www.aydinyayinlari.com.tr ÷/5&(3\"- 7. MODÜL ·/÷7&34÷5&:&)\";*3-*, ÖRNEK 2 y = x3 (SBGJLUFZ= x3 ile ÖRNEK 4 y = x2 y = x2FôSJMFSJBSBTO- y y = x2 -QBSBCPMÐWFZ= x + 1 EBLBMBOCÌMHFOJOBMB- EPôSVTVJMFTOSMCÌMHFOJOBMBOLBÀCS2EJS OOCVMVOV[ x y y =x2 – 1 x2 - 1= x + 1 y=x+1 x2 - x - 2 = 0 (x - 2) (x + 1) = 0 –1 O 1 x x = Y= -1 –1 x2 = x3 j x = Y= 1 1 34 1 # ^ x2 - x3 h dx = x x - 11 1 34 =-= 0 0 3 4 12 22 a ^ x + 1 h - ^ x2 - 1 h k dx = 2 # #A = ^ - + x + 2 h dx x –1 –1 32 2 = -x x =d -8 +2+4n + + 2x 32 –1 3 -d 1 + 1 - 2 n = 9 32 2 ÖRNEK 3 y = 1 x2 + 1 ÖRNEK 5 2 y = x2 - 1 ve y = x2 ve y = 2x - x2 FôSJMFSJBSBTOEBLBMBOCÌMH FOJOBMBOLBÀCS2EJS FôSJMFSJJMFTOSMCÌMHFOJOBMBOLBÀCJSJNLBSFEJS y y = x2 +1 x 2 2 2 + 1 = 2 - 1 x2 = 2x - x2 x j 2x2 =Y x x 2 y O1 x = Y= 1 O x 2= y =x2 – 1 2 x2 = 4 x=±2 22 1 #A = f f x + 1 p - ^ x2 - 1 h p dx #A = ^ 2x - 2 h - x 2 dx 2 –2 x 0 22 32 #A = f - x + 2 p dx = - x + 2x 11 2 21 2 6 –2 #A = ^ 2x - 2 h dx = 2 - 3 =1- = –2 2x x x 3 33 0 0 =d -8 +4 n-d 8 -4 n 66 - 16 32 16 = +8= = 6 63 1 16 49 9 1 2. 3. 4. 5. 12 3 23
TEST - 22 ÷LJ'POLTJZPOVO(SBGJLMFSJ\"SBTOEB,BMBO#ÌMHFOJO\"MBO 1. y = x2 + 1 ve y = 9 - x2 5. y QBSBCPMMFSJ BSBTOEB LBMBO CÌMHFOJO BMBO LBÀ g(x) = x2 – 3x br2EJS x A ) 32 # 36 C ) 16 % ) 64 E ) 32 f(x) = –x2 + 6x 35 33 Denklemleri f ( x ) = -x2 + 6x ve g(x) = x2 - 3x PMBOFôSJMFSJOTO SMBE ôCÌMHFOJOBMBOLBÀCS2 EJS \" # 257 C ) 243 % ) 567 E ) 600 8 887 2. Denklemi y = x2 -YPMBOFôSJJMFEFOLMFNJ 6. 3x + y = 3 y = 4 - x2PMBOFôSJBSBTOEBLBMBOCÌMHFOJOala- EPôSVTVWFZ= x2 - 4x +FôSJTJJMFTOSMCÌM- OLBÀCS2EJS HFOJOBMBOLBÀCS2EJS \" # $ % & A ) 17 # 8 C ) 5 % & 1 6 3 2 6 3. y = x3 7. y = x2 ve y2 = 4x 2 4 FôSJTJ JMF Z = 4x dPôSVTV BSBTOEB LBMBO CÌM FôSJMFSJBSBTOEBLBMBOBMBOLBÀCS2EJS genin BMBOLBÀCS2EJS \" # $ % & A) 8 # 16 C ) 25 % 32 40 3 3 E) 3 33 4. y y y = ax2 1 8. y = x –1 x Ox O 1 –2 õFLJMEFLJUBSBMCÌMH FOJOBMBOLBÀCS2EJS (SBGJLUFLJUBSBMBMBO 1 br2 ise a LBÀUS 24 A ) 4 # 5 $ % 7 8 A) 1 # $ 3 % & 5 33 3 E) 2 2 2 3 1. D 2. B 3. D 4. B 50 5. C 6. & 7. B 8. D
÷LJ'POLTJZPOVO(SBGJLMFSJ\"SBTOEB,BMBO#ÌMHFOJO\"MBO TEST - 23 1. ôFLJMEFCJS¿JGU¿JOJOEJLпHFOõFLMJOEFLJBSB[JTJWF 4. y = x2 + a y UBSBM CËMHF JTF FLJN ZBQUó CËMHFZJ HËTUFSNFLUF- Ox EJS y g(x) = 2x + 8 y = 2x Ox #JSJODJCÌMHFEF Z= x2 +BQBSBCPMÑJMFCVQB- f(x) = 4–x2 SBCPMÑOPSJKJOEFOHFÀFOUFôFUJPMBOZ=YEPô- #VOBHÌSF FLJNZBQMBOCÌMHFOJOBMBOLBÀCJ- SVTVWFZFLTFOJBSBTOEBLBMBOöFLJMEFLJUBSBM SJNLBSFEJS CÌMHFOJOBMBOOWFSFOJOUFHSBMBöBôEBLJMFSEFO IBOHJTJEJS A ) 74 # 76 C ) 77 % 79 E) 80 2 1 33 333 #A ) ^x2 + 2hdx ## ^x2– 1hdx 00 1 2 #C ) ^x2 + x – 1hdx #% ^x2 + xhdx 0 0 2. y = 4 - x2 1 QBSBCPMÑJMFZ= - 2x +EPôSVTVOVOTOSMBE- #E ) ^x – 1h2 dx ôCÌMHFOJOBMBOLBÀCJSJNLBSFEJS 0 5. f ( x ) = 4x3 A ) 43 # 32 $ % 26 14 FôSJTJOJOY=OPLUBTOEBLJUFôFUJ YFLTFOJWF 33 E) FôSJ BSBTOEBLJ CÌMHFOJO BMBO LBÀ CJSJN LBSF- EJS 33 \" # $ % 16 E ) 4 3 3. y 6. y d 16 A A 4x –4 y=16–x2 O ôFLJMEFEEPóSVTVWFZ= 16 - x2QBSBCPMÐ O 2x A ( - WF OPLUBMBSOEBLFTJõJZPSMBS õFLJMEFG Y = x2 - 2x +FôSJTJOJOUFQFOPLUB- #VOBHÌSF UBSBMBMBOLBÀCS2EJS T\"PMEVôVOBHÌSF UBSBMCÌMHFOJOBMBOLBÀCJ- SJNLBSFEJS A ) 74 # 76 C ) 79 % 80 E ) 82 33 3 33 \" # 14 $ % 10 E ) 3 3 3 1. B 2. B 3. D 51 4. & 5. D 6. D
TEST - 24 ÷LJ'POLTJZPOVO(SBGJLMFSJ\"SBTOEB,BMBO#ÌMHFOJO\"MBO 1. y 3. ôFLJMEFLJ HSBGJLUF Z = ' Y GPOLTJZPOVOVO HSBGJóJ A1 A2 y = f(x) WFSJMNJõUJS y = g(x) y x y = F(x) O1 47 –4 O x 2 5 :VLBSEBLJHSBGJLUFZ= f ( x ) ve y = g ( x ) fonksiyon- #VOBHÌSF CPZBMCÌMHFMFSJOBMBOMBSUPQMBNO MBSOOHSBGJLMFSJWFSJMNJõUJS\"1 ve A2J¿JOEFCVMVO- WFSFOJOUFHSBMBöBôEBLJMFSEFOIBOHJTJEJS EVLMBSCËMHFMFSJOBMBOMBSPMNBLÐ[FSF 5 7 A) # F_ x i dx # _ f_ x i - g_ x i idx = 6 ve –4 1 25 7 # # F_ x i dx + # F_ x i dx # _ f_ x i - g_ x i idx = - 8 –4 2 4 –4 2 PMNBLÑ[FSF \"1 +\"2LBÀCJSJNLBSFEJS C) # F_ x i dx - # F_ x i dx \" # $ % & 25 –4 % # F_ x idx 5 25 E) - # F_ x i dx - # F_ x i dx –4 2 4. y g(x) = x2– 2x 2. y f(x) = x2 – 2x + 13 Ox g(x) = x2 + 4x + 1 F(x) = –x2+4x–4 x :VLBSEBLJõFLJMEF F ( x ) = -x2 + 4x - 4 ve g ( x ) = x2- 2x O1 3 GPOLTJZPOMBSOOHSBGJLMFSJWFSJMNJõUJS :VLBSEBLJHSBGJLUFG Y = x2- 2x + 13 ve g ( x ) = x2 + 4x +GPOLTJZPOMBSOOHSBGJLMFSJWFSJM- #VOBHÌSF CPZBMBMBOMBSUPQMBNLBÀCJSJNLB- SFEJS NJõUJS #VOB HÌSF CPZBM CÌMHFMFSJO BMBOMBS UPQMBN LBÀCJSJNLBSFEJS \" # $ % & \" # $ % & 1. \" 2. C 52 3. C 4. B
÷LJ'POLTJZPOVO(SBGJLMFSJ\"SBTOEB,BMBO#ÌMHFOJO\"MBO TEST - 25 1. ôFLJMEFZ=' Y QBSBCPMÐOÐOHSBGJóJWFSJMNJõUJS 3. ôFLJMEF Z = ' Y GPOLTJZPOVOVO HSBGJóJ WFSJMNJõ- y tir. y 10 6 8 –2 O 5 x a b y = F(x) O6 x y = F(x) #VOBHÌSF UBSBMCÌMHFMFSJOBMBOMBSUPQMBNLBÀ 6 CJSJNLBSFEJS # F_ x idx = 32 \" # 16 C) 17 % & 19 3 3 3 0 PMEVôVOBHÌSF UBSBMCÌMHFOJOBMBOLBÀCJSJN- LBSFEJS \" # $ % & 2. y 4. ôFLJMEFZ=' Y GPOLTJZPOVOVOHSBGJóJWFSJMNJõUJS y = F(x) (k ` R+) S1 S2 y = g(x) y F(x) = x2+k x 5 S2 aO b c S1 y=5 :VLBSEBLJõFLJMEFZ= F ( x ) ve y = g ( x ) fonksiyon- O x MBSOOHSBGJLMFSJWFSJMNJõUJS41 ve S2J¿JOEFCVMVO- 8 EVLMBSCËMHFMFSJOBMBOMBSPMNBLÐ[FSF cc S1 ve S2J¿JOEFCVMVOEVLMBSCËMHFMFSJOBMBOMBSPM- # _ F_ x i - g_ x i idx = 3 # _ g_ x i - F_ x i i dx ab NBLÐ[FSF PMEVôVOBHÌSF S1 PSBOLBÀUS 8 S2 S = 4S ve # F_ x i dx = 34 1 2 0 A) 3 # C) 5 % & PMEVôVOBHÌSF 41 kaÀCJSJNLBSFEJS 2 2 \" # $ % & 1. & 2. & 53 3. D 4. C
·/÷7&34÷5&:&)\";*3-*, 7. MODÜL ÷/5&(3\"- www.aydinyayinlari.com.tr %0ó364\"-)\"3&,&5130#-&.-&3÷ %Ñ[HÑO%PôSVTBM)BSFLFU %Ñ[HÑO:BWBöMBZBO)BSFLFU Konum (x) &óJN= tana = Dx Konum (x) )[ 7 Dt 7 =)[ Dx a Dt Zaman (t) 0 Zaman (t) Dt Zaman (t) )[ 7 Alan = V. Dt öWNF B Dt Zaman (t) 7 –a Dt Zaman (t) %Ð[HÐO ZBWBõMBZBO IBSFLFU HSBGJLMFSJOEF FóJN WF öWNF B I[B[BMNBLUBPMVQJWNFTBCJUWFOFHBUJGUJS FóJNJ FóJNJ ,POVNm[BNBO )[m[BNBO öWNFm[BNBO Zaman (t) BMBO BMBO %Ð[HÐO EPóSVTBM IBSFLFU HSBGJLMFSJOEF LPOVN–[B- ÖRNEK 1 NBOHSBGJóJOJOFóJNJBOMLI[ I[–[BNBOHSBGJLMF- SJOEFJTFFóJNBOMLJWNFZJWFSJS )[ LNTB :BOEBLJ HSBGJLUF CJS BSBDO 120 I[m[BNBO HSBGJóJ WFSJMNJõ- %Ñ[HÑO)[MBOBO)BSFLFU 80 tir. 0 4 ;BNBO TB #VOB HÌSF CV BSBDO TBBU JMF TBBU BSBTOEB- )[ 7 LJLPOVNEFôJöJNJLBÀLN EJS Konum (x) 7 Denklemi y = -10x +ÀLBS 5 # ^ - 10x + 120 h dx = 160 3 0 Zaman (t) Dt Zaman (t) öWNF B ÖRNEK 2 a Konum (m) :BOEBLJ õFLJMEF CJS IB- Dt Zaman (t) x = 40 + 3t2 SFLFUMJOJOLPOVNm[BNBO %Ð[HÐOI[MBOBOIBSFLFUHSBGJLMFSJOEFFóJNWFI[ 40 BSUNBLUBPMVQJWNFTBCJUWFQP[JUJGUJS 0 ;BNBO TO HSBGJóJWFSJMNJõUJS #V IBSFLFUMJOJO [BNBOB CBóM LPOVN EFOLMFNJ x = 40 + 3t2 PMEVóVOB HËSF IBSFLFUMJOJO a) TBOJZFEFLJBOMLI[LBÀNTOEJS b) TBOJZFEFLJBOMLJWNFTJLBÀNTO2 EJS a) x' = 6t =W U= 2 jNTO b) v' = 6 j a =NTO2 54 1. 160 2. a) 12 b) 6
÷OUFHSBM KARMA TEST - 1 1. # 2x dx 4. y = F ( x ) fonksiyonunun x =OPLUBTOEBLJUFóFUJY x4 + 2x2 + 1 FLTFOJJMFMJLWFY=OPLUBTOEBLJUFóFUJJTFY integrBMJOJOFöJUJBöBôEBLJMFSEFOIBOHJTJEJS FLTFOJJMFMJLB¿ZBQNBLUBES A) - 1 + c B) 1 + c 4 x2 + 1 x2 + 1 #VOB HÌSF # F' (x) .F'' _ x i dx JOUFHSBMJOJO TP- C) - 1 + c D) 1 + c 2_ x2 + 1 i 2_ x2 + 1 i 2 OVDVLBÀUS A) 1 B) 2 C) 3 D) 4 E) 5 E) 2 + c x2 + 1 5. ôFLJMEF Z = ' Y GPOLTJZPOVOVO HSBGJóJ WFSJMNJõ- UJS y 3 y = F(x) 2. # x2 dx –2 O x 4 x6 - 4x3 + 4 JOUFHSBMJOJOFöJUJBöBôEBLJMFSEFOIBOHJTJEJS A) - 1 + c B) - 1 + c –6 2_ x3 - 2 i 3_ x3 - 2 i C) - 2 + c D) - 3 + c 4 x.F'_ x i - F_ x i x3 - 2 x3 - 2 # E) 2_ x3 - 2 i + c #VOB HÌSF –2 2x2 dx integralinin 3 TPOVDVBöBôEBLJMFSEFOIBOHJTJEJS A) - 13 B) - 12 C) - 11 12 11 10 - 10 -9 D) E) 9 8 11 E) 6 6. N>PMNBLÐ[FSF 3. # F_ x idx = 8 PMEVôVOBHÌSF m 3 # _ - x2 + 4x - 3 i dx 3 1 # 72 - F_ 4x - 1 iA dx JOUFHSBMJOJOBMBCJMFDFôJen CÑZÑLEFôFSJLBÀUS 2 45 1 A) B) 1 C) D) E) 2 integralinJOTPOVDVLBÀUS 3 33 A) 2 B) 3 C) 4 D) 5 55 4. A 5. E 6. C 1. A 2. B 3. A
KARMA TEST - 2 ÷OUFHSBM 1. y r y= 3x 3. 6 sin x dx x2 + y2 = 16 # x r x4 + x2 + 1 O – :VLBSEBLJõFLJMEFY2 + y2 =¿FNCFSJWF 6 y = 3 x EPóSVTVWFSJMNJõUJS JOUFHSBMJOJO TPOVDV BöBôEBLJMFSEFO IBOHJTJOF 5BSBMCÌMHFOJOBMBOOWFSFOJOUFHSBMBöBôEBLJ- MFSEFOIBOHJTJEJS FöJUUJS 1 A ) 4r2 B ) 3r - 1 C ) 5r 5 2 6 A) # a 3 x - 16 - x2 k dx D) 1 & 0 2 4. #JSIBSFLFUMJJMLI[OEBOCBõMBZQEPóSVTBMCJSõF- B) # a 3 x - 16 - x2 k dx LJMEF I[MBOBSBL TO EF NT I[B VMBõNõUS %BIBTPOSBI[OEPóSVTBMCJSõFLJMEFB[BMUBSBL 0 TOEFNTI[BEÐõÐSNÐõUÐS4POSBTBCJUI[- 1 MB TO HJUNJõ WF EBIB TPOSB EPóSVTBM CJS õFLJMEF ZBWBõMBZBSBLTOEFEVSNVõUVS C) # a 16 - x2 - 3 x k dx #VIBSFLFUMJOJO[ ]TBOJZFMFSBSBTOEBBME- 0 2 ôUPQMBNZPMLBÀNFUSFEJS D) # a 16 - x2 - 3 x k dx \" # $ % & 0 π 3 E) # a 16 - x2 - 3 x k dx 0 2. y ôFLJMEFLJUBSBMCËM- 5. :BUBZEÑ[MFNEFIBSFLFUFEFOCJSDJTNJOJWNF- O ab HFOJOBMBO [BNBOEFOLMFNJB U = 3t -WFU=BOO- EB I[ NT BMEô ZPM N PMEVôVOB HÌSF y= k b iMF CVMB- CV DJTNJO ZPM – [BNBO EFOLMFNJ BöBôEBLJMFS- x k ln f p EFOIBOHJTJEJS a x A) S (t) = t3 + t2 – 8t CJMEJóJOFHËSF 2 y \"CÌMHFTJOJOBMB- B) S (t) = t3 – t2 + 8t + 20 2 y = 2x OLBÀCJSJNLBSF- C) S (t) = t3 - t2 + 8t + 3 y = 2 EJS x D) S (t) = t3 + t2 - 8t + 3 OA x E) S (t) = t4 – 3t2 - 4t + 20 2 \" MO # MO $ MO F & MO F % MO F 1. D 2. D 56 3. E 4. C 5. B
÷OUFHSBM KARMA TEST - 3 1. y = ' Y GPOLTJZPOVOVO UBONM PMEVóV BSBMLUB 5. ' Y GPOLTJZPOVOVOEJGFSBOTJZFMJE' Y PMNBL \" Y Z OPLUBTOEBLJUFóFUJOJOFóJNJ x EJS Ñ[FSF y 3 - x3E' Y =YWF' = 5 ' Y GPOLTJZPOV # OPLUBTOEBO HFÀUJôJ- PMEVôVOB HÌSF ' - BöBôEBLJMFSEFO IBOHJ- TJEJS OFHÌSF F_ 13 iEFôFri BöBôEBLJMFSEFOIBOHJ- TJEJS A) -4 B) -3 C) -2 D) - & \" # $ % & 6. # 3 2x + 5 - 3 dx 2x + 5 2. f( x ) = 5x4 + 4x3 +YWFG = 2 integraliOEFV6 =Y+EFôJöLFOEFôJöUJSNF- TJZBQMSTBBöBôEBLJJOUFHSBMMFSEFOIBOHJTJFM- PMEVôVOBHÌSF G EFôFSJLBÀUS EFFEJMJS A ) 12 17 C) 6 11 13 A) # _ u4 - 3u2 i du B) 3 # _ u4 - 3u2 idu B) D) E) C) # _ 3u2 - u4 idu D) 6 # _ 3u2 - u4 i du 2 22 E) 6 # _ u4 - 3u2 idu J 9 N d K O 3. f'' ( x ) = 12x2 - 6x + 2 7. dx KK # _ x 2 - 2x + 1 idx OO GPOLTJZPOVJÀJOGh =WFG = -JTF L3 P G - LBÀUS integSBMJOJO TPOVDV BöBôEBLJMFSEFO IBOHJTJ- A ) - # - $ -7 D ) -5 E ) -4 EJS A) -2 B) - $ % & 4. d # d ^ d f^ x hh 2 dx # x . x - 1 dx JGBEFTJOJOFöJUJBöBôEBLJMFSEFOIBOHJTJEJS –1 \" Gh Y EY # Gh Y + c C ) f'( x ) integralJOJO TPOVDV BöBôEBLJMFSEFO IBOHJTJ- EJS % Ghh Y & Ghh Y EY A) 3 B) 5 C) 11 D) 13 E) 15 2 3 6 62 1. A 2. D 3. E 4. E 57 5. B 6. B 7. C C
KARMA TEST - 4 ÷OUFHSBM 1. # F2 _ x i dx = x5 - 2x3 + x + c 5. (FS¿FM TBZMBS LÐNFTJOEF UBONM WF TÐSFLMJ CJS ' 53 10 PMEVôVOBHÌSF ' Y GPOLTJZPOVOVOHSBGJôJJMF fonksiyPOVJ¿JO # F_ x i dx = 12PMEVóVOBHËSF Y FLTFOJ BSBTOEB LBMBO CÌMHFOJO BMBO LBÀ CJ- SJNLBSFEJS 4 A) 2 B) 1 C) 4 D) 2 E) 8 4 3 33 # _ 5 - F_ 3x - 2 i i dx 2 integSBMJOJO TPOVDV BöBôEBLJMFSEFO IBOHJTJ- EJS \" # $ % & 2. F''( x ) = 4x +PMNBLÐ[FSF ' Y GPOLTJZPOVOB Af 1, 49 pOPLUBTOEBOÀJ- 6 [JMFOUFôFUJOFôJNJPMEVôVOBHÌSF ' EFôF- SJBöBôEBLJMFSEFOIBOHJTJEJS \" # $ % & 3. ôFLJMEF' Y GPOLJTZPOVOVOUÐSFWJOJOHSBGJóJWFSJM- 6. ' HFS¿FM TBZMBS LÐNFTJ Ð[FSJOEF UÐSFWMFOFCJMJS CJS NJõUJS GPOLTJZPO WF HFS¿FM TBZMBS LÐNFTJOEF TÐSFLMJ CJS y GPOLTJZPOPMNBLÐ[FSF YHFS¿FMTBZTJ¿JO F'( x ) =H Y FõJUMJóJTBóMBOZPS 2 F'(x) ' =WF' =PMEVôVOBHÌSF 3 # F_ x i.g_ x i dx 2 integralinJOEFôFSJBöBôEBLJMFSEFOIBOHJTJEJS \" # $ % & O1 x ' - =PMEVôVOBHÌSF ' EFôFSJBöBôEB- LJMFSEFOIBOHJTJEJS A) 3 B) 4 C) 5 D) 6 E) 7 7. # F'_ x i dx = # 4 dx WFF_ 0 i = 1 F2_ x i 4 4. NQP[JUJGHFS¿FLTBZPMNBLÐ[FSF Z=NYEPóSVTV PMEVôVOB HÌSF ' EFôFSJ BöBôEBLJMFSEFO JMFZ= 3x2QBSBCPMÐBSBTOEBLBMBOTOSMCËMHFOJO IBOHJTJEJS BMBOCJSJNLBSFPMEVóVOBHËSF NEFôFSJBöBô- EBLJMFSEFOIBOHJTJEJS \" # $ % & A) - 1 B) - 1 C) - 1 D) 1 E) 1 8 4 22 4 1. C 2. A 3. E 4. D 5. A 6. D 7. B
÷OUFHSBM KARMA TEST - 5 8 13 4. ôFLJMEFZ=' Y GPOLTJZPOVOVOHSBGJóJWFSJMNJõUJS 1. # F_ x i dx = 10WF # F_ x i dx = 6 y 35 1 PMEVóVOBHËSF –1 x O 3 5 13 y = F(x) # F_ x idx - # F_ x i dx 38 JöMFNJOJOTPOVDVBöBôEBLJMFSEFOIBOHJTJEJS A) - # $ % & 2 #VOB HÌSF # x.F_ x i dx JOUFHSBMJOJO TPOVDV –1 BöBôEBLJMFSEFOIBOHJTJEJS A) 17 B) 15 C) 12 D) 8 E) 5 18 16 13 9 6 2. y y = F(x) m O S2 S3 rx S1 np :VLBSEBLJõFLJMEFZ=' Y GPOLTJZPOVOVOHSBGJóJ 5. F_ x i = 3x - 8 PMEVóVOBHËSF WFSJMNJõUJS 4x - 3 41 =CJSJNLBSF 42 =CJSJNLBSF 3 43 =CJSJNLBSF # _ fof i_ x idx PMNBLÑ[FSF BöBôEBLJJGBEFMFSEFOIBOHJTJyan- MöUS 2 pr integSBMJOJO TPOVDV BöBôEBLJMFSEFO IBOHJTJ- EJS A) # F_ x i dx = - 2 B) # F_ x i dx = 15 3 B) 2 5 E) 7 mm A) C) D) 3 rr 2 22 C) # F_ x i dx = 9 D) # F_ x i dx = - 7 6. F ( -4 ) =' WF nm 2 p # F'_ - 2x i dx = 8 E) # F_ x i dx = - 5 –3 r PMEVôVOB HÌSF ' BöBôEBLJMFSEFO IBOHJTJ- 3. 4 d_ x4 + x3 i EJS # 2 x2 integSBMJOJO TPOVDV BöBôEBLJMFSEFO IBOHJTJ- A) –6 B) -4 C) –2 D) 2 E) 4 EJS \" # $ % & 1. B 2. E 3. C 59 4. A 5. C 6. B
KARMA TEST - 6 ÷OUFHSBM 1. :VLBSEBLJ õFLJMEF Z = ' Y QBSBCPMÐ \"0#$ EJL- 4. y EËSUHFOJOJO#LËõFTJOEFOHF¿NFLUFEJS F(x) = x2 y y = F(x) A x C O2 B x S2 T S1 AO 1BSBCPMÐO UFQF OPLUBT 5 - PMVQ 5 OPLUBT :VLBSEBLJ õFLJMEF ' Y = x2 QBSBCPMÐOÐO Y = 2 BQTJTMJ\"OPLUBTOEBOUFóFUJ¿J[JMNJõUJS [AC]Ð[FSJOEFEJS41WF42J¿JOEFCVMVOEVLMBSCËM- #VOBHÌSF UBSBMCÌMHFOJOBMBOLBÀCJSJNLBSF- HFMFSJOBMBOMBSPMVQ41 =42EJS EJS #VOB HÌSF # OPLUBTOO PSEJOBU BöBôEBLJMFS- A) 1 B) 1 C) 1 D) 2 E) 1 EFOIBOHJTJEJS 6 4 3 3 A) 24 B) 6 32 D) 36 & 5 C) 55 2m 5. # 1 - 3 x dx 2. # F_ 2m - x i dx 1-6 x JOUFHSBMJOEFU6 =YEFôJöLFOEFôJöUJSNFTJZBQ- m MSTB BöBôEBLJ JOUFHSBMMFSEFO IBOHJTJ FMEF FEJ- integrali BöBôEBLJMFSEFOIBOHJTJOFFöJUUJS MJS m 2m A) 3 # _ t2 + t i dt B) 6 # _ t6 + t5 i dt A) # F_ u i du B) # F_ u i du C) 6 # _ t2 + t idt D) 3 # _ t6 + t5 idt 0 m m m C) - # F_ u idu D) # F_ u i du 0 2m 3m E) 3 # _ t3 + t2 i dt E) # F_ u i du 2m 4 6. # x2.F_ x i dx = 4x4 + 6x3 3. # F_ u i du = 18PMEVóVOBHËSF PMEVôVOB HÌSF ' BöBôEBLJMFSEFO IBOHJTJ- EJS 3 1 \" # $ % & # x2.F_ 4 - x3 i dx 0 JOUFHSBMJOJO TPOVDV BöBôEBLJMFSEFO IBOHJTJ- EJS A) -54 B) - $ -6 D) 6 E) 54 1. C 2. A 3. D 60 4. D 5. B 6. C
÷OUFHSBM KARMA TEST - 7 1. F_ 2x - 1 i = 4x PMEVóVOBHËSF 4. (FS¿FMTBZMBSLÐNFTJOEFUBONMBSUBOWFTÐSFLMJ x+5 CJS'GPOLTJZPOVJ¿JO ' = Ff 1 p = 6 F ( 1 ) = Ff 3 p = 12 3 22 # d _ F_ 3x i i F ( 2 ) = 14 1 FõJUMJLMFSJWFSJMNJõUJS JOUFHSBMJOJO TPOVDV BöBôEBLJMFSEFO IBOHJTJ- 2 EJS #VOBHÌSF # F_ x i dx inteHSBMJOJOEFôFSJBöB- 3 4 5 6 7 A) B) C) D) E) 0 4 5 6 7 8 ôEBLJMFSEFOIBOHJTJPMBCJMJS 2. ' Y GPOLJTZPOV\" WF# OPLUBMBSOEBO \" # $ % & HF¿NFLUFEJS #VOBHÌSF 2 dx - 2 x.F'_ x i dx # # 1 F_ x i 1 F2_ x i JOUFHSBMJOJO TPOVDV BöBôEBLJMFSEFO IBOHJTJ- EJS 5. NWFOQP[JUJGHFS¿FMTBZMBSWFN<OPMNBLÐ[F- 1 1 1 2 5 SF EJLLPPSEJOBUTJTUFNJOEFZ=NY2 +OFóSJTJJMF A) B) C) D) E) x = Y=WFZ=EPóSVMBSBSBTOEBLBMBOCËM- HFZ=OEPóSVTVJMFBMBOMBSPSBOPMBOJLJCËMHF- 6 3 2 3 6 ZFBZSMZPS #VOBHÌSF m PSBOLBÀUS n 3. (FS¿FM TBZMBS LÐNFTJOEF UBONMBOBO ' Y = x2 A) 2 3 C) 2 3 5 QBSBCPMÐOÐOCJSLTNWFSJMNJõUJS[ ]BSBMóOEB- 15 B) 9 D) E) LJTBZMBSJ¿JOZ=' Y HSBGJóJCJSJNLBSFMFSFCËMÐO- 16 4 18 NÐõUÐS y F(x) = x2 O 12 x 6. (FS¿FMTBZMBSLÐNFTJOEFUBONMWFJLJEFGBUÐSFW- #VOB HÌSF öFLJMEFLJ UBSBM CÌMHFMFSJO BMBOMB- MFOFCJMJSCJS'GPOLTJZPOVJ¿JO S UPQMBNOO UBSBONBNö CÌMHFMFSJO BMBOMBS UPQMBNOBPSBOLBÀUS F ( 2 ) = ' = 'h =WF'h =FõJUMJL- MFSJWFSJMJZPS 3 #VOBHÌSF # x.F' '_ x i dx JOUFHSBMJOJOTPOVDV 2 BöBôEBLJMFSEFOIBOHJTJEJS 1 3 557 \" # $ % & A) B) C) D) E) 4 8 12 19 24 1. D 2. A 3. D 61 4. D 5. B 6. C
KARMA TEST - 8 ÷OUFHSBM 4 5. (FS¿FMTBZMBSLÐNFTJOEFUBONMWFTÐSFLMJCJS' Y 1. # ^ 8x3 + x2 - 12x + 1 hdx 8 -4 GPOLTJZPOVJ¿JO # F^ x hdx = 6 PMEVóVOBHËSF JOUFHSBMJOJO TPOVDV BöBôEBLJMFSEFO IBOHJTJ- 5 EJS 3 \" # 151 C) 152 D) 51 E) 154 # ^ 2 + F^ 3x - 1 hhdx 33 3 2 JOUFHSBMJOJOTPOVDVLBÀUS A) 3 B) 4 C) 5 D) 6 E) 7 2. 4 x.dx # –1 3x + 4 JOUFHSBMJOJOTPOVDVLBÀUS A) 2 B) 4 C) 5 D) 13 E) 15 6. y =Y-Y2FôSJTJJMFY-FLTFOJBSBTOEBLBMBO 22 LBQ BMCÌMHFOJOBMBOLBÀCS2EJS \" # $ % & 3. N`( - PMNBLÐ[FSF 7. ôFLJMEFLJ HSBGJL EJL пHFO õFLMJOEFLJ CJS QBSL WF 3 UBSBM CËMHF ¿JNMFOEJSJMNJõ BMBO HËTUFSNFLUFEJS 1BSLOJ¿JOEFLJ¿JNMFOEJSJMNFNJõCËMHFG Y = 1 - x2 # 2x - m dx - m2 = 13 fonksiyonuna aittiS \"QTJTi – 1 PMBO \" OPLUBT GPOLTJZPOUFóFUOPLUBTES 2 -2 y PMEVôVOBHÌSF NLBÀUS A A) -1 B) - 1 $ % 1 E) 1 2 2 4. F^ x h = * 2x - 1 , x ≤ 2 –1 O x 2 3x - 2 , x > 2 #VOB HÌSF ÀJNMFOEJSJMNJö CÌMHFOJO BMBO LBÀ 1 CJSJNLBSFEJS PMEVôVOBHÌSF # F'^ 2x + 1 hdx JOUFHSBMJOJOTP- OVDVLBÀUS 0 5 B) 3 C) 7 D) 4 9 A ) 59 B ) 35 C ) 87 D ) 115 E ) 123 A) E) 48 32 60 96 48 2 22 1. C 2. A 3. C 4. B 62 5. B 6. C 7. D
÷OUFHSBM <(1m1(6m/6258/$5 1. ôFLJM EF FõLFOBS 3. #JSNÐIFOEJTCJSZÐ[NFIBWV[VUBTBSMBZQCVOMBS пHFOCJ¿JNJOEFLJCJS F ( x ) = -2x2 +YWFH Y = 4x2 -YGPOLTJZPO- MBSOOBSBTOEBLJTOSMCËMHFEFPMBDBLõFLJMEFLP- IBWV[VO CJS LFOBS PSEJOBU TJTUFNJOEF CJSJNJ NFUSF PMBDBL õFLJMEF NPEFMMJZPS NFUSFEJS #V IB- F(x) WV[VO UBCBOOO UB- NBNOO GBZBOTMBSMB LBQMBNBT MJSB NFUSF UVUNBLUBES ôFLJM #V IBWV[VO UBCB- :Ð[NF IBWV[V O 3JFNBOO UPQMBN g(x) NBOUóZMBõFLJMEF- LJHJCJHFOJõMJóJNFU- SF PMBO BMU BSBMLMBSB BZSMBSBL PMVõUVSVMBO EJLEËSUHFOMFS ZBSE- :Ñ[NFIBWV[VOVOEFSJOMJôJ NFUSFPMBDBôOB HÌSF UBTBSMBEôZÑ[NFIBWV[VLBÀN3TVBMS NZMBLBQMBOBDBLUS \" # $ % & ôFLJM #VOB HÌSF öFLJM EFLJ HJCJ CJS LBQMBNB JÀJO ÌEFOFDFLUVUBSLBÀMJSBPMNBMES \" # $ % & 2. 4 m ôFLJMEFLJ QBSL ¿JNMFOEJSJ- 4. ôFLJMEF LPPSEJOBU TJTUFNJOEF NPEFMMFONJõ BMU WF MFDFLUJS)FSCJSLTNBSB- ÐTU TOSMBS QBSBCPM õFLMJOEF PMBO CJS UÐOFM HËSÐM- 8 m TOEBLJV[BLMLMBSCJSCJSJOF NFLUFEJS,PPSEJOBUTJTUFNJOEFCJSJN NFUSF LBCVMFEJMJQËM¿FLMFOEJSJMNJõUJS 12 m FõJU WF 2 3 NFUSFEJS ¥JNMFOEJSNF JõMFNJ 3JF- y 36 32 16 m NBOO ÐTU UPQMBN NBOU- óZMBZBQMSTB9NFUSFLB- x 12 m SF 3JFNBOO BMU UPQMBN 46 8 m NBOUóZMB ZBQMSTB : –6 –4 O NFUSFLBSFLTN¿JNMFOEJ- SJMNJõPMVZPS 5ÑOFMJOZBOZÑ[FZMFSJOJONFUSFLBSFGJZBU5- 4m PMBO LBQMBNB NBM[FNFTJ JMF LBQMBONBT EVSV- NVOEBLBQMBNBNBMJZFUJLBÀ5-PMVS #VOBHÌSF 9-:BöBôEBLJMFSEFOIBOHJTJEJS A) 8 3 B) 12 3 C) 16 3 \" # $ D) 18 3 E) 24 3 % & 1. D 2. E 63 3. A 4. D
<(1m1(6m/6258/$5 ÷OUFHSBM 1. # t #PMNBLÐ[FSF 4. :VSUEõUVSMBSEÐ[FOMFZFOCJSUVSBDFOUBTOONÐõ- CJSIBWV[EBLJTVZVO[BNBOBHËSFCPõBMNBI[ UFSJTBZTOOBSUõI[ V' ( t ) =-U MJUSFEBLJLB F' ( t ) = + t )2 .ÐõUFSJTBZTZM GPOLTJZPOV GPOLTJZPOVJMFWFSJMNJõUJS JMFWFSJMNJõUJS #VIBWV[CPöBMNBZBCBöMBELUBOEBLJLBTPO- ' U \"DFOUBOO NÑöUFSJ BMNBZB CBöMBELUBO U SBIBWV[EBOLBÀMJUSFTVCPöBMNöUS ZMTPOSBLJNÑöUFSJTBZTOHÌTUFSEJôJOFHÌSF CVBDFOUBOOZMTPOSBLJNÑöUFSJTBZTBöBô- \" # $ % & EBLJMFSEFOIBOHJTJEJS \" # $ % & 2. #JS LËZEFLJ JOTBO OÐGVTV EÐS 4 U CV LËZEF 5. 4 U U TBOJZFEF CJS CBMPOVO ZFSEFO ZÐLTFLMJóJOJ ZBõBZBOJOTBOOÐGVTVOVHËTUFSNFLUFEJS HËTUFSNFL Ð[FSF CV CBMPOVO ZFSEFO ZÐLTFLMJóJOJO EFóJõJNI[4h U =-U NFUSFTBOJZF GPOLTJ- /ÐGVTVOUZMEBBSUõI[ 4h U =+U LJõJZM ZPOVJMFUBONMBOZPS GPOLTJZPOVJMFWFSJMNJõUJS #VCBMPOZFSEFONFUSFZÐLTFLMJóJOEFLJCJSQMBU- #VOBHÌSF ZMTPOSBCVLÌZÑOOÑGVTVLBÀLJ- GPSNB LPOVQ CVSEBO ZVLBS EPóSV ZÐLTFMNFZF C- öJPMVS SBLMZPS \" # $ % & #BMPOVOTBOJZFEFZFSEFOZÑLTFLMJôJLBÀNFU- SFPMVS \" # $ % & 3. #JSEPóSVCPZVODBIBSFLFUFEFOCJSDJTNJOJMLI[ 6. #JSGJSNBOOU ZM BOOEBLJBOBQBSBNJLUBS' U JMF V=NTO UTBOJZFTPOSBLJJWNFTJ CFMJSMFONJõUJS a ( t ) = t +NTO2EJS F'_ t i = 3 t ZMCJO5- PMBSBLUBONMBOBOGPOLTJ- ZPOJTFOFUZBUSNBLõPMBSBLBEMBOESMNBLUBES )BSFLFUMJOJOUBOOEBLJI[EFOLMFNJ #V GJSNBOO BOB QBSBT CJSJODJ ZMEBO TFLJ[JODJ V = V0 + # a_ t iEUEJS ZMBLBEBSLBÀCJO5-EFôJöNJöUJS #VOB HÌSF DJTJN IBSFLFUF CBöMBELUBO TB- \" # $ % & OJZF TPOSB CBöMBOHÀ OPLUBTOEBO LBÀ NFUSF V[BLMLUBCVMVOVS \" # $ % & 1. C 2. A 3. E 64 4. D 5. B 6. D
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