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1、/2023年 A B B A B A B A A B B A B x A x x B A B x A x x A A A A A A A A A)(A C AU)(A C AUU)(A C CU UA U U A A CU A x U x x,B A B A A B A B)(B A CU)()(B C A CU U)(B A CU)()(B C A CU U A B B A)(B A CA)(B A CB)(B A CU)()()()(B A card B card A card B A card N N N Z Q R C A(1)nn 2n 21n 21n 22n A x x f y),
2、(x x A x x f y y),(B /D)(x f y 2 1,x x D 2 1x x)(1x f)(2x f)(x f D 2 1x x)(1x f)(2x f)(x f D D x x 2 1,2 1x x)()(2 1x f x f)(),(x g x f)()(x g x f)(x f)(x f)(x g f y M)(x f)(x g)(x g f y)(x f)(x g)(x g f y)(x f b a,)(x f)(a f)(b f)(),(a f b f)(x f y I M I x M x f)(x I M x f)(0 M)(x f y)(x f y)(x f x
3、)(x f)(x f)(x f x)(x f)(x f)(x f /0)()(x f x f)(x f 0)()(x f x f)(x f y)(x f 0 x)0(f 0()y f x T x()()f x T f x()y f x T()y f x)(x f 0,0,|()()hhhhy f x y f x h|0,0,|()()kkkky f x y f x k|0 1,1,()()y f x y f x 0 1,1,()()AAy f x y Af x()()xy f x y f x()()yy f x y f x()()y f x y f x 1()()yxy f x y f x()
4、(|)yyyy f x y f x()|()|xxy f x y f x /5)(x f y),(b a A b x a f x f 2)2()()(x f y a x)()(x a f x a f)2()(x a f x f)(x f y)2(2 x a f b y),(b a A)(x f y)2(x a y a x)1,0(a a a yx 0(log a x ya)1 a x y a xn x a n n n N 0 0 n n a an n n)0()0(|aaaaa an n)1,0(*n N n m a a an mnm)1,0(1 1*n N n m aaaan mnmnm 0
5、 0 0 s ra as ra s ra)(rsa rab)(r rb a Q s r b a,0,/)1,0(a a a yx x R),1()1,0(a 1 a 01 a R R),0(),0(R R)1,0()1,0(b a,)1,0()(a a a x fx)(),(b f a f)(),(a f b f 0 x 1)(x f)(x f R x)1,0()(a a a x fx a f)1(N ax)1,0(a a x a N N xalog a N Nalog 0 a 1 a x N N aax log N lg 71828.2 e N ln N ab)1,0,0(log a a N
6、 N ba logaaN N a a N b aba log a a b)0,0,1,0(N M a a N M N Ma a alog log)(log N MNMa a alog log)(log 654321-1-4-2 2 4 601654321-1-4-2 2 4 601/N n Nanalog)(log)0,1,0,0,0(logloglog N b b a abNNaab 1 log log a bb a c c ba b alog log log b bananlog log bmnbanamlog log 0(log a x ya)1 a x),0(R x y2log 2 5
7、log5xy 0(a)1 a x 0 x 1 a 01 a),0(),0(R R R R)0,1()0,1(x y)(R a),0()1,1(0),0 1 1 0 0),0(32.521.510.5-0.5-1-1.5-2-2.5-1 1 2 3 4 5 6 7 801132.521.510.5-0.5-1-1.5-2-2.5-1 1 2 3 4 5 6 7 8011/x y y x x x)(D x x f y 0)(x f x)(D x x f y)(x f y 0)(x f)(x f y x 0)(x f)(x f y x)(x f y 0)(x f)(x f y)0(2 a c bx
8、ax y 0 02 c bx ax x 0 02 c bx ax x 0 02 c bx ax x 3)(x f b a,)(a f)(b f 0)()(b f a f),(b a)(x f)(0 0b x a x 0)(0 x f.)(x f b a,)(x f 0)()(b f a f)(x f b a,)(x f b a,)(x f b a,()()0 f a f b)(x f b a,/E D C B A ABCDE AD ABCD S D C B A ABCD /x x y y c h h l ch S h c S 21 rl S h c c S)212 1 l R r S)2 l
9、r r S)l r r S)22 2 121r l r l r r S V Sh 2V Sh r h 13V Sh h r V231 1()3V S S S S h 2 211()()33V S S S S h r rR R h 24 R 334R V /a b O a a b b a b a b 2,0 a a b a b a a a b a b a a b b a b p a b a b a a b a /l l l b a b l l,P b a l PA A PO O A AO 0 90,0 l P AB Q O O AOB 180,0 /l x x x l l 0 90,0 90 k
10、 180,90 90,0,tan k),(1 1 1y x P),(2 2 2y x P 1 21 2x xy yk)(2 1x x)(1 1x x k y y),(1 1y x k x b kx y k b y x),(1 2 1 21 211 21y y x xx xx xy yy y),(),(2 2 1 1y x y x x y)0(1 abbyax a x b y x y)0(02 2 B A C By Ax A B C 1l 01 1 1 C y B x A1 1,(B A 0 2l 02 2 2 C y B x A2 2,(B A 0 2 1/l l 01 2 2 1 B A B
11、 A 01 2 2 1 C B C B 01 2 2 1 C A C A /1l 2l 01 2 2 1 B A B A 01 2 2 1 C B C B 01 2 2 1 C A C A 2 1l l 02 1 2 1 B B A A),(1 1 1y x P),(2 2 2y x P 22 122 1 2 1)()(y y x x P P),(0 0y x P l 0 C By Ax 2 20 0B AC By Axd 01 C By Ax 02 C By Ax 2 22 1B AC Cd),(b a r 2 2 2)()(r b y a x 2 2 2r y x 1 0 42 2 F E
12、 D 02 2 F Ey Dx y x)2,2(E D 242 2F E D 2 r b a,F E D,r b a,F E D,d r r d P r d P r d P y x x y 0 0 0 /2 r d r d r d 2 1,r r d d 2 1,r r 2 1r r d 2 1r r d 2 1 2 1r r d r r 2 1r r d 2 10 r r d),(0 0y x k k1 0y y 0 x x),(0 0y x)(0 0 x x k y y k 0 x x 0 C By Ax 0:2 2 F Ey Dx y x C 0)(2 2 C By Ax F Ey Dx
13、 y x 1C 01 1 12 2 F y E x D y x 2C 02 2 22 2 F y E x D y x /1 1 12 2F y E x D y x)1(0)(2 2 22 2 F y E x D y x 1C 01 1 12 2 F y E x D y x 2C 02 2 22 2 F y E x D y x 0)()(2 1 2 1 2 1 F F y E E x D D r d l 2 2 2)2(rld x y x nx x x x,3 2 1 1 K N n /nx x xxn 2 1 nx x x x x xs sn2 2221 2)()()(k)3,3(s x s
14、x S S S S S S /S n n A n A A f n A Ann A A A f n A P A A n A n Ann A B A B A B A B A B A B A B P A B P A P B A B A B P A B P A P B P A P B P A A B P A B P A P B A B A B P A B P A P B P A P B A B A B A B A B A B A B B A A P A A /A x 360 360 90,k k k 360 90 360 180,k k k 360 180 360 270,k k k 360 270
15、360 360,k k k x x 180,kk y 180 90,kk x y Z k k,45 180 Z k k,225 180 x y Z k k,45 180 Z k k,135 180 90,kk Z k k,45 2 2 2 2 2 2,kk rad /2 360 180 902 1180 1801 57.3 r l C S lr 2 C r l 21122S l r r 2,0 0 30 45 60 90 120 135 150 180 270 360 0 6 4 3 2 23 34 56 32 2 sin 0 21 22 23 1 23 22 21 0 1 0 cos 1 2
16、3 22 21 0 21 22 23 1 0 1 tan 0 33 1 3 3 1 33 0 0)0,0(O),(y x P),(y x P r 220 r x y ry sin rx cos xy tan sin MP cos OM tan AT 22sin cos aa 2sin 1 cos 2cos 1 sin),2(tancossinZ k k a aaa cot1tan 三角函 数线：MP、OM、AT /sin yx cos yx tan yx R R,2x x k k 1,1 1,1 R 22xk k max1 y 22xk k min1 y 2 x k k max1 y 2 xk
17、 k min1 y 2 T 2 T T 2,222kk k 32,222kk k 2,2 k k k 2,2 k k k,22kk k(,0)k k 2x k k,02kk x k k,02kk sin yx 2|sin|yx|sin|y x k 2 sin|yx)2 sin(k sin cos 2k cos tan 2k tan)kZ)sin(sin cos cos tan tan /)sin(sin cos cos tan tan)sin(sin cos cos tan tan)2sin(cos cos2 sin)2tan(cot)2sin(cos cos2 sin)2tan(cot co
18、s)23sin(sin)23cos(cot)25tan(y sin(x)A y=sinx y=sin(x+)y sin(x)y sin(x)A|y=sinx y sin x y sin(x)y sin(x)A,0),sin(x x A y A 0 A 2 T 21 Tf x 0 x x)0(2 f BC AB AC BA AB 0 AC AB CB A A A A A n A n A nA 0 b a b a b a /b a/(0)a b b 12,ee a 2 1,1 1 2 2a e e.,0,180 ab,ab ab x y,ij a,xy a xi y j(,)xy a(,)a x
19、y 1 1 2 2(,),(,)a x y b x y 0 b,ab 1 2 1 20 x y y x a b O OA a OB b AOB a b 0,180 a b 2 b a a b a b a|b a x y b x y 2 1 2 1y y x x b a a b cos,a a b b a cos,b a b a x y b x y a 2211xy 1 2 1 22 2 2 21 1 2 2x x y y abab x y x y b a 1 2 1 20 x x y y b a/01 2 2 1 y x y x a a a a 22 2 2222)(b b a a b a b
20、 a a b c /a b b a a b a b a b)a b c a c b c sin cos cos sin sin sin cos cos tan tan1 tan tan sin()cos()tan()a b a b 2 2b ab 2 2b aa 2sin cos 22cos sin 22cos 1 21 2sin k k 1 cos2 1 cos2 1 cos1 cos 2 2 2tan 1tan 2cos sincos sin 22 sin 222 22 2tan 1tan 1cos sinsin cos2 cos sinAa sinbB sinCc R R ABC a 2
21、 sinA R b 2 sinB R c 2 sinC R A 2 a R2aR B 2bR C 2cR a b c sinA sinB sinC 2sin sin sin sin sin sina b c a b cRA B C A B C /2 2 22 cos a b c bc A 2 2 22 cosB b a c ac 2 2 22 cosC c a b ab A bca c b22 2 2 B acb c a22 2 2 C 2 2 22a b cab A B C B C A S a h a h a a S ab C ac B bc A abc R S r a b c r 2 d
22、b A a,A b a,A 2b a)(1 N n d d a an n)()1(1 N n d n a an),()(N n m d m n a am n.),(N q p n m q p n m n ma aq pa a p n m 2 p n ma a a 2 na d c na c d na c c cd na nb 2 1,d d pan qbn p q pd /qd na nb 2 1,d d n nb a 2 1d d n S n 2)(1 na a n dn nna 2)1(1 d na n nS na S n a n n S n a a a a n a n a n S n n
23、 S n a n S n S n S n n n,2 3 2 n n n n nS S S S S n n nT S,n nb a,n nnnnbaTS 1 21 2 2 q)0(q b G a,G b a,ab G 2)(1 N n q qaann)(11 N n q a ann 11 nnq a a xq c y 0,11 a q 0,1 01 a q na 0,11 a q 0,1 01 a q na 1 q na /0 q na na q)0(q),(N n m q a am nm n),(2 N k q p n m k q p n m 2k q p n ma a a a a na)0
24、(c a cn q na q n nb a,2 1,q q n nb a 2 1q q n 11 1(1)(1)(1)11nnnna qSa a q aqqqq q na n,2 3 2 n n n n nS S S S S n na 1 q nnnqqaqaqq aS 1 1 1)1(1 1 1 nnq B A S 0 B A 0 a b a b 0 a b a b 0 a b a b),(R b a),(1 R b R a b aba),(1 R b R a b aba),(1 R b R a b aba a b b a a b a c b c,a b b c a c /,a b c d
25、a c b d,0 a b c ac bc,0 a b c ac bc 0,0 a b c d ac bd 0,1nna b a b n n 0,1nna b a b n n b aab b a1 10,b ab a1 10 0,0 m b a)0(m bm am bbbm am b 2 2)21(43)21(a a 211,(1)k k k 211,(1)k k k 2 2 1 2,21 k k k k k k*12(,1)1k N kk k k 24 b ac 0 0 0 2y ax bx c)0(a 2ax bx 0 c 0 a 1,22bxa 12xx 122bxxa /20 ax b
26、x c 0 a 12x x x x x 2bxxa R 20 ax bx c 0 a 12x x x x 02 c bx ax 0)0 4,0(2 ac b a 02 c bx ax a()0()()0()()()0()0()0()fxf x g xgxf x g xfxgx gx“”2()0()(0)()fxf x a af x a 2()0()(0)()fxf x a af x a 2()0()0()()()0()0()()fxfxf x g x gxgxf x g x 2()0()()()0()()fxf x g x gxf x g x /()0()()()0()()fxf x g x
27、gxf x g x 1 a()()()()f x g xa a f x g x 01 a()()()()f x g xa a f x g x 1 a 01 a()0log()log()()0()()aafxf x g x g xf x g x()0log()log()()0.()()aafxf x g x g xf x g x 20 ax bx c 0 a 0,0 c b 0 a 00 a 02 c bx ax 0 a 0,0 c b 0 a 00 a a x f)(a x f max)(a x f)(a x f max)(a x f)(a x f min)(a x f)(a x f min)
28、(aR bR 222 a b ab 222abab b a 0 a 0 b 2 a b ab 2abab b a 22222aba b ab /0,(2 21 122 2 b ab a b aabb a b a b a,x y x y s xy xy 24s xy p xy xy 2 p pq p q q p pq p q M x)(x p M 0 x)(0 x p)(,x p M x)(,0 0 x p M x)(,0 0 x p M x)(,x p M x)()()(q p q p)()()(q p q p /,M x y F F F F P M MF MF a F F c a c a c
29、 a c P a c P a c P x a y b a b y a x b a b a x a b y b b x b a y a A a A a,B b B b A a A a B b,B b,A A a B B b F F c ace 221ab)1 0(e a b c 2 2 2c b a a 2axc 2ayc),(0 0y x P P x y x a y b P x y x a y b P x y x a y b 1F 1d 2F 2d 1212FFedd /P x y F F r PF r PF x a y b a b r a ex r a ex y a x b a b r a
30、ey r a ey P x y PF F r PF r PF F PF PF F S 2tan22 1 b SF PF x a y b a b r r P S b c y y b P S bc l b a AB x a y b a b A x y B x y M x y l k x x 2 122 124)(1 x x x x k l k y y 2 122 124)(11 y y y yk AB k AB b x a y 2 1,F F)2 2 0(22 1c F F a a 2 1,F F 2 1F F c 2 a 2 c F F a a MF MF M P 2 2,22 1 2 1 x
31、y /)0,0(12222 b abyax)0,0(12222 b abxay)0,(),0,(2 1c F c F),0(),0(2 1c F c F c F F 22 1)(2 2 2b a c,a a x,a a y)0,(a),0(a b 2 a 2 x y 0,0)0(122 eabace 0 byax)(xaby 0 aybx)(xbay 2axc 2ayc)0(2 2 y x 2 e x y),(0 0y x P)0,0(12222 b abyax P 1220220 byax P 1220220 byax P 1220220 byax 1F 1d 2F 2d 1212FFedd
32、 /P F F S PF F b F PF b P F F PF a c PF c a b a a P A B A B PA PB PA PB b a F l)0(22 p px y)0(22 p px y)0(22 p py x)0(22 p py x p F l O y x)0,2(pF)0,2(pF)2,0(pF)2,0(pF 1 e 2px 2px 2py 2py R y x,0 R y x,0 R x y,0 R x y,0 AB)0(22 p px y),(1 1y x A),(2 2y x B)0,2(pF l /2px l BD l AC,N M,CD AB,22 1p y y
33、 422 1px x 090 CFD BN AN AB NF,p x x AB 2 1 cos 1 pAF cos 1 pBF 2sin2 pAB l 5 p BF AF2 1 1 AB AF BF y O sin 2 422 1py ypS SCOD AOB MN R MN AB MN RF4121 0 1 a a a a b C C a b a b ab /a a 0 a a 0 a a 0 a 0 a a a b a b a b aa a 0 bb/ab ab C x y x y C x y C C 1 x y z C x y z a b a b a b,ab,0,ab a b,2ab
34、a b ab a b cos,a b a b a b ab cos,a b a b a b 0 ab a a b a cos,b a b a b e 1cos,e a a e a a e 2 0 a b a b 3 a b a baba b a b 2a a a /a a a 4cos,ababab 5a b a b 1 a b b a 2 a b a b a b 3 a b c a c b c a b c p,x y z p xa yb zc a b c,p p xa yb zc x y z R a b c,a b c a b c 1e 2e 3e 1e 2e 3e 1e 2e 3e x y
35、 z xyz p p,x y z 1 2 3p xe ye ze x y z p 1e 2e 3e,p x y z p xyz,x y z 1 1 1,a x y z 2 2 2,b x y z 1 2 1 2 1 2,a b x x y y z z 1 2 1 2 1 2,a b x x y y z z 1 1 1,a x y z 1 2 1 2 1 2a b x x y y z z a b 1 2 1 2 1 200 a b a b x x y y z z 0 b 1 2 1 2 1 2/,a b a b x x y y z z 2 2 21 1 1a a a x y z 1 2 1 2
36、1 22 2 2 2 2 21 1 1 2 2 2cos,x x y y z z ababab x y z x y z 1 1 1,x y z 2 2 2,x y z 2 2 22 1 2 1 2 1d x x y y z z l l a a a b a b /a b a b a b R 0 a b a b a b a a n a/aa 0 a n a n/a a a n a n a b/ab ab 0 a b a b b a,b ab ab a,cos cos 2,0 l l a n n an an a,cos sin 2,0 l m n n mn mn m,cos cos,0 P A n
37、nn PAn PA PA d,cos xfxyxx f x x fx xx f x f)()()()(1 11 21 2 x /0 x x yc y nyx*nN 1ny nx xya 0,1 aa lnxy a a xye xye logayx 0,1,0 a a x 1lnyxa ln yx 1 yx sin yx cos yx cos yx sin yx fx gx()()()()f x g x f x g x()()()()()()f x g x f x g x f x g x Cf x Cf x 2()()()()()()0)()()f x f x g x f x g xgxgxgx
38、 21()gxg x g x x u xy y u)(x f y xx f x x fxyx x)()(lim lim0 00 0)(x f y 0 x)(x f y 0 x)(0 x f0|x xy)(0 x fxx f x x fxyx x)()(lim lim0 00 0/f x()fx()fx x()fx x f x f x()fx()fx/()fx f x f x f x)(x f b a,)(x f b a,)(x f(),()f a f b bi a i a b|,C a bi a b R a bi c di c a d b 0000bZ aba()()()()bi a z),(
39、b a),(b a x y /z OZ bi a z 或 2 2b a bi a z 12z a bi c di z 12()z z a c b d i()()a bi c di ac bd ad bc i 2 2 2 2()()()()a bi a bi c di ac bd bc adic di c di c di c d c d c di a bi a bi 0 b(1);(2)2,2;z z z z a z z bi 2222(3);(4);(5)z z z z a b z z z z z R 4 1 4 2 4 3 4 4(6),1,1;n n n ni i i i i i 22 1
40、 1 1(7)1;(8),11 2i i ii i i i iii)9(23 1 i 0 12 1,3 3 2 3 1 3 n n n n 1m 2m n nm nm m m N 2 1 n 1m 2m n nm nm m m N 2 1 n n m m n m r bi a z /n n m m n m mnA n n m m n m mnC n n m m n m 1 2 1 m n n n n Amn!m nnAmn!n Ann 1!0!1 2 1mm n n n nCmn!m n mnCmn m nnmnC C 10 nC mmmnmnA C A 11 mnmnmnmA A A m n
41、nmnC C 11 mnmnmnC C C N n b C b a C b a C b a C a C b an nnr r n rnnnnnnnn 2 2 2 1 1 0 /N n N r n r b a C Tr r n rn r,01 nx 1 0 2 2 1 1 01 x C x C x C x C xnnnnnnnnn 1 x nn n n nn nC C C C 2 1 02 1 1 X X X X X X 1 X nx x x,2 1 2 X ix np p p,2 1 X 1x 2x ix nx P 1p 2p ip np X X X P p 1 p /1 0 p X p N
42、M N n n n X m nNm nM NmMCC Cm X P l m 0 l n M X X n M N,A B A B A B P A B D A B A B B P A B P A B A B A B A B A B n n A n k knC A k k n k n kp p 1 A p n A k n k p p C k Pk n k kn n,2,1,0 1 /A X A p q 1 n A k k n k knq p C k X P n k,2,1,0 X X k n P nnq p C0 0 1 1 1 nnq p C k n k knq p C 0q p Cn nn 0
43、2 2 2 1 1 1 0 0q p C q p C q p C q p C q p C q pn nnk n k knnnnnnnn X p n,p n B X,X nx x x,2 1 np p p,2 1 n np x p x p x X E 2 2 1 1 X p X E np X E NnMX E X nx x x,2 1 np p p,2 1 n np X E x p X E x p X E x X D2222 121 X pq X D p q 1 p q npq X D 1 X D X D X /b X aE b aX E)()()()(2X D a b aX D R x e x fx,21 222 2 x 6827.0)(X P 9545.0)2 2(X P 9973.0)3 3(X P i i iy y e a x b y /a b,niinii ix xy y x xb121)()(niinii ix n xy x n y xb1221 x b y a 2K n ad bc a b c d a c b d 2K k X Y a b c d 5 X Y)(2k K P k a b c d 5 2K k 2.706 k 0090 X Y 3.841 k 0095 X Y 6.635 k 0099 X Y 2.706 k X Y