ò‹Ñïå‚@óibïä öa‹èÜa@óábÐ@óîíäbq
‘ì‹«@‹Ð
pbïšbî‹ÜaZò†b¾a p@Ë@l@‘2@óåÜaZõín¾a 2HZŒb−fia@ò‡à
Zßìÿa@æî‹ánÜa 3 x +1− 3 x +1 x +5 −2 1 1 lim n cos − cos ، lim ، lim أ ا ت ا: ∞n →+ x →3 x x −3 n + 1 x →0 n 2 n n . lim 3 Arc tan 3 ∞n →+
ZðäbrÜa@æî‹ánÜa
1 a n +1 − a n = n +1 ﺃﺤﺴﺏ a nﺒﺩﻻﻝﺔ . n ﻝﺘﻜﻥ ) ( a nﻤﺘﺘﺎﻝﻴﺔ ﻋﺩﺩﻴﺔ ﺒﺤﻴﺙ3 : a0 = 1 ZsÜbrÜa@æî‹ánÜa ﺍﻝﺠﺯﺀ:A
x +1 f x = − 1; x ≥ 1 ( ) x ﻝﺘﻜﻥ fﺩﺍﻝﺔ ﻋﺩﺩﻴﺔ ﺒﺤﻴﺙ: f ( x ) = x + 1 ;0 ≤ x ≺ 1 x -1ﺃﺤﺴﺏ ) lim f ( xﻭ ) . lim f ( x ∞x →+
x →0
ﻓﻲ . x 0 = 1
-2ﺃﺩﺭﺱ ﺍﺘﺼﺎل ﺍﻝﺩﺍﻝﺔ f -3ﺒﻴﻥ ﺃﻥ ﺍﻝﺩﺍﻝﺔ fﺘﺯﺍﻴﺩﻴﺔ ﻗﻁﻌﺎ ﻋﻠﻰ ﺍﻝﻤﺠﺎل [∞. I = [1, + −1 -4ﺒﻴﻥ ﺃﻥ fﺘﻘﺎﺒل ﻤﻥ ﺍﻝﻤﺠﺎل Iﻨﺤﻭ ﻤﺠﺎل Jﻴﻨﺒﻐﻲ ﺘﺤﺩﻴﺩﻩ ﺜﻡ ﺤﺩﺩ ) f ( xﻝﻜل xﻤﻥ . J x +1 = f ( x ) − xﺜﻡ ﺍﺴﺘﻨﺘﺞ ﺇﺸﺎﺭﺓ f ( x ) − xﻋﻠﻰ . I -5ﺒﻴﻥ ﺃﻥ1 − x ; ∀x ∈ I : x
)
(
ﺍﻝﺠﺯﺀ:B
1+ un − 1; ∀n ∈ ℕ = u n +1 u ﻝﺘﻜﻥ ) (u nﺩﺍﻝﺔ ﻋﺩﺩﻴﺔ ﺒﺤﻴﺙ: n u0 = 2 -1ﺒﻴﻥ ﺒﺎﻝﺘﺭﺠﻊ ﺃﻥ. ∀n ∈ ℕ : u n ≥ 1: -2ﺃﺩﺭﺱ ﺭﺘﺎﺒﺔ ﺍﻝﻤﺘﺘﺎﻝﻴﺔ ) (u nﻭﺍﺴﺘﻨﺘﺞ ﺃﻥ. ∀n ∈ ℕ : u n ≤ 2 :
-3ﻫل ﺍﻝﻤﺘﺘﺎﻝﻴﺔ ) (u nﻤﺘﻘﺎﺭﺒﺔ؟ﺇﺫﺍ ﻜﺎﻥ ﺍﻝﺠﻭﺍﺏ ﺒﻨﻌﻡ ﺃﺤﺴﺏ ) (u n
. lim
∞x →+
ﺍﻝﺠﺯﺀ:C
1 +v n ; ∀n ∈ ℕ = v n +1 2 ﻝﺘﻜﻥ ) (v nﺩﺍﻝﺔ ﻋﺩﺩﻴﺔ ﺒﺤﻴﺙ: 1 = v0 2 π v n = cos ﻝﻜل nﻤﻥ ℕﺜﻡ ﺃﺤﺴﺏ ) . lim (v n -1ﺒﻴﻥ ﺃﻥ -2. ∀n ∈ ℕ : 0 ≤ v n ≤ 1 :ﺒﻴﻥ ﺒﺎﻝﺘﺭﺠﻊ ﺃﻥ: n ∞x →+ 3.2