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ÁÁÊ 92ÿ2 Í72 Äîâ3äêîâå âèäàííÿ Àâòîðñüêèé êîëåêòèâ: Ìàòåìàòèêà: Î. Ì . Ðîãàí3í, Î . Ì . Òèòàðåíêî Ô3çèêà: Î. Ï. Áàëüâà Õ3ì3ÿ: Î.  . Ìoøêîâà,  . Ã. Òèùåíêî Á3îëîã3ÿ: Þ. Î . Ñàäîâíè÷åíêî Ãåîãðàô3ÿ: Î. À. Áàéíàçàðîâ, Þ . 2 . Êàíäèáà ÍÎÂ2ÒÍ2É ÏÎÂÍÈÉ ÄÎÂ2ÄÍÈÊ ØÊÎËßÐÀ Â3äïîâ3äàëüíèé ðåäàêòîð Î. Êàøèð3íà Êîðåêòîð Ã. Òðóá3öèíà Êîìïþòåðíå ìàêåòóâàííÿ Ï. Øèðí3í Äèçàéí îáêëàäèíêè Â. Êóë3ê Ï3äïèñàíî äî äðóêó 29.01.10. Ôîðìàò 84 108 1/16. Ãàðí3òóðà Øê3ëüíà. Äðóê îôñåòíèé. Óì.-äðóê. àðê. 60,48. Íàêëàä 20 000 ïðèì. Âèäàííÿ çä3éñíåíå çà ë3öåíç3oþ ÔÎÏ Øàï3ðî Ì.  . «ÒÎÐÑ2Íà ÏËÞÑ» Ñâ3äîöòâî ñåð3ÿ ÄÊ 1 2143 â3ä 01.04.05 ð. 61057, ì. Õàðê3â, âóë. Ñóìñüêà, 13 Ç ïèòàíü îïòîâèõ ïîñòàâîê çâåðòàòèñÿ: Òåë. (057) 717-10 -26, 719-98-73 E-mail: torsing_or@rider.com .ua Êíèãà ïîøòîþ: 61057, ì. Õàðê3â, à/ñ «Êíèæêîâà ë3ãà» òåë.: 8 (057) 7-199 -880 2íòåðíåò-ìàãàçèí: www.torsing.com .ua Í72 Íîâ3òí3é ïîâíèé äîâ3äíèê øêîëÿðà. Õàðê3â: ÒÎÐÑ2Íà ÏËÞÑ, 2010. 5 7 6 ñ. ISBN 978-611-030 -075 -9 . Ó íîâ3òíüîìó äîâ3äíèêó øêîëÿðà ïîäàíà ïîâíà 3íôîðìàö3ÿ ç îñíîâíèõ øê3ëüíèõ ïðåäìåò3â: ìàòåìàòèêè, ô3çèêè, õ3ì3¿, á3îëîã3¿, ãåîãðàô3¿. Âèäàííÿ ï3äãîòîâëåíå â3äïîâ3äíî äî ñó÷àñíèõ âèìîã øê3ëüíî¿ ïðîãðàìè. Ìàòåð3àë âèêëàäåíèé ïîñë3äîâíî, çà òåìàìè, êîæíèé ðîçä3ë ì3ñòèòü çàâäàííÿ äëÿ ñàìîêîíòðîëþ, 3ëþñòðàòèâíèé ìàòåð3àë (ñõåìè, òàáëèö3, ãðàô3êè), ùî äîçâîëèòü ó÷íÿì ï3äãîòóâàòèñÿ äî êîíòðîëüíèõ òà ñàìîñò3éíèõ ðîá3ò 3 óñï3øíî ñêëàñòè âèïóñêí3 3ñïè òè. ÁÁÊ 92ÿ2 © Àâòîðñüêèé êîëåêòèâ, 2009. © ÔÎÏ Øàï3ðî Ì. Â., ìàêåò, 2010 ISBN 978-611-030-075 -9
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BCa ABc ACb ABc BCa ACb ACb BCa sin , cos cos sin 1 cos 1 sin 2 2 3 62. 2 1 sin180 3 cos180 sec180 tg60 . 2 A 1 1 031 3313231. 2 1 A 231. A 4 sin , 5 2 cos 1sin. 2 2 4 16 9 3 cos 1 sin 1 1 , 5 25 285
sin 4 3 4 tg : , cos 55 3 1 3 ctg . tg 4 3 4 3 cos ,tg , ctg 5 3 4 2 2 2 sin ctg . 1 sin A 2 2 2 2 2 2 2 sin sin cos ctg 1. 1 sin cos sin A 2 2 1 sin 1 cos 2 2 A 2 2 cos sin cos sin 2 2 2 2 A 3 4 222 2 cos cos , 2 2 sin sin . 2 2 cos sin . 2 2 A cos sin . 2 2 A 2 1 2 m m m m m m m 1 1 2 2 1 2 3 2 2 2 2 . 2 1 2 m 2 1 ; 2 m m m 3 2 2 .
2 2 sin cos 1 2tg ; ctg sin cos sin cos sin cos sin sin cos cos sin c 2 2 2 2 1 2 ctg oos cos sin sin cos 2 2 1 2 1 2 2 2 sin cos : cos sin cos sin sin cos cos sin 2 2 2 2 sin 2tg . cos a b a bab 22 22 2 2 2 2 . sin cos sin cos 2 2 2 2 sin cos sin cos sin cos 2 2 2 2 2 2 1 2 2 2 2 sin cos sin cos 0. sin sin cos cos 2 2 2 2 1 1 sin cos cos sin 2 2 2 2 2 2 2 2 sin cos sin cos 0. sin 1 cos 1 cos sin ac bd
2sin36 cos36 cos72 sin72 cos72 2sin36 2sin36 A sin 180 36 2sin72 cos72 sin144 sin36 1 . 4sin36 4sin36 4sin36 4sin36 4 1 cos 36 cos 72 . 4 sin 4 sin 5 sin 6 . cos 4 cos 5 cos 6 A 2sin5 cos sin 5 sin 4 sin 5 sin 6 cos 4 cos 5 cos6 2cos5 cos cos 5 A sin5 2cos 1 tg5 . cos5 2cos 1 tg5 . 4 3 4cos2 cos 4 . sin A 4 4 8 sin 8. sin A tg tg A sin( ) cos sin( ) cos cos( ) sin sin cos( )
1 sin(2 ) sin sin(2 ) sin 2 1 sin(2 ) sin sin(2 ) sin( ) 2 3 sin sin 4 sin 2. 3 sin sin 2 sin 1 cos 2 2 2 13 6 4 sin cos sin cos . 6 4 1 2 2 sin . cos 2 2 12 6 2 3 2 3 2 2 sin sin cos sin cos cos 6 2 22 4 2 sin sin sin cos cos cos 2 2 4 4 2 2 4 sin 2sin cos cos 4 2 2 4 4 2 2 4 2 sin 2 sin cos 2cos sin 2sin cos cos 4 4 4 4 3 sin 3cos = 3(sin cos ).
, 2 2 2 1x sin 1 x x 2 2 1 1 1 x x 2 ; 1 x x 2 1x x 2 1 x 2 1 x x 2 1x x 1 x cos , 1 x x 2 2 1 sin = 1, 1 x x 2 1x x 2 . 1 x x
2 1 x 2 1x x 2 1 x x 1. x 2 1. x 1 x x 2 1 x x 2 1 1x 1 (0) x x 2 1 x x 2 1 1x 1 x 2 1x 2 1 1x 2 1 x x 1 . x 2 1 1x 2 1 x x 1 x cos arccos( 0, 8) 2 2 1x cos arccos( 0, 8) 2 2 1(0,8) 1 0,64 0,36 0,6 Â3äïîâ3äü: 0,6.
sin 2 0. 4 x sin 2 02 2 4 4 4 82 n x x n xn x . 82 n x nZ 1 2 1 1 2 cos arccos 2 2, 2 2 3 x x n nnZ 1 1 2 arccos arccos 2 2 33 1 arccos 2 3 2 1 12 cos cos cos arccos , 3 3 32 23 2 0 3 2 1 cos 32 2 2 . 3 nnZ 3 tg2 5 3 x 3 3 tg2 2 arctg 5 3 5 3 6 x x n n 11 11 2 , 56 30 602 n x n n x nZ 11 . 602 n nZ 2 1 ctg 2 43 x 2 1 1 3 ctg 2 ctg 2 43 4 3 3 x x
3 2 arcctg , 4 3 3 2 arcctg , 4 3 x nnZ x kkZ 7 7 2 43 12 242. 2 11 11 2 43 12 242 n x n n x k x k k x 7 11 ; , . 242242 n k nkZ x n k nkZ 7 242 11 242 ; , . 7 11 . 242 242 n k nZ kZ 1 7 , 242 n x n Z2 11 , 242 k x k Z 1 7 242 n x 2 11 242 n x 1 a 1 a 2 xn 2 n 2 2 n 4 n 4 n 2 2 n 4 n 4 n 2 2 x n
1 arcsin 1 k k 2 ()0 () fx x ()0 ()0 fx x ()0 ()0 fx x cos 0 cos (2cos3 1) 0 2cos3 1 0 x x x x , 2 . 2 , 93 x nnZ k x kZ 2 ; , . 2 93 k n nkZ
2 ; . 2 93 n n nZ , 2 x nnZ 1 cos 3 2 x 1 arccos 2 2 2 3 k k 2 , 93 k x kZ 2 x n 2 , , 93 k x nZkZ 71 16 25 16 25 4 2 1 4 t 25 4 1 4 2 11cos41 1 cos 2 cos 4 4 2 4 2 x x x , 62 n x n Z
2 1 cos 2 1 1 2 cos 2 cos 2 4 2 1 cos 2 2 x x x x , 6 . , 3 x kkZ x mmZ 62 n nZ , . 2 n x nZ
1 2,3 1 tg1 15 15 tg 2 2 y x x y 4 . 15 arctg , 2 x n nZ x n 15 ; arctg . 4 2 n nnZ tg 2 x 2 2tg 2 sin 1tg 2 x x x 2 2 1tg 2 cos 1tg 2 x x x tg 2 x tg 2 x t 2 2 2 2 1 2 1. 1 1 t t t t 1 2 t 1 tg 22 x 1 1 arctg 2arctg 2 , 2 2 2 x n x nnZ 1 2arctg2;2 , . 2 n knkZ
2 2 sin cos sin( ) a xb x a b x 2 2 cos , a ab 2 2 sin b ab 3 sin cos 1 x x 3 12 2 2 3 sin cos 1 x x 31 sin cos 1 2sin cos sin cos 1 22 6 6 x x x x 1 2 sin 1 sin (1) 6 62 6 6 n x x x n (1) , 6 6 n x nnZ 3 cos 2 1 sin 2 6 (1) . 6 6 n nnZ 3 sin cos 1 3sin (1cos)0 x x x x 2 3 2sin cos 2 cos 0 2cos 3sin cos 0 22 2 2 2 2 x x x x x x 2, 2, cos 0 2 . 2, 3tg10 3sin cos 0 3 2 2 2 x x nnZ x n x x x x kkZ 2;2 , . 3 n knkZ tg , 2 x t 2 2 2 2 1 1 3 3 3 1 tg 3 23 1 1 3 t t x t t t 2, 3 x kkZ
2;2 , . 3 k nnkZ (1) 6 6 n nnZ sin 2 x sin 0 2 x 2 2 2 sin cos 0 2 2 x x x x 2 2 2 , sin 0 2 2 2 2 cos 0 , . 2 n x x nZ x k x kZ 4 4 ; , . 2 2 n k nkZ , cos 3 0 63 12sin 0 (1) , . 6 k n x nZ x x x kkZ , 63 n x nZ (1) , 6 k x kkZ
(1) 6 k x k 63 n x 0 0 . 63 n nZ 1 1 cos 12 cos 6 cos 8 cos 6 cos 12 cos 8 0 2 2 x x x x x x sin10 0 2sin10 sin2 0 sin2 0 x x x x , 10 10 2 , . 2 n x nZ xn xk k x kZ 2 kp x , , 10 n x nZ , 10 n x nZ 10 n x 2 k x . 10 n nZ 10 n ,, 2 k nkZ
2 1 cos2 sin = 2 2 1 cos2 cos 2 2 2 2 3 cos 3 cos 4 cos 5 . 2 x x x 2 1 cos2 cos 2 x x 1cos61cos81cos103 2 2 2 2 x x x cos 6 cos 8 cos10 0 cos10 cos6 cos 8 0. x x x x x x 2cos8 cos2 cos 8 0 cos8 (2cos2 1) 0 x x x x x , cos 8 0 168 2cos2 1 0 , . 3 n x nZ x x x kkZ ; , . 1683 n knkZ 1 1 , sin 2 6 , . 2 63 sin 1 2, 2 n x nnZ x m x mZ x x kkZ 2 . 63 m mZ
1 2 2 1 (à) 51380 1,6 (á) t t t t cos sin 4 4 11 sin cos 1 2 sin cos 1 22 x x x x 1 2 sin 1 sin 4 4 2 x x (1) (1) , . 4 4 4 4 n n x n x nnZ sin 2 sin cos 22 x x x 2 2 sin 2 x 2 2 sin cos 2 sin 0 2sin cos sin 0 22 2 2 2 2 x x x x x x 2, 2, sin 0 2 . 1tg 0 2, cos sin 0 2 2 2 2 x x mmZ x mmZ x x x x kkZ sin cos 2 sin 2, 4 x x x 2 (1) . 4 4 n nnZ
cos sin . x x cos sin sin cos cos sin , x x x x x x kx k 0 0 2 2 2 2 2 kkZ x x cos cos 2 10 2 2 2 51 2 2 kx kkZ x nnZ x , arccos , arccoos , , 51 2 2 nnZ 2 2, . 2 kx kkZ 51 arccos 2 . 2 nnZ cos sin sin cos sin . x x x x x 0 2 2 sin 1 x cos 1 x 7 16 7 16 3 sin 5 cos 3 sin 5 cos 358. x x x x 7 16 sin 1 sin 1 cos 1 cos 1 x x x x x
1 . 54 1 arctg 5 1 1 1 25 sin arctg sin cos arctg sin arctg cos 4 5 4 5 5 4226 x 12 5 1 2 2 26 213 213 13 2 13 x 2 . 13 1 1 , 4 t 1 1 arcsin sin , 4 4 x x arcsin , 2 2 x 4 ;. 22 1 sin 4 x 1 sin . 4
2 , 3 cos cos 1. xy x y 2 2 , , 3 3 2 cos cos 1 2cos cos 1 3 3 3 y x y x x x x 2 , 2, , 3 3 2 2, . 3 3 y x y nnZ x n x nnZ 2, 2, . 3 3 x ny nnZ 2,2 3 3 n nnZ sin cos 0, 5, cos sin 0,5. x y x y 11 sin 0. 22 xy 1 1 sin 1. 2 2 xy , , sin 0, 42 , . 2 sin 1 , 2 42 n xyn x k xy nkZ n xy k xy y k , , . 42 42 n n k knkZ
3 cos 2 x 3 2 y 3 cos 2 x 6 x , 6 x ; 66 3 cos 2 x 2 2, . 6 6 nx nnZ 2 2, . 6 6 nx nnZ
arctg3. 2 x arctg 3 , . 2 nx nnZ ; arctg 3 2 nZ n n 21 ; 2 221 ; 21 21 ; 5 5 21 . 21 1 2 11 2 12; 2 1 3. 2 ;1; 2 B 4cos 2 x y ; 2 11 sin . 2 1 ; 2 6 3 3 1tg . tg ctg
, ; 4 x nnZ , ; 4 x nn Z , . 4 x nnZ 1 . 2 22, ; 9 nn Z 2 , ; 93 n nZ 22 , ; 93 n nZ 22 , . 33 n nZ . 2 , ; 2 nn Z , ; nnZ 1 cos 2 x 5 2 2, ; 6 6 nx nn Z 2 2, ; 3 3 nx nnZ 5 2 2, ; 3 6 nx nn Z 2 2, . 6 6 nx nnZ 1 sin 2 x 2 2, ; 6 6 nx nn Z 5 2 2, ; 6 6 nx nnZ 5 2 2, ; 6 6 nx nn Z 7 2 2, . 6 6 nx nnZ 1 tg arctg0,5 . 2 5 2; 5 2; 5 2; 52.
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log log ab a a x b 2 1 (121) . 11 x x 1 1 11 11 121 1 11 11 11 11 11 2 2 2 2 1 2 x x x x x x 2 2 1 2 2 2 0 (2)0 0, 2. x x xx x x x x 35225 x x 2 35 35 35 15 15 x x x x x x x 2 2 15 15 2 4. 2 x x x 3 1 2 27 9 x x 3 3 3 1 33 x x 9 3 3 2 22 42 x x x 42 33 3 3 1 2 33 42 3 2 27 9 3 3 3 3 x x x x x x 3342 17 333242 . 2 3 13 x x x x x 17 . 13
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1 33 108 x x 1 33 1083331083(13)108 x x x x x 3 108 3 273 3. 4 x x 1 1 53565100. x x x 1 1 2 5 55 x x 11 1 55 55 x x x 1 2 1 1 5535655100 x x x 553651005 2100 12 1 x x 1 5 5 11 2. x x x 1 2 1 2 333 44 4 x x x x x x 3 4 3213 21 21 log . 13 4 13 13 4 x x x x 3 4 21 log . 13 2 0 x x AaBaC 55 1, , 0. 51 x x x x 2() () 0 fx fx Aa Ba C () fx a y 2 0 AyByC
2 2 2 3123270. x x 2 3xy 3 3 2 2 2 2 2 x x y 2 1 2 12270 3, 9 y y y y 2 2 2 2 1 33 1 , , . 2 2 39 x x x x x x 1; 2. 2 2 3 3 x x 1 2 22. x x 2 20 2 , 222 1. 220 x y y y x y yy 22 0 xx x x Aa Bab Cb ab x 2 22 2 0 0. xx x x x x x x x a Bab b b b A C AB C a a a a a 2 x b y a 9 16x7 12x16 9x0 (: 16x 0) 2 12 9 3 3 97 16 097 16 0. 16 16 4 4 x x x x 3 0 4 x y 1 9 16 y 2 9 3 93 2. 16 4 164 x y x
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3 64 36 4 4 100 100 27 27 3 3 x x x x 3 100 yy 4 0 3 x y y y y y y y y y 2 250 20 250 2 2 2 4 3 4 2 2 log 2 3 x y x 4 3 log 2. 6810 6 8 3 4 1 1 10 10 5 5 101010 x x x x x x x x x x 2 3 3 , 5 5 x 2 2 2 4 4 3 4 3 4 1 5 5 5 5 5 5 x x x 2 3 3 , 5 5 x 2 2 2 4 4 3 4 3 4 1 5 5 5 5 5 5 x x x () () () () x gx fx fx ()0 ()1 () () fx fx x gx
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log log . 2 2 2 3 68 x x x x x log log 2 2 2 2 2 3 68 368 30 680 20 21 x x x x x x x x x x x x x 2 980 0;3 4 1;8 8. 3 3 0 1 2 x x x x x x x x x x x log log log . 2 3 4 0 x log log log log log log log 2 3 4 2 3 4 2 0 1 x x log log log log log log 3 4 3 4 3 4 1 3 3 x x x 100 lg lg 20 5 3 3 3 log 1 log 31log 1 31 10 1 30 3 x x x x x x x x log3 2 2 2 431 1 433 1 40 1 x x x x x x x x x 0; 4 0. 1 x x x x 2 4 4 4 4 4 16 2log8log4 log8log log 2. 8
4 4 3 3 log log 2 2 1 5 1 30 3 3 5. 10 1 1 x x x x x x x x x x x x 3 log12 2 9 55. x x 9 3 3 3 12 3 3 3 3 2 12 2 12 2 12 12 log log log log x x x x x 2 3 2 2 log12 2 12 55 9 55 120 x x x x x 2 2 2 144 55 460 1 1 2 2 x x x x x x x 210 2 10. 1 2 x x x 2 10. log log log log log ; a a a a a x x x x x 2 2 22 2 2 2 2 2 2 4 log log log log log . a a a a a x x x x x 23 32 2 2 2 2 3 3 9 log log log log , a n m a m n a n n a n x x m x m x 2 log log 2 log log n n m n k n n n a a a a x x k xm x 25 04 29 3 3 , , ,. log log x x 5 2,5 2 1 25 0,4 . 52
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log()log () a a fx gx 1 ()0 1 log()log () ()0 ()()0; () () a a a fx a fx gx gx fx gx fx gx 0 1 ()0 0 1 log()log () . ()0 0()() () () a a a fx a fx gx gx fx gx fx gx log log . , , 02 2 023 x x log log , , 02 2 02 2 2 2 3 0 30 3 0 30 x x x x x x x x x 0 0 0 3. 300 3 x x x xx x 2 3 3 log 2 log 30. x x 2 1 3 3 3 3 3 230131log 3 log3 log log 3 tt t x x 1 27. 3 x 1 ;27 3 x
4 2 1 log 1 1 log 2 x x 2 2 4 2 2 log log 1 log log . log 4 2 2 x x x x 1 1 1 2 1 12 2 0 0 1 221 2 21 t t t t t t 2 2 2 1 2 2 2 2 1 1 log 1 1 log log 0 2 2 1 1 log 2. 2 2 log log 2 x t x x t x x x x 0 1 2 2 ; ; . () ()Vx Ux a lg2 100. 10 x x lg2 lg2 100 lg lg100 lg 2 lg 2 10 10 10 x x x x x x lg2lg12. x x 2 2 12 300 30lg3 t t t t t x 3 lg1 lg lg10 1 1000. x x
() ()1 () () log () 0()1 0()() A x A x fx x fxA x fx x () () ()1 0()() log ()log () 0()1 0()() x x x gx fx fx gx x fx gx log . 2 2 561 x x x 2 2 2 1 0 2 021 560 0 562 562 x x x x x x x x x x 1 1 0 0 2 2 . 2;3 1 2 1 6 3 6 x x x x x x x x 2 1 21 2 1 ;1 6; . 1 2 562 6 x x x x x x x x 1 ;1 6; . 2 x 0 1 1 1; 1. 6 610 20 x x x x x x
1 1 1 1 1 1 6 . 6 6 4 1 612 4 x x x x x x 1 1 1. 1 612 4 x x x x x x 11 ; 1; . 64 x 2 1 81 9 x x 2 2 3 1; x x 3 (0,7) 1; xx 3 arctg 3? x 3 2 1 3 logloglog( 1)0 x 2 2 log(3 ) log( 2) 1? x x 39? x 1 2 16. x 1 2 log 21. x 237, 322318 x y x y 3 log 10? x
21 41 1 x x 2 1 1 24 2 0 x x a a , xa () . fxb lim() . xa fxb lim() 0 0:, , xa fxb xxaxa () . fxb xa () fxb lim(), xa b fx
lim() () lim()lim() xa xa xa fx gx fx gx lim() () lim()lim(); xa xa xa fxgx fx gx lim() () lim ,lim()0; ()lim() xa xa xa xa fx fx gx gx gx lim () lim(), xa xa kfx k f x lim() (). xa fx fa 1 ; 2 y x 0 0 0 ()()( )() fxfxfx xfx 0 ( ) fxx 0 ( ) f x x 0 0 0 0 0 0 () ( )() ()lim lim . x x fx fxxfx fx x x
2 1 1 , x x 2 3 1 2 , x x x x 1 2 ,x x 3 2 3 1 3 ,x x 4 2 4 1 4 . 2 1 . cos x 2 1 . sin x 1 , ln xa 1 . x () (), const cfx cfxc 2 . f fgfg g g () () (), x fgx fggx () fgx 0 () tg. fxk 0 0 0, yyyxxx y
3 (;3)(3;); x x
1 2 x x
[1;3] [1;3] min (2) 25, max (3) 0. yy yy [1;3] min (2) 25; yy [1;3] max (3) 0. yy ;. 62 , sin 0 1 2, . cos , 3 2 xnnZ x x kkZ x ; 62 x 3 x , 3 y , 6 y : 2 y 2 1 13 2 cos cos 2 ; 3 3 3 2 22 y 2 31 1 2 cos cos 2 3; 6 6 6 22 2 y 2cos cos 0(1) 1. 2 2 y , 2 x . 3 x ; 62 min 1; 2 yy ; 62 3 max . 32 yy 1 ; cos 3 ln10 x 3sinlg3 ; ln 10 x x sin lg3 ; x x 2 ln1tg. y x
2x ye 2 3 2 y x x ;. 22 2 a y x
cos
F
r r r r rrr x y . rrt .
MM 12 r1 r2 rrr 2 1 v s t ñåð. . v v r t ñåð. . v r
dr dt , v r t dr dt t lim . 0 v1 v2 v12 v vv 12 1 2
v vv 2 1 v2 v1 v v1 v r1 , r r2 r r r 2 1 . t, r t r t r t 2 1 v v v v v v x x x y y y 2 1 2 1 , . vv const
v const. v r t . v r t v s t . x vt x , a v v r. a v t dv dt dr dt lim . 2 2
a v t dv dt t lim . 0 a v R n 2 . a a an 2 2 . a v ; a v . a v t vv t 0 . a v t vv t 0 , a v t . a v t 0 .
v vv ñåð. , 0 2 sv t vv t v t vat tvt at ñåð. . 0 0 0 0 2 2 2 2 2 xx vt at 0 0 2 2 . rrvt at 0 0 2 2 . T l t r v êîëà 2 . 1 2 T v r .
v vt R . a v t v R 2 . a R T 42 2 t . 2 2 T .
v R T R 2 2 ; a v R R 2 2 .
rxyz (,,) rrvttt , r r F Fð,
F FFF F i i n n ð 112 ... . a F m a F m x x ,a F m .
F F 1 2 . ma ma 11 22 a a m m 1 2 2 1 . FG mm r 12 2 , FG Mm R ò 3 3 2 , g GM R 3 3 2 .
p. p pmv .
h gt 2 2 v gh 2. v0 ag const, htv gt 0 2 2 sin . hx g v x tg cos . 20 2 2 2
g v b 20 2 2 cos ,
a v r 2 . gG M r 3 3 2 . v G M r G M Rh 3 3 3 () . v G M R 1 3 3 . 2 v v 2 1 2 112 ,
F1 F2
Fâ . Fç . FF i i â ç 0 F F i i â ç 0. Fi ç 0. Fi ç 0; Mk 0. p v: pmv .
a vv t 2 1 , v1 v2 mvv t F ( ). 2 1 ppFt 2 1 , pp p 2 1 pFt . F F12 F21 , F1 F2. ppFFFFt 1 2 12 21 1 2 ( ). F F 21 12 , ppFFt 1 2 1 2 ( ). p Ft ñèñò F p mv mv ñèñò 11 22 const .
F, r AFr cos, x H sin Amg H mgH g cos( ) sin . 90 xl H sin ,
F, F F kxkxk xx ïð.ñåð. 0 0 2 2 (). Fïð ñåð . . x A k xxx x kx kx ïð. 2 2 2 0 0 0 2 2 ()() . Fïð . x, r F
x vt at 1 2 2 Amavt at mat vat 1 2 1 2 2 2( ). a vv t 2 1 . A mvv vvv mv mv ( )( ) . 2 1 1 2 1 2 2 1 2 2 2 2 2 E mv ê 2 2 . AEE E k k k 2 1 . A kx kx 0 2 2 2 2 . E kl ð 1 2 2 ().
AEE EE E ð ð ð ð ð 1 2 2 1 ( ) . EEAA ê ê ï íï 2 1 , Eð1 Eð2 . ( )( ) . EE EE A ê ð ê ð íï 2 2 1 1 N A t N Fr t Fv cos cos .
m V , p F S . 1 V V p ,
P PF ïîâ À í 0 , FÀ
r kx kx 0 2 2 2 2
m M N r (êã) [ìîëü êã] [ìîëü ] 103 1 1 A , M m m M m r r 0 0 0 1 12 1 C à.å.ì . , 1 12 1 16610 0 27 mC à.å.ì . êã. ,
N NA . m M . NN N m M A A
v Z n N V , Z nvS x 1 2 , 20 0 2 mvZ mnvS x x . F mv t ()FmnvS x 0 2 . F mnvS x 0 2 . vx 2 . v v v x y z 2 2 2 . v v v v x y z 2 2 2 2 . v v v v v x y z x 2 2 2 2 2 3 . v v x 2 2 1 3 . F nmvS 1 30 2 . p F S mnv 1 30 2 . E mv ê : 0 2 2 pnE 2 3ê .
pV m M RT . n N VV m M N 1 A, pV m M kNT A, pV T pV T 11 1 22 2 const.
p T
VV T 0 273 15 , V T const
EkT 3 2 N m M NA U m M RT 3 2 . U pV 3 2 . EkT 3 2 kT 2
5 2 kT U m M RT pV 5 2 5 2 . U im M RT i pV 2 2 ,
F F FF FpS AFhpShpV . F F AA pV .
U im M RT 2 0,
v s t .
UU U Ui i n 1 2 3 1 0 ... .
QQQ Qi i n 1 2 3 1 0 ... . pV m M RT p F S mnv 1 30 2
q1 q2, qqqq 1 2 1 2 Fk qq r 1 2 2
k 1 40 , F qq r 12 0 2 4
E F q E F E Fk qq r 0 2 . E F q k q r 0 2 . EEE 12 3 , , EEEE 1 2 3 ..... E
E E1 E
E1 E0 Fk qq r 12 2 . E r: FqE . F F r: AFrFr Fh () cos F r: r E ri F E A AFhFh i i . F F
A q E q n A EE n n ( )1 EE n n 1 U A q 1 2 . F,
E F E E U d . C q .
C q U S d 0 , 1111 1 2 3 CCCC E qUq C CU n 22 2 2 2 0 2 2 E
I q t . I q t qnSlv l q nvS 0 0 .
I U R . R l S ,
RS l . 1 .
U U R R 1 2 1 2 111 1 2 1 2 1 2 RRR R RR RR , I I R R 1 2 2 1 U A q . P A t . U R 2 .
I U R QUt R 2
I Rr
B B F Il B F Il
1 1 Òë Í Àì B
B
FqE eë FFF åë Ë
B B n B
B B B B
i I R i i i i t i t
E v B i L A q Fl q q vBl q vBl |||| ||sin || sin I Rr vBl Rr i sin
i t BS t sin i vBl sin is t L I t mv 2 2 W LI ì 2 2
Fk qq r 1 2 2
2 2 T .
WWW kxmvkx mv p k x m m 2 2 2 2 2 2 2 2
x k m x
a s s g l s 0 2 k m T m k ;
0 2 g l T l g ;
1
W LIq C q C LI m m 2 2 2 2 222 2
q LC q 1 0 1 LC T LC 2 2 0
i u R U t R I t m m cos cos
I U R m m I I I m m 2 071 , U U U m m 2 071 , p IU mm 2 piuIU t mm cos 2 ()
X C C 1 I U XC
I U L m m I U XL 1 LC I U R m m U U I L C Cm Lm m , , L C
E t 0 B E E
E B E EBZ v c
E
sin sin n sin sin v v n 1 2
sin sin arcsin 0 0 90 1 1 n n ;
D F 1 111 1 ddf , (4.23)
11 1 dd D . (4.24) à H h à d d = 1 T l g 2 T m k 2 0 k m ; T 2
5 5 ï1 ï2 1 1 n n n 2 1 1 2 n n n 1 2
x 460210 160 01510 23 23 , ,
D a b D a b MN MN M M 11 22 1 2 .
ABCD. 12 1 kCD kAB [][] [][] k k CD AB 1 2 [][] [][] . k k k 1 2 ð, k CD AB ð [][] [][] .
kp [] [][] NH NH 3 2 2 2 3 1 2 1 2 923 2 2 H Cl HCl ,
2 2 2 HgOHgO t 2 2 3 3 2 2 KClO KCl O MnO,t
3 2 2 2 3 HN NH , CO Cl COCl 2 2.
1 1 p 1 1 H ( )01 n
19 40 K18 40 Ar,24 54 Cr 26 54 Fe e 1 1836
1 2 ; 1 2 .
NH NHH 4 3 íàãð îõîëîä . . .
E ll r 12 2 ,
SO4 2 ,
NaClNaCl 0 0 1 1 .
HCl , MgN 3 2 2 3 ,NaS 2 1 3 ,Cl2 0 . H2 0 ,F2 0 ,Fe 0 , Na 0 . ClO27 2 . Cl O 2 7 7 2 .
40 40 1 ã ã V m d .
N N äèñ. çàã. , NaNOKClNaClKNO 3 3
ÄÐH O 2 10 14 , CH 107 , lgCH 7 lg . CH 7
2 2 2 2 2 2 NaCl HO H Cl NaOH åëåêòðîë3ç
CrO NaOH NaCrO HO 23 2 2 2 2 t 2 3 3 23 2 FeOH FeO HO () t Al OH HAlO KOH KAlO HO ()333 3 3 2 3 3 Sn OH HSO NaOH NaSnO HO ()424 4 4 2 4 4
Me H Me H 0 2 2 n n n ,
4 4 2 3 2 2 2 HNO NOHOO t Ba NO HCl HNO BaCl ( )32 3 2 2 2
NaCO SiO NaSiO CO 2 3 2 2 3 2 t 4 2 4 4 3 23 2 2 2 KNO AlO KAlO NO O t BaCO BaO CO 3 2 t
î
î î Al OH HCl AlCl HO t ()3 3 2 3 3 AlO HCl AlCl HO t 23 3 2 6 2 3 AlOH NaOH NaAlO HO t ()3 2 2 2 AlO NaOH NaAlO HO t 23 2 2 2 2
2 2 2 ZnO ZnO t 4 3 2 2 23 AlO AlO t Zn OH ZnO HO ()2 2 t 2 3 3 23 2 Al OH AlO HO () t
ZnSO Cu CuSO Zn 4 4
2 2 2 NaCl Na Cl åëåêòðîë3ç ðîçïëàâ (êàòîä) àíîä () 2 4 3 23 2 AlO Al O åëåêòðîë3ç ðîçïëàâ (êàòîä) àíîä ()
3 2 2 2 3 HN NH P,t,êð - 2 2 2 2 3 SOO SO t,êð - NO NO 2 2 2 t 45 2 2 25 PO PO t COCO 2 2 t
Cl HO HCl HClO 2 0 2 1 CleCl 0 1 1 CleCl 0 1 1 SO HSO MgSO HO MgMe H 6 3 2 6 4 6 4 2 2 ( ) äî
SiO 4 2PO 2 5 5 SO 6 3ClO2 7 7 î
2 2 3 .
1 2 1 2 1 2 1 2 2 ,
HOOHH 2 HCOH HCO H 3 3 î
CH CH HCl HCCHCl HCCHCl HCl HC CHCl 2 2 3 2 = =
HOCH HC COOCH HO 25 3 25 2 HNH CH COOH H CH CONH CHCOOH HO 2 2 2 2 2 N .