻧﺴﺒﺖ ﺑﻪ ﻛﺪام ﻣﻮرد ﻣﺘﻘﺎرن اﺳﺖ؟
(2ﻣﺤﻮر xﻫﺎ
(1ﻣﺒﺪاء ﻣﺨﺘﺼﺎت
(3ﻣﺤﻮر yﻫﺎ
4
3
ﺑﺎﺷﺪ ﻣﻘﺪار ورودي ﻛﺪام اﺳﺖ؟
(2
7
2
.109اﮔﺮ ﺧﺮوﺟﻲ از ﻣﺎﺷﻴﻦ ﺷﻜﻞ ﻣﻘﺎﺑﻞ
y=x (4
ﺧﺮوﺟﻲ → x
(1
.110
11
9
atgx + b 3 x + 1 : x ≥ 0
f (x ) =
) g (x
:x < 0
(1
.111اﮔﺮ
(1
(2
−a
f (cos x ) = cos 2x
sin 2a
ﻓﺮد ﺑﺎﺷﺪ،
آن ﮔﺎه
(3
−17π
?=)
4
a
19
8
4 (4
(f
(3
a
2
(4
a
2
−
? = ) f (sin a
(2
− sin 2a
(3
− cos 2a
(4
cos 2a
→ → 2x − 2ورودي
ﻓﺼﻞ :2رﻳﺎﺿﻲ ﭘﺎﻳﻪ 151
saati.com
.112اﮔﺮ ] ، f (x ) = [xﺗﺎﺑﻊ ﺑﺎ ﺿﺎﺑﻄﻪي
(1ﻫﻤﺎﻧﻲ
)) f (x − f (x
ﭼﮕﻮﻧﻪ اﺳﺖ؟
(2ﺛﺎﺑﺖ
(3ﻳﻚ ﺑﻪ ﻳﻚ
(4ﻏﻴﺮﻣﺘﻨﺎوب
.113اﮔﺮ }) f = {(1, 2),(2, 3),(3, 1آن ﮔﺎه ﻛﺪام ﺗﺎﺑﻊ ﻫﻤﺎﻧﻲ اﺳﺖ؟
(1
f2
.114داﻣﻨﻪ ﺗﺎﺑﻊ
(1
R
(2
−f
] f (x ) = cos2 x − [cos x
(2
(3
fof
(4
fofof
ﻛﺪام اﺳﺖ؟
R −Z
(3
N
(4
Z
