تصحيح الإمتحان الوطني في مادة الرياضيات 2010 التمرين 5

ردادإ ناراسا نرا : ادا لوا ءزا : ( ) x g x xe = + 2 1 4 1 -نأ ن: ( ) ( ) x x IR:g' x x e " Î = + 2 4 2 1 د: ( ) x g x xe = + 2 1 4 اذإ : ( ) ( ) ( ) x x x x g' x xe ' e xe e x = + = + = + 2 2 2 2 1 4 4 8 4 1 2 ن !و ( ) ( ) x x IR:g' x x e " Î = + 2 4 2 12 -نأ ن: "#دازg ; -   + ¥     1 2 و g "#$; -   -¥     1 2 د( ) ( ) x g' x x e x x - = Û + = Û + = Û = 2 1 0 2 1 0 2 1 0 2 نx x IR: e " Î 2 0 ≻ ن !و ناذإ x - ³ 1 2 ن x+ ³ 2 1 0 اذإ ( ) g ' x ³ 0 ن و g "#داز; -   +¥     1 2 ناذإ ن اذإ ن و g "#$x - £ 1 2 x + £ 2 1 0 ( ) g ' x £ 0 ; -   - ¥     1 23 -أ -نأ ن: g e -   = -     1 2 1 2 نأ ند%و g -       1 0 2 ≻ g e e e -   ´  -   - -     = + = - = -         1 2 1 2 1 1 2 1 4 1 2 1 2 2 داذإ g e -   = -     1 2 1 2 نأ e ; » 2 7 ن e 2 ≻ e2 1 ≺ اذإ e - 2 1 0 ≻ اذإ g -       1 0 2 ≻ اذإ 3 -ب -ج)ا : د"#دازg ; -   + ¥     1 2 و g "#$; -   -¥     1 2 اذإ g د ل*g -       1 2 x - = 1 2 دن !و ( ) g x g -       1 2 ≻ g -       1 0 2 ≻ و د( ) x IR : g x " Î 0 ≻ اذإ +ا ءزا : ( ) ( ) x f x x e x = - + + 2 2 1 1 ب),-1 ( ) xl im f x ® + ¥ ( )  ( ) ( ) x x x x xlim x lime limf x lim x ®+¥ ®+¥ ®+¥ ®+¥ - =+¥  =+¥ ⇒ =+¥  + =+¥  22 1 1 اذإ ( ) xlim f x ®+¥ = +¥ د( ) ( ) x f x x e x = - + + 2 2 1 1 اذإ ( ) x x f x xe e x = - + + 2 2 2 1 نأ و ( ) xlim f x ®-¥ = -¥ ن نأ ن-2 ( ) xlim f x ® -¥ = -¥ ( ) x n x xlim xe lim e ® -¥ ® -¥  =  =  2 2 2 0 0 xlim x ® -¥  + = -¥  1 نأ ن-3 ( ) ( ) x IR: f ' x g x " Î = د( ) ( ) x f x x e x = - + + 2 2 1 1 اذإ ( ) x f' x e = 2 2 x x xe e + - 2 2 4 2 ( ) x xe gx + = + = 2 1 1 4 ن !و ( ) ( ) x IR: f ' x g x " Î = ( ) ( ) x I R : f ' x g x " Î = 4 -ج)ا د( ) x IR : g x " Î 0 ≻ ق) ب), دو ( ) x IR: f ' x " Î 0 ≻ اذإ f IR . ن !و "#داز5 -ب),( ) x f x lim x ®+¥ ( ) ( ) x x f x x e x x e x x x x - + + -   = = + +     2 2 2 1 1 2 1 1 1 دن !و ( ) x x x x x x lim x f x lime lim x lim x ®+¥ ®+¥ ®+¥ ®+¥  -    =      =+¥ ⇒ =+¥    + =       22 1 2 1 1 1 ( ) x f x lim x ® +¥ = +¥ يأج)ا -6 ( ) x f x lim x ®+¥ = +¥ نأ ",ا ن ( ) f C بارا رو,ها #/ ر ل*. ب),-7 ( ) ( ) ( ) xlim f x x ® -¥ - + 1 د( ) ( ) ( ) x x x f x x x e xe e - + = - = - 2 2 2 1 2 1 2 نأ و x x lim xe ®-¥  =  2 2 0 ( ) ( )   - + = x xlime ®-¥ =  2 0 ن xlim f x x ®-¥   1 0 ج)ا -8 ( ) ( ) xlim f x x ®-¥   - + =   1 0 نأ م*)ا ن ( ) : y x D = + 1 ",#بر*( ) f C راو-¥ م*)ا 2ط*ط*+اد,إ جوز دد,-9 ( ) D ",او ( ) f C ن( ) E x ; y 2ط*ط*( ) D ( ) f C و ( ) E D Î ( ) f E C Î اذإ و ( ) ( ) ( ) x f x y x e x x = Û - = Û - = Û = 2 1 2 1 0 2 1 0 2 دنx e2 0 ≻ y = + = 1 3 1 2 2 !و ن( ) E D Î ن و E ;       1 3 2 2 10 ( ) D ",او ( ) f C م*)#)ا 24وا سرد-10 ( ) ( ) x f x y x e - = - 2 2 1 دx x IR : e " Î 2 0 ≻ و ةر/إ اذإ ( ) ( ) f x y - اد,ا ةر/إ ھ ( ) x - 2 1 * ناذإ x - 2 1 0 ≻ ن x 12 ≻ ن !و ( ) ( ) f x y - 0 ≻ ",ا ن و ( ) f C م*)ا قو دو( ) D* ناذإ ن ن !و ",ا ن و ( ) f C م*)ا ت,دو( ) D x- 2 1 0 ≺ x 12 ≺ ( ) ( ) f x y - 0 ≺ نأ ن-11 ( ) T : y x = سل ( ) f C 0 د( )( ) ( ) y f ' x f = - + 0 0 0 ( ) f ' = 0 1 و و ( ) f = 0 0 y x = اذإ د:ھ ( ) T ",#( ) f C 0 ",#نأ ن-12 ( ) f C <و$ أ فط:ا ط*- 1 2 د( ) ( ) ( ) ( ) ( ) ( ) x f ' x g x f '' x g' x f '' x x e = ⇒ = ⇒ = + 2 4 2 1 ( ) f '' x x x - = Û + = Û = 1 0 2 1 0 2 ن !و ",ا ن و ( ) f C <و$ أ فط:ا ط*!- 1 2( ) f C ( ):y x D = + 1 13 -",ا ء/إ ( ) f C ( ) T : y x =نأ ن-14 ( ) x e x e dx - = - ∫12 2 0 2 1 1 2 .ءاز#ل:)24( ) ( ) x u' x e v x x  =  = -  2 2 1 اذإ ( ) ( ) x u x e v ' x  =  =  2 122 ن !و ( ) ( ) x x x e e x e dx x e e dx   - = - - = - + = -     ∫ ∫ 1 1 1 2 2 2 2 2 2 1 1 1 2 1 2 1 1 2 2 2 2 2 0 0 0 ( ) x e x e d x - = - ∫12 2 0 2 1 1 2 ن ونأ ن-15 ( ) ( ) e cm A D = - 2 6 2 د( ) A D ",ا نرو$,ا ىو)ا ز, ,)ھ ( ) f Cم*)او ( ) T : y x = سا ل ( ) f C 0 ن*)او x = 0 و x = 12 ( ) ( ) ( ) ( ) e A f x x dx cm cm e cm D   = - ´ = - + = -     ∫12 2 2 2 0 1 4 4 1 6 2 2 2 اذإ ( ) ( ) e cm A D = - 2 6 2 ن و