κυριακόπουλος κ. αντώνης υποψηφιακαί ασκήσεις άλγεβρας 1ο, πραγματικοί - μιγαδικοί αριθμοί (1975)
κυριακόπουλος ολοκληρώματα
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ΣΥΝΑΡΤΗΣΕΙΣ ΠΟΥ ΟΡΙΖΟΝΤΑΙ ΑΠΟ ΟΛΟΚΛΗΡΩΜΑΤΑ. ΣΧΕΤΙΚΑ ΘΕΜΑΤΑ ΑΝΤΩΝΗΣ ΚΥΡΙΑΚΟΠΟΥΛΟΣ Μαθηµατικός – Συγγραφέας – µέλος του ∆.Σ. της Ε.Μ.Ε. – Πρόεδρος της Συντακτικής Επιτροπής του περιοδι- κού «Ευκλείδης Β΄»
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Transcript of κυριακόπουλος ολοκληρώματα
1. . .. ... - 2. . f(x)dx f f(x)dx , , , f(x)dx = f(t)dt = f()d = ...F(x) = f(t)dt .21. :) . x - .) f x, , , - . : 2. xF(x) = f(t)dt f . , :x , x t. To x - F. t ( ) , , x, . :. F f . , F :F(x) = f (x), x ., f ( - F). 3. - , :. f G f , , : f(x)dx = G() G() .= ,F(x) = f (t)dt ,F(x) = f (t)dt 3 , (, , , , ). -, :2xt02F(x) e dt , , . , - . .3. . -:g(x)h(x) f, g h . :xg(x)F(x) = f (t)dt . F.B f . , g - h. x F , , x(B) f - h(x) g(x). 4. F. f g h - : g(x) h(x) , xI .F(x) = f (t)dt + f (t)dt = f (t)dt + f (t)dt .(x) = f (t)dt, x , : (x) = f (x), x .F(x) = f (t)dt = G g(x) G h(x) .4:, F . - F : . . , xI : g(x) h(x) g(x)h(x) :x, :F(x) = (h(x)) + (g(x)) ., :F(x) = (h(x)) h(x) + (g(x)) g(x) = f (h(x)) h(x) + f (g(x)) g(x) . . G f , :G(t) = f (t) , t ., :( ) ( )g(x)h(x), :F(x) = G(g(x)) g(x) G(h(x)) h(x) = f (g(x)) g(x) f (h(x)) h(x) .4. 1. :x20tF(x) dtt 1= . 5. 1 1 + 88( )( )( )F(x) = 9 t tdt.5.t . : f (t)2t 1= :A = (,1)(1,1)(1,+) . x 0 F , , f - 0 x. : 1x1. F : AF = (1,1) .. F AF :x2F (x) f (x)x 1 = =.. :x22tF(x) dtt 1 = AF = (,1) :x23tF(x) dtt 1= AF = (1,+) . , :x2F (x)x 1 =. 2. :x21. . :f (t) 9 t2 = t - : = [3, 3]. : g(x) = x B = [0,+) . x F , , x f - 6. 1 g(x) = x . : x 0 x 0 x 0 F(x) = f (t)dt = G( x) G(1). = = = = .F(x) e ln t dt + 860 x 93 x 3 x 3 x 9., F AF = [0,9] .. g(x) = x (0,+) ., F I = (0,+)[0,9] = (0,9] . G f , :G (t) f (t) 9 t2 = = t, t ., :x1, :IF (x) G ( x)( x) 9 x x2 x9 x x2 x . () 0. :x 0F(x) F(0)F (0) limx 0 ++ =(00) = ( )x 0F(x) F(0)lim(x) + =x 09 x x 3lim2 x 2 + = = . : 9 x x, 0 x 9F (x) 2 x3, x 02 == 3. :x 2t1x 1. . f (t) et n t = :80A = (,0)(0,+) . ( )( ) 7. g(x) = x 2 B = = = + = + F(x) f (t)dt f (t)dt f (t)dt f (t)dt f (t)dt.(x) = f (t)dt, x(,0) , : (x) = f (x), x(,0) . = + = + = + 21 171h(x)x 1= = (,1)(1,+) . : = (,1)(1,+) . x F , , x() f h(x) g(x). , :x 1 x 1x 1 x 1x 2 0 x 2 0x 2 x 2 (x 1 x 2)1 10 0 x 1 x 1x 1 x 1., F : AF = (,1)(2,+) .. i) 1I = (,1) . g(x) = x 2 1h(x)x 1= I1 1 = (,0) , xI1 . , F - I1 . (,0) = 1 . xI1 :1x 2 x 2 x 1 x 21 1 x 1 x 1 :x, I1 :1F(x) (x 2).x 1, I1 :1 1F (x) (x 2)(x 2)x 1 x 11 1 f f(x 2) (x 1) x 11x 1 x 22e ln e lnx 2(x 1) x 1 = + .ii) 2I = (2,+). . 8. .F(x) 1 d dt= f (t) = 1d .F(x) = f (t)dt = [1,+) .F(x) = f (x) = 1d = ()d F(x) = f (x) = x2 1 .()d0 x(1,+) .85. 1. :x t21 1) F.) F .. ) :t21 : 2() = 1 -:(,1][1,+) , f : = [1,+) . f - f (t) = t2 1 . , f , , :x1) xA , :x x21 1 F [1, + ) ( ). x(1,+) . [1,x] , 2() = 1 0 , = = 1. , :x1: F(x)0 , x(1,+) F - = [1,+) . F [1,+) x(1,+) :F(x) = x2 10 ., F = [1,+) . 9. = = F(x) = x t f (tx)dt .F(x) = (xt)f (xt) xdt.9 2 :x2x1F(x) dt.1 t =) F.) F , : 2x0 .. ) :21f (t)1 t= (1, 1). x F , :1 x 1 1 x 1 x , 1 x 1 1 x 1 2 . A , F : F= .2 ) G f (1,1), :21G (t) f (t) , t (1,1)1 t., xI , :F(x) = G(x) G(x) , :F(x) = G(x) (x) G(x)(x) = ... =x x = 0,x x x x0 , 2x0 ., F . 3 f, . :120. , x :10 10. : = xt , d = xdt . t = 0 1 = 0 t =1, 2 = x . , :F(x) = f ()d : F(x) = xf (x) .F(x) = f (x t)dt. = = .F(x) f()d f()dF(x) G() = = G(x ) G(x ). = + .= + . d F(x) d d d= = = + = x x d d = .10x0 4 f, . , , . :. : = x t , d = dt . t = : 1 = x t = , : 2 = x . , :x x x x G f , :G() = f (), ., :[ ]x x :F(x) = G(x )(x ) G(x )(x ) = f (x ) f (x ) . 5 :2)1(tx)F(x) dt, x (0, )t)x2x(xt)F(x) dt, x (0, )t ) x0. = xt, o d = xdt. t =1 1 = x t = 2 2 = 2x . , (0, +) :2x 2x 1 2xx x x 12x x1 1 11. = = .= , = + + (1)3 f (t)dt f (t)dt 2x 2x 1.= + . (4) = + . (1)11, (0,+) , :2x x 2x xF (x) (2x)2x x x) x0 = xt :3 F(x) d2xx3 3 2 x 2x = .F (x)x 6 f : , x :x x20 1 f . (1) , x :3f (x) f (x)(x) = 4x + 23f (x) + f (x) = 4x + 2 (2). (2) x x x :3f (x) + f (x) = 4x + 2 . (3) (2) (3) ( f( x)), :1f (x) 2x , x2. (4) (1) f .3. (1) : 2x 2 2x+ + , -2 , (4). (1) ( ) f. 7 f :[0,+) , : f (1) = 1 ( ) ( )101f tx dt f x , x (0, )4. f :[0,+) -. x0 : = tx, d = xdt. 12. t = 0 1 = 0 t = 1 : 2 = x. (1) - x0:1 xx x( ) ( ) ( ) ( ) xf tx dt = f x f d = f x.4 40 0 f (0,+) () :3 2( ) ( ) ( ) ( ) ( ) ( ) ( ) 4f x = f x + xf x xf x 3f x = 0 =0= = = : f (x) = x3 , x[0,+) (2).f (0) lim f (x) lim x 0,x 1 2 t . (1)f x f dt= + 2 t x2 x 2 xx 1 1 x 1 f()= + = + .f x 2 f() xd f x 2 d2 x 2 = + = + = = = x f (x) 2xf (x) 1 f (x)12x f x 3x f x6x ( )3f x = 0x ( ) ( ) ( ) 33f xc c f x c xx= = . f(1) = 1, c = 1 : f (x) = x3 , x0 . 3x 0 x 0+ + . , (2) . 8 f : (0,+) , x0 :( )2 x2x. f : (0,+) -t. x0 := x1d dt= . xt = x 1 =1 t x2 = 2 = x. , (1) :( ) ( )1 1 f (0,+) :2 2 f (x)f (x) x 2 xf (x) x 2f(x) xf (x) 2f(x) xxx f (x) 2xf (x) x3 2 = 24 2(ln x)x x x ( ) 2 2f (x)2ln x c c f (x) x ln x cxx= + = + . 13. (1) : f (1) = 0 , : c = 0 :f (x) x2 ln x, x (0, ) = + . (2). , (2) . . (1)= = . t = x 1 =1 t =1 2 = x. , 1 1= + = + (2)f x 2lnx f()d f (x) 2ln x f ()dx x xf (x) = 2x ln x + f ()d . (3)+ = + + = + = + = + = 13 9 f : (0,+) , x0 :( )12x1 xf x 2lnx f dtt t. f : (0,+) -x. x0 := , t1d xdt2t (1) :( )x x1 1x1 (2) f (0,+) , (3) () :( ) 2 2lnxf x xf (x) 2ln x 2 f (x) f (x)x x2 2lnx 1f (x) dx 2 dx 2 (ln x) ln xdxx x x2ln x ln2 x c (c ) = + + . (1) : f(1) = 0, c = 0 :f (x) = 2ln x + ln2 x, x(0,+) . (4). , (4) . 10 0. - f : , x : 14. = + . (1)f x 3x e f(x t)dt = 2 = 0 . , (1) := = + (2)f x 3x e f()d f (x) 3x e ef()de f (x) = 3x e + e f()d . (3)f (t)dt f (t)dt xf (x) + = . (1)14( )x2 t0. f : . x . : = x t , d = dt . xt = 0 = x 1 t( )0 x2 x 2 x x 0xx 2x 0 (2) f . , (3)(), :exf (x) exf (x) 6 x 2 x x 2 + = xe + 3x e + e f (x)f (x) = 6x + 3xf (x) (6 2 = x + 3x )dx f (x) 3 2 3 = x + x + c . (1) x = 0 f(0) = 0, c = 0 :f (x) = 3x2 + x3 . (4). , (4) . 11 0 f [0, ] f (0) = 0 f (x)0 , x[0,] . x[0,] :x f (x)10 0 : f , f1 f().. f [0, ], - 1-1 - f1 f ([0,]) = [f (0), f ()] = [0, f()] . : 15. x f (x)= + F(x) f (t)dt f (t)dt xf (x). F(x) = f (x) f (t)dt = .g(x) f (t)dt = = =g (x) 2 f (t)dt f (t)dt 2f (x) f (t)dt 2F(x). = = =g(x) 0 f (t)dt 0 f (t)dt 01510 0 F [0, ] :F (x) f (x) f 1(f (x)) f (x) f (x) xf (x) = + = f (x) + xf (x) f (x) xf (x) = 0 . F(x) = c , x[0,] (c) F(0) = 0, c = 0. F(x) = 0, x[0,] . , (1) x[0,] . 12 f :x0 . :f (x) = 0, x . :x 20. F . g :x x x0 0 0 g . g(0) = 2F(0) = 0, x , :x 0g(x) g(0) = 0g(x) 0 x 0g(x) g(0) = 0g(x) 0 . g 0 -, g(0) = 0 . , x : g(x) 0 g(x) 0 , :x 2 x20 0 x . = =0f (t)dt 0 f (x) 0 16. 1. . : .2. . : 3. Louis Brand: .4. G. Thomas R. Finney: .5. M. Spivak: .6. . : .7. ...: .16 30/3/2006 - . - . , . - , - . . - . .. - .
