FONKS\u0130YON<\/p>\n
TANIM: A ve B gibi bo\u015f olmayan iki k\u00fcme i\u00e7in, A n\u0131n her eleman\u0131n\u0131 B\u2019nin bir ve yaln\u0131z bir eleman\u0131 ile e\u015fleyen A\u2019dan B\u2019ye bir f ba\u011f\u0131nt\u0131s\u0131na A \u2018dan B\u2019ye FONKS\u0130YON denir. <\/p>\n
K\u0131saca, A\u2019dan B\u2019ye bir ba\u011f\u0131nt\u0131n\u0131n fonksiyon olmas\u0131 i\u00e7in,<\/p>\n
a) x A i\u00e7in (x, y) f olacak bi\u00e7imde y B olmal\u0131. <\/p>\n
b) A k\u00fcmesinin bir eleman\u0131 B k\u00fcmesinin birden fazla eleman\u0131 ile e\u015flenemez.<\/p>\n
A k\u00fcmesinin f fonksiyonunun TANIM K\u00dcMES\u0130 ve B k\u00fcmesine f fonksiyonunun DE\u011eER K\u00dcMES\u0130 denir. <\/p>\n
f fonksiyonu x A\u2019y\u0131 y B\u2019ye e\u015fliyorsa y\u2019ye x\u2019in g\u00f6r\u00fcnt\u00fcs\u00fc denir ve f: x y veya y = f (x) bi\u00e7iminde g\u00f6sterilir. <\/p>\n
TERS FONKS\u0130YON:
\n f: A B ye, f: x y = f (x) fonksiyonu birebir ve \u00f6rten fonksiyon olsun. B A ya ve y x fonksiyonuna f in tersi denir ve f-1 \u015feklinde g\u00f6sterilir. <\/p>\n
f: A B f-1 : B A
\n f: x y = f (x) f-1 : y x = f-1(y)<\/p>\n
\u00d6RNEKLER:
\n 1. f: R R, f (x) = x + 5 ise f-1(x) nedir?
\n \u00c7\u00f6z\u00fcm:<\/p>\n
2. R+ R ye f (x) = x2 + 2 fonksiyonunun tersini bulunuz (x > 0)
\n \u00c7\u00f6z\u00fcm:<\/p>\n
B\u0130LE\u015eKE FONKS\u0130YON:
\n f: A B ve g: B C birer fonksiyon ise A\u2019daki her eleman\u0131 f ve g fonksiyonlar\u0131 ile C\u2019nin elemanlar\u0131na d\u00f6n\u00fc\u015ft\u00fcren fonksiyon f ile g\u2019nin bile\u015fkesi denir. <\/p>\n
\u00d6ZELL\u0130KLER\u0130:
\n 1) fog gof
\n 2) (fog)oh = fo(goh
\n 3) fof-1 = f-1 of = I ( I birim fonksiyon)
\n 4) foI = Iof = f
\n 5) (f-1)-1 = f
\n 6) (fog)-1 = g-1of-1
\n 7) (fogoh)-1 = h-1 o g-1 o f-1
\n 8) fog = h f = hog-1 ve g = f-1 o h<\/p>\n
\u00d6RNEKLER:
\n 1. R R\u2019ye iki fonksiyon, f (x) = 2x \u2013 1 ve g (x) = x + 1 ise (gof)( – 1) nedir?
\n \u00c7\u00f6z\u00fcm:
\n (gof)(- 1) = g(f(- 1)) = g(2.(- 1) \u2013 1 )
\n = g(- 3) = – 3 + 1 = – 2
\n 2. f ve g : R R\u2019ye
\n f (x) = 3x + 2 ve g(x) = ise, (fog)(x) ve (gof)(x) fonksiyonlar\u0131n\u0131 bulun.
\n \u00c7\u00f6z\u00fcm:<\/p>\n
3. f ve g : R R\u2019ye
\n f (x) = 2x + 1 ve (gof) (x) = 3x + 2 ise, g(x) nedir?
\n \u00c7\u00f6z\u00fcm:
\n (gof of-1)(x) = (3x + 2) of-1<\/p>\n
g (x) = (3x + 2) of-1
\n f (x) = 2x + 1 f-1 (x) = dir.<\/p>\n
4. f ve g : R R\u2019ye f (x) = ve (fog)(x) = 6x + 1 ise g(x) = ?
\n \u00c7\u00f6z\u00fcm:
\n (f-1o fog)(x) = f-1 o (6x + 1)
\n g (x) = f-1 o(6x + 1)
\n f (x) =
\n g (x) = (3x + 1) o (6x + 1)
\n g (x) = 3. (6x + 1) + 1 = 18x + 4
\n 5. f ve g : R R\u2019ye
\n (gof-1) (x) = ve g-1 (x) = 3x \u2013 1 ise f (x) nedir?
\n \u00c7\u00f6z\u00fcm:
\n (g-1ogof)(x) = g-1 o <\/p>\n
L\u0130M\u0130T
\n B\u0130R FONKS\u0130YONUN L\u0130M\u0130T\u0130
\n TANIM
\n A R ve f: A \u2013 {xo} R \u2018ye bir fonksiyon F(x) olsun. x de\u011fi\u015fkeni xo R say\u0131s\u0131na yakla\u015ft\u0131\u011f\u0131nda f(x) fonksiyonu da t R\u2019ye yakla\u015f\u0131yorsa t ger\u00e7el say\u0131s\u0131na x, xo\u2019a yakla\u015f\u0131rken f(x) fonksiyonunun limiti denir ve lim f(x) = t
\n x xo
\n \u015feklinde g\u00f6sterilir. <\/p>\n
SA\u011eDAN VE SOLDAN L\u0130M\u0130T:
\n SA\u011eDAN L\u0130M\u0130T:
\n y = f(x) fonksiyonunda x, xo R de\u011ferine sa\u011f taraftan yakla\u015f\u0131rken f de bir t1 R de\u011ferine yakla\u015f\u0131yorsa t1\u2019e fonksiyonun sa\u011fdan limiti denir ve lim f(x) = t1 bi\u00e7iminde
\n x x+o
\n g\u00f6sterilir. <\/p>\n
SOLDAN L\u0130M\u0130T:
\n y = f(x) fonksiyonunda x, xo R de\u011ferine sol taraftan yakla\u015f\u0131rken f de bir t2 R de\u011ferine yakla\u015f\u0131yorsa t2 ye fonksiyonun soldan limiti denir ve lim f(x) = t2
\n x x-o<\/p>\n
\u00d6RNEK:
\n x2 + 1, x 0 ise,
\n x + 1 , x < 0 ise,\n\n fonksiyonun x = 0 noktas\u0131nda limiti nedir?\n\n \u00c7\u00d6Z\u00dcM:\n lim f(x) = lim (x2 + 2) = 02 + 1 = 1\n x 0+ x 0+\n\n lim f(x) = lim (x + 1) = 0 + 1 = 1\n x 0- x 0-\n\n O halde lim f(x) = 1 dir.\n x 0\n\n\n L\u0130M\u0130T TEOREMLER\u0130:\n\n 1) lim (f(x) g(x)) = lim f(x) lim g(x)\n x x0 x x0 x x0\n\n 2) lim (f(x).g(x)) = lim f(x).lim g(x)\n x x0 x x0 x x0\n\n 3) lim c = c (c R)\n x x0\n\n 4) lim (c.f(x)) = c . lim f(x)\n x x0 x x0 \n\n 5) g(x) 0 ve lim g(x) 0 ise\n x x0 \n\n 6) n N+ olmak \u00fczere\n\n 7) n tek do\u011fal say\u0131 ise, \n\n\n 8) n \u00e7ift do\u011fal say\u0131 ve f(x) 0 ise\n\n BEL\u0130RS\u0130ZL\u0130KLER VE L\u0130M\u0130TLER\u0130\n\n A) BEL\u0130RS\u0130ZL\u0130\u011e\u0130N\u0130N L\u0130M\u0130T\u0130:\n\n \u00d6RNEK:\n\n ifadesinin de\u011feri nedir?\n\n\n \u00c7\u00d6Z\u00dcM:\n\n B) BEL\u0130RS\u0130ZL\u0130\u011e\u0130N L\u0130M\u0130T\u0130:\n\n \u00d6RNEK:\n\n limitinin de\u011feri nedir?\n\n \u00c7\u00d6Z\u00dcM:\n\n Pay\u0131n derecesi paydadan b\u00fcy\u00fck oldu\u011fundan\n\n \u00c7\u00d6Z\u00dcML\u00dc TEST\n\n 1. de\u011feri a\u015fa\u011f\u0131dakilerden hangisidir?\n\n A) \u20132 B) \u20131 C) 0 D) 1 E) 2\n\n \u00c7\u00f6z\u00fcm 1.: \n\n d\u0131r. O halde,\n\n Cevap: B\n\n 2. limitinin de\u011feri nedir?\n\n A) B) C) D) E) \n\n \u00c7\u00f6z\u00fcm 2.:\n\n\n Cevap: C\n\n T\u00dcREV VE UYGULAMALARI\n\n TANIM: y = f(x) fonksiyonu [a, b] kapal\u0131 aral\u0131\u011f\u0131nda tan\u0131ml\u0131 ve s\u00fcrekli, x0 (a,b) olsun.\n\n limiti bir ger\u00e7el say\u0131 ise,\n\n bu limite y = f(x) fonksiyonunun x = x0 noktas\u0131ndaki T\u00dcREVi denir ve f\u2019(x0) \u015feklinde g\u00f6sterilir. \n\n \u00d6RNEK:\n\n f : R R, f(x) = -x2 + 2 fonksiyonunun x0 = 1 noktas\u0131ndaki t\u00fcrevi nedir?\n\n \u00c7\u00d6Z\u00dcM: \n\n f(1) = - 12 + 2 = 1\n f\u2019(1) \n\n \u00d6RNEK:\n\n f(x) = |x2 \u2013 4| fonksiyonu verilir. \n\n a) f\u2019(2) = ? b) f\u2019(1) = ?\n\n \u00c7\u00d6Z\u00dcM:\n\n a) f (2) =|22 \u2013 4| = 0 oldu\u011fu i\u00e7in fonksiyonun x = 2 noktas\u0131nda t\u00fcrevi yoktur.\n\n b) \n\n T\u00dcREV ALMA KURALLARI:\n\n 1) c R olmak \u00fczere \n f (x) = c f\u2019(x) = 0\n 2) f (x) = x f\u2019(x) = 1\n 3) f (x) = cx f\u2019(x) = c\n 4) f (x) = c . xn f\u2019(x) = c . n . xn-1\n 5) f (x) = c . un f\u2019(x) = c . n . un-1 . u\u2019x\n 6) f (x) = u v f\u2019(x) = u\u2019x v\u2019x\n 7) f (x) = u . v f\u2019(x) = u\u2019x . v + v\u2019x . u\n 8) f (x) = u . v . t f\u2019(x) = u\u2019x . v. t + v\u2019x . u . t\n + t\u2019x . u . v\n 9) f (x) = \n 10) f (x) = \n\n \u00d6RNEKLER:\n 1. f (x) = 5 f\u2019(x) = 0\n 2. f (x) = f\u2019(x) = 0\n 3. f (x) = x5 f\u2019(x) = 5x4\n 4. f (x) = x f\u2019(x) = 1\n 5. f (x) = 2x f\u2019(x) = 2\n 6. f (x) = \n\n 7. f (x) = x4 \u2013 x3 + 2x \u2013 3 fonksiyonunun t\u00fcrevi nedir?\n \u00c7\u00d6Z\u00dcM:\n f\u2019(x) = 4x3 \u2013 3x2 + 2\n\n 8. f (x) = (3x2 + 5)11 fonksiyonunun t\u00fcrevi nedir?\n \u00c7\u00d6Z\u00dcM:\n f\u2019(x) = 11 (3x2 + 5)10 . (3x2 + 5)\u2019\n = 11(3x2 + 5)10 . 6x\n = 66x (3x2 + 5)10\n\n 9. f (x) = fonksiyonunun t\u00fcrevi nedir?\n \u00c7\u00d6Z\u00dcM:\n\n olur. \n\n TR\u0130GONOMETR\u0130K FONKS\u0130YONLARIN T\u00dcREV\u0130:\n A) \n 1) f (x) = Sinx f\u2019(x)=Cosx\n 2) f (x) = Cosx f\u2019(x) = - Sinx\n 3) f (x) = tanx f\u2019(x) = 1 + tan2x\n\n 4) f (x) = Cotx f\u2019(x) = - (1 + Cot2x)\n\n \u00d6RNEKLER:\n 1. f (x) = Secx f\u2019(x) = ?\n \u00c7\u00d6Z\u00dcM:\n\n 2. f (x) = Cosec f\u2019(x) =?\n \u00c7\u00d6Z\u00dcM:\n\n\n B. \n 1) f (x) = Sin[u[x]] f\u2019(x) = u\u2019(x) . Cos[u(x)]\n 2) f (x) = Cos [u(x)] f\u2019(x) = - u\u2019(x) . Sin [u(x)]\n 3) f (x) = tan [u(x)] f\u2019(x) = u\u2019(x) [1 + tan2u(x)]\n\n 4. f (x) = Cot[u(x)] f\u2019(x) = -u\u2019(x) [1 + Cot2u(x)]\n\n \u00d6RNEKLER:\n 1. f (x) = Sin3x f\u2019(x) = 3Cos3x\n 2. f (x) = tan(x2 \u2013 1) f\u2019(x) = ?\n \u00c7\u00d6Z\u00dcM:\n f\u2019(x) = (x2 \u20131)\u2019 . [1 + tan2(x2 \u2013 1)]\n f\u2019(x) = 2x [1 + tan2 (x2 \u2013 1)]\n 3. f (x) = Sin (tan x) fonksiyonunun t\u00fcrevi nedir?\n \u00c7\u00d6Z\u00dcM:\n f\u2019(x) = Cos (tanx) . (tanx)\n\n 4. f (x) = 2Sin3 x + 3Cos2x f\u2019(x) = ?\n \u00c7\u00d6Z\u00dcM:\n f\u2019(x) = 2.3.Sin2x . (Sin x)\u2019 + 3.2 Cosx . (Cosx)\u2019\n f\u2019(x) = 6Sin2x . Cosx + 6 Cosx . ( - Sin x)\n\n \u0130NTEGRAL\n TANIM:\n f: [a,b] R ve F:[a, b] R ye tan\u0131ml\u0131 iki fonksiyon olsun, [a,b] i\u00e7in, F\u2019(x) = f(x) yaz\u0131labilirse F(x)\u2019e f(x)\u2019in ilkel fonksiyonu yada integrali denir. \n F\u2019(x) dx = F(x) veya\n f(x) dx = F(x) \u015feklinde g\u00f6sterilir. \n\n \u00d6RNEK: \n f (x) = 2x2 f\u2019(x) = 4x 4xdx = 2x2\n f (x) = 2x2 \u2013 1 f\u2019(x) = 4x 4xdx = 2x2 \u2013 1 \n f (x) = 2x2 + 3 f\u2019(x) = 4x 4xdx =2x2 + 3\n\n BEL\u0130RS\u0130Z \u0130NTEGRAL \u00d6ZELL\u0130KLER\u0130:\n A. f\u2019(x) dx = f(x) + C\n B. d[f (x)] = f (x) + C\n C. f (x)dx = f (x) dx ( R)\n D. [f (x) g(x)] dx= f(x) dx g (x)dx\n E. [ f (x) dx] = f (x)\n F. d[ f (x)dx] = f(x) dx\n\n \u00d6RNEKLER:\n 1. 2x dx = x2 + C \n 2. d(3x2) = 3x2 + C\n 3. 5x4dx = 5 x4dx\n 4. (x3 + x)dx = x3 dx + x dx\n 5. [ 2x dx] = 2x\n 6. d (x3dx) = x3dx\n\n\n\n \u00d6RNEKLER:\n 1. \n 2. 12dx = 12x + C\n 3. \n 4. (x3 + x2 \u2013 2)2 (3x2 + 2x)dx = ?\n \u00c7\u00d6Z\u00dcM 4:\n x3 + x2 \u2013 2 = u (3x2 + 2x) dx = du\n\n\n TR\u0130GONOMETR\u0130K \u0130NTEGRAL:\n A. Cos x dx = Sin x + C\n B. Sin x dx = - Cosx + C\n C. Sec2x dx = (1 + tan2x) dx\n\n D. Cosec2x dx = (1 + Cot2x) dx =\n = \n\n \u00d6RNEKLER:\n 1. Cos2x . Sin x dx = \n \u00c7\u00d6Z\u00dcM:\n Cosx = u -Sin x dx = du\n Sin x dx = - du\n u2 . (-du) = - u2 . du\n\n\n\n 2. Sin 3x dx = ?\n \u00c7\u00d6Z\u00dcM:\n\n 3. Cos (2x + 1) dx = ?\n \u00c7\u00d6Z\u00dcM:\n\n\n LOGAR\u0130TM\u0130K VE \u00dcSTEL \u0130NTEGRAL: \n A. \n B. \n C. eu du = eu + C\n D. \n\n \u00d6RNEKLER:\n 1. \n 2. tan x dx = ?\n \u00c7\u00d6Z\u00dcM:\n\n Cos x = u - Sin x dx = du\n Sin x dx = - du\n\n = - ln |u| + C = - ln |Cos x| + C\n 3. ex dx = ex + C...\n<\/p>\n","protected":false},"excerpt":{"rendered":"
FONKS\u0130YON TANIM: A ve B gibi bo\u015f olmayan iki k\u00fcme i\u00e7in, A n\u0131n her eleman\u0131n\u0131 B\u2019nin bir ve yaln\u0131z bir eleman\u0131 ile e\u015fleyen A\u2019dan B\u2019ye bir f ba\u011f\u0131nt\u0131s\u0131na A \u2018dan B\u2019ye FONKS\u0130YON denir. K\u0131saca, A\u2019dan B\u2019ye bir ba\u011f\u0131nt\u0131n\u0131n fonksiyon olmas\u0131 i\u00e7in, a) x A i\u00e7in (x, y) f olacak bi\u00e7imde y B olmal\u0131. b) A …<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1404,1403],"tags":[7408,7407],"class_list":["post-3179","post","type-post","status-publish","format-standard","hentry","category-matematik-odevleri","category-odevler","tag-birim-fonksiyon","tag-fonksiyonlar-limit-turev-integral"],"_links":{"self":[{"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/posts\/3179","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/comments?post=3179"}],"version-history":[{"count":0,"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/posts\/3179\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/media?parent=3179"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/categories?post=3179"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/tags?post=3179"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}