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- よりお きちや
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8 N Z, I Q R C N Z Q R C R i R {,,} N A B X N Z Q R,, c,,, c, A, B, C, L, y, z,, X, L pq p q def f () lim f ( ) f ( ) ( ), p p q r q r p q r p q r c c,, f ( ) " or " All Eist ( ) N ( ) R y, y y, y y, y y [ ] i,, y y Q y def, dog Q R R C
9 () [] g ( ) lim f ( ) N {,,, L}, S, S,!, C,,, e, i i e i c di
10 R C i def i,, c,l, y, z,l i w, z,,, w f y ( y ) f '( ) dy d Q R,e r (cos i si ) i (z) (cos i si ) cos i si rg ( )( ) w r(cos isi ) z V R AB OP p, (, y) y z e e e v f (u),, st s t ( ) ( ) OP l OP m OP soa tob cos e e e M d g (,, ) (,, ) e h c f i A, B, CLO, E (,, ) A O, E f ( u) Au A A ( d) A( dce ) c d d A A c d d c c A tr (A) (A) A ke pa qeopq A AB BA AB O A O, B O AkE pa qeo p q A E A
11 , (, y) A, B A B A B def f (to) def, A f ( ) f ( ) f oto y B; A f ( ) y def f, y; f ( y) f ( ) f ( y), f ( k) kf ( ) def ( f o g)( ) f ( g( )) y f ( ) f ( y) A A f ( ) A f : V V, lim def A B [ A B] A B def S, lim S ( r, ),, def A B A B { } { z { f ( )} l C, F( t,, y) [] y y y f g : f ( ) g( ) s, t, s', t' R, def c,, c, d R; i c di s s' d s t s' t' t t' ' def ' y y ' y y ' z z ' z z ' r(cos i si ) R(cos ' i si ') ' r R ' ' ' def ( k ) ' ' 6 k c d c' d' c c ' si si d d ' log log A P
12 A 4 6 Q R [Q.] d c A ( d) A,e,L, L C C,,, L, L 4 A A E O L 4 lim : d E,E, L d c c d 4 8 ; L A A E O 4 5 A L 5 7 9, y R, r y 8 6 A P(, y) yi r O O (, y) yi N Z Q r cos O y r si OA, OB P( r, ) [] r B O X O A R C
13 y y p y c ( ) ( ) y y tu t t y y k( ) y y t s t AP t AB s t ( s t ) ( t) t s t z t z z r cos( ) p y y z z c AP sab t AC su tv s t uc ( s t u ) y cz d ( ) ( y ) r r cos t r y r si t r t y r t y y c ( c 4 4 CP r, AP BP r ( ) ( ) k k z r z z z r z k z k r r cos ( ) ( y ) ( z c) r ) FP : PH e : P(, y) P ( r, ) (, R) y cy f gy h [] (, R) c,, c R c c c X E (, R) c X E 4c X E c 4c z z c 4c 4c,, z A A X X ( 4c) i 4c. z y, y t t y
14 OP,, w z z z (cos isi ) cos isi w r(cos isi ) kr( )( ) z d r ( X, Y) ( ), ) ( r, ) ( y y f () t g() y h(t) F(si,cos,si cos ) y' y' f ( ) g( ) f '( ) g'( ) *
15 S ( ) d 6 OP OT TC CP log( ) d si d ( t) f ( t) dt ( t) f ( t) dt q * * z k p, * t S rs lim r ( ) e ( ) [ ] p f ( ) f () S S { } { S } S S (,, L) f ( ) lim p p X * A PA Q E A A L A ( ) P(), ( )[ P( k) P( k )] y c AB p
16 (cos isi ) cos isi I S c R c ccos A si A si B si C [] si( ) si cos cos si si( ) si cos cos si cos( ) cos cos si si cos( ) cos cos si si t t t t t( ) t( ) t t t t z zz z yi (, y R) f ( ) g( ) d f ( ) d g ( ) d, g f t; ( tf g) t; t f t fg D / 4 f g g g f t tf g f kg :, ( ) 5 si d 4 5 si d 4 I si d si si d si ( ( ) y y si ( cos) ( )si cos d ( )( I S csi A c cos )' d I) I I ( ) si lim A A d A d c E O c d ( ) ( ) A ( A) d c A [] cos A
17 d d c A d c c c d u d c A y v u y v c d f () S' ( ) f ( ) S() f ( t) dt f ( ) d y def S( h) S( ) hf ( t) S'( ) lim lim lim f ( t) f ( ) h h h h t p, p q ( p, q R) q p A ke pa qe O ( p, q R) q p p' A ke pa qe p' A q' E ( p, q, p', q' R) q q' p p' A ke A pa qe A p' A q' E ( p, q, p', q' R) q q' w r w r(cos isi) z z z z z z z z z,, cos 6 i si 6 cos( 6) i si( 6) z r(cos isi ) ( r, 6) z r (cos isi ) r'(cos ' isi ') r r' ( r, r' ) r r' ( r, r' ) ' 6 ' 6 k k c ( ) c,, c d cd,,,,, i cos i si ( yi)(cos isi ) 9 i cos si 9 p si cos y y q d S () (, f ( ))
18 cos cos si cos si cos si si cos si( ) cos si cos si cos( ) (, ) (cos,si ) si( ) cos( 9) si cos si( ) y L y L S i cos k k k cos( 9) (cos i si ) si D k f ( k ) [] 4 5 S S S 6 6 ( )( ) d ( ) ( ) d ( ) 6 d f () d f ( t) dt f () y c y d d P( X r) P C p ) r A B, C, P r r r ( p ( r,,, L) C cz d c def m E( X ) i pi V( X) E( X E( X Y) E( X) E( Y) ) m i E( X ) E( X ) ( X ) def E( XY ) E( X ) E( Y ) V ( X ) V ( X ) V ( X ) r y ( X ) ( Y ) ( X ) ( X ) p r B(, p) E( X ) p V ( X ) p( p), s t : AP s AB t AC (, ),
19 P ABC OP OA OB OC, S ( ) ( ) ( ) ( ) C C C C, C C C C ( ) C p C p q C p q Cq ( p q) (,,, ) LL def L (,,, L) LL ( ) L ( ) S S S ( ) c c k k ( ) k k ( )( ) k 6 k ( ) k f ( ) g( ) ' ( ) f g' ( ) t t t t cos si t t t t y r r cos r ( t ) y t r si t ( r,) cos t rt y t t ( t ) (t) ( t ) si t d t t d dt si cos t y (t ) R ( ) ( ) R cos si cos si t t si cos si cos t t t lim f ( ) lim f ( ) f ( ) f ( ) k k lim k k ( L ) k ( ) def f ( ) d k
20 7 d LL d 4 log d log C (log ) d ' (log ) d (log ) log( ) d ( )log( ) ( ) C log cos si si d d ( si C td d log cos C cos si d ( cos )' d cos ( cos ) d cos si C ) 4 d d si 8 [] : C d t d C f f( ) f( ) ( ) f'( c) c [] [ f : f ( ) d f ( c)( ) c ; r cos y r y r si LL ( ) z z z LL z z ( z ) E A A LL A ( E A) ( E A ) if ( E A) OP OT TP P P P P L P P OT E A A A L A OT ( E A A A L A ) OT ( E A) E A ( E A) OT ( E A) ( E { R(45)} ) OT ( E A) def f () : k( k ) :, y R, f ( y) f ( ) k y f ( ) f () f '( ) k, ( k ) f ( ) f ( ) f ( ) f '( c)( ) f '( c) k k k k lim k lim lim O T P P O
21 d d S yd yd yd c t d t d t d ( t t ) y dt y dt y dt t dt t dt t 4 ( t t dt ) S t d t d t 4 d y dt y dt y dt t ( t t ) dt t dt t c d dt ( t t4 ) t 4 d t 4 y dt g t f t dt t dt ( ) ' ( ) t p() : p( ) : p( ) : y : p(, y) y, : p(, y) p( ) q( ) ; p( ) q( ) ; p( ) q( ) f () lim f ( ) f ( ) lim h f ( h) h lim f ( )( ) i i i i f ( ) f '( ) f ( ) d def h si e e lim( h).78l lim h AB E BA B A E i i def R r S sr ( ) def
22 y ' z k t X i C t z A A w w z r z y ( ) d e t C dy y t D dt dt
23 i e, e,,log f ( ) p q p q S f ( ) i i, e,log,log A B A B A B y' z A d c : c : d (,, L) (,L) k k ( k,, L) } { ➄ S L k lim() k S L lim S lim( L ) lim f ( ) f ( ) def f ( h) f ( ) h k y k
24 lim AB y z yi r(cos isi ) re i A c d dy f '( ) d ( p q) q ( p q) q C e si d e cos d i i e e e e sih( ) cosh( ) y si si y dy d d dy cos y (si )' y t t y dy (t d d t y dy cos y d si C d t C L 4 L 4 L log 4 L )' L 4 4,,,, L,,,, L 4 4,,,,L
25 m, m m cosmcosd m si msi d cos si m m d m cos,cos,cos, L,cos m, L si, si, si, L, si m, L cos si i A A 5A E O 4 A P Q P Q E P P Q Q PQ O QP O AP AB AC P,, f ( )cos r f ( ) y f ( )si C C e,,, t,log, L
26 S S f ( ) d e i r [ cm] [ cm ] 6. l f '( ), f ( ) d y z
27 p p p p ( p ) p , AB BA f o g g o f lim f ( ) f (lim ) lim f ( g( )) f (lim g( )) lim lim lim f ' (lim f )' lim lim
28 l f (, y) y c P, ) f (, y) f, y ) ( y ( y c P l : y log log log log ( ka ) ka [] z ( z ) ( ka) A ( A B)( A B) A B k E AE EA log d C log d log C si si d si C d C si cos cos si d d ( si ) C cos si cos( ) z z z z z z si( ) cos( 9 ) (, ) cos si si( 45) (cos,si ) cos si cos( ( 45)) cos si si cos si( 5) cos( 5 9) si cos si( 45) AB BC AB BC cos6 ( )( ) d ( ) 6 f '( ) f () f '( ) f () dy d f ''( ) f () y f ''( ) f () y 8.4L
29 f '( ) f () f ''( ) 4 f ( ) V { f ( ) g( )} d V { f ( )} d { g ( )} d f ( ) g( ) f '( ) g'( ) f g ; f ( ) g( ) f ( ) si f ( ) 7 f () f (7) 7 si si lim lim V ( X ) V ( X ) V ( X Y) V ( X ) V ( Y ) V ( X ) V ( X ) X, Y V ( X Y) V ( X ) V ( Y ) k L lim f ( ) 4 5 f ( ) d k { } S k f ( ) k f ( ) d ( L ) d ( L ) d A B A B A A B A B A B A B si si dy d
30 [ P( k) P( k )] k P(k) k k P( k ) k N P(k) k N; P( k) k k [ P( k) P( k )] k N; [ P( k) P( k )] P(k) f ( ) g( ) ( ) () () (),,, () () R r () ()
31 () Z N Q e R C p (mod p) N ,9, L p p L p k k k c. def 7 7 c 7 d 7 e 7 f ( 7) 7, 7
32 m, m m e,log, t h lim lim ( h) h m, m m m, N; m m ( ) m m t m,log, t e +/!+/!+ -/+/5-/7+ i i e e [ e ] /4. + *(4.-) A c d d d c A A E O A A ( d) A ( d c) E O c d d, d c 5 A A 5E O A c d A A E O d d c
33 ,, c, d A E A A E O d 4, d c 4 A A A E O c d A A ( d) A ( d c) E O c d A A E O ( d) A ( d c) E A E pa qe O ( d ) A ( d c ) E O p q A d d c p, q A E d d c d q p ; A E p d d c A c d d c A E d A c d d t s t, s A t( t) t, s t s d c A d c d A ke A A E O d c d d d d d d d, d, (, d) (,),(, ) t s A t( t) t, s A E, E t s A p, q, p', q' R pa qe p' A q' E p p' q q' A ke ' A p, q, p', q R pa qe p' A q' E p p' q q' A p, q, p', q' R A pa qe A p' A q' E p p' q q' A ke
34 A ke A ke A ke A A E O ( ke) ke E O ( k k ) E O k k k, A E, E A ke A A ( d) A ( d c) E O c d A A E O A A E A ( d) A ( d c) E A E ( d) A ( d c) E d A ke d c d A d c c d d A E, E d c A c d (,, L) M 4 (,L), 5, (,,4, L) (,,,4, L) (,,,4, L) (,,4, L) M 4 M 5 M
35 P() P() P(),,4, L P( ), P(), P(4), L P() [ P( k) P( k )] k,,4, L P( ), P(), P(4), L P() P() [ P( k) P( k )] k,4, L P( ), P(), P(4), L P() P() [ P( k), P( k ) P( k )] k,,4, L P( ), P(), P(4), L P() P() [ P( k) P( k )] k,,4, L P( ), P(), P(4), L [ P( k) P( k )] k P(k) k k P( k ) k N P(k) k N; P( k) k N P(k) P( k ) R; R; R; p( ) q( ) p( ) q( ) y' y y' ' y p, q, r, p', q', r' R p q r p' q' r' p p' q q' r r' p, q, r, s, p', q', r', s' R r s, r' s', p q p' q' r s r' s' p p' q q' r r' s s' p q r s p' q' r' s' p, q, p', q' Q p q p' q' p p' q q' p, q, r, p', q', r' Q p q r p' q' r' p p' q q' r r', s, t, s', t' R
36 s t s' t' s s' t t', s, t, s', t' R s t s' t' s s' t t',, c s, t, u, s', t', u' R s t uc s' t' u' c s s' t t' u u' A p, q, p', q' R pa qe p' A q' E p p' q q' A A ke p, q, p', q' R pa qe p' A q' E p p' q q' A p, q, p', q' R A pa qe A p' A q' E p p' q q' A A E A ( ) A E A ( A E) A Ay y A(, y) (, y) ( AB B ( B ) A E k k, y) A E k ( k ) k si si log log A ke A ke pa qe p p A p, q, p', q' R pa qe p' A q' E p p' q q' A A ke p, q, p', q' R pa qe p' A q' E p p' q q' A p, q, p', q' R A pa qe A p' A q' E p p' q q'
37 A A E A ( ) A E A ( A E) A Ay y A(, y) (, y) ( AB B ( B ) A E k k, y) A E k ( k ) k ka O k A O X E X E X E X X X E O 4c ( p)( q) X X E O ( X E)( X E) O X E, E X X ( )( ),, X E O ( ) X X E O, ( X E) O X E E O ( X E)( X X E) O ( )( ),, y, y y L, L i i i i i i f f ( ) f ( ) f ( ) f ( ) f, f (( ) ) ( ) f ( ) f ( ) f ''( ) i
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P P N p() N : p() N : p() N 3,4,5, L N : N : N p() N : p() N : p() N p() N p() p( ) N : p() k N : p(k) p( k ) k p(k) k k p( k ) k k k 5 k 5 N : p() p() p( ) p q p q p q p q p q p q p q p q p q x y p q
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1 1.1 1.1.1 A 2 P Q 3 R S T R S T P 80 50 60 Q 90 40 70 80 50 60 90 40 70 8 5 6 1 1 2 9 4 7 2 1 2 3 1 2 m n m n m n n n n 1.1 8 5 6 9 4 7 2 6 0 8 2 3 2 2 2 1 2 1 1.1 2 4 7 1 1 3 7 5 2 3 5 0 3 4 1 6 9 1
More information名古屋工業大の数学 2000 年 ~2015 年 大学入試数学動画解説サイト
名古屋工業大の数学 年 ~5 年 大学入試数学動画解説サイト http://mathroom.jugem.jp/ 68 i 4 3 III III 3 5 3 ii 5 6 45 99 5 4 3. () r \= S n = r + r + 3r 3 + + nr n () x > f n (x) = e x + e x + 3e 3x + + ne nx f(x) = lim f n(x) lim
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8 ) ) [ ] [ ) 8 5 5 II III A B ),,, 5, 6 II III A B ) ),,, 7, 8 II III A B ) [ ]),,, 5, 7 II III A B ) [ ] ) ) 7, 8, 9 II A B 9 ) ) 5, 7, 9 II B 9 ) A, ) B 6, ) l ) P, ) l A C ) ) C l l ) π < θ < π sin
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1 1 1.1 64 A6, 1) B1, 1) 65 C A, 1) B, ) C 66 + 1 = 0 A1, 1) B, 0) P 67 A, ) B1, ) C4, 0) 1) ABC G ) A B C P 64 A 1, 1) B, ) AB AB = 1) + 1) A 1, 1) 1 B, ) 1 65 66 65 C0, k) 66 1 p, p) 1 1 A B AB A 67
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yoshi@image.med.osaka-u.ac.jp http://www.image.med.osaka-u.ac.jp/member/yoshi/ II Excel, Mathematica Mathematica Osaka Electro-Communication University (2007 Apr) 09849-31503-64015-30704-18799-390 http://www.image.med.osaka-u.ac.jp/member/yoshi/
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