(1) θ a = 5(cm) θ c = 4(cm) b = 3(cm) (2) ABC A A BC AD 10cm BC B D C 99 (1) A B 10m O AOB 37 sin 37 = cos 37 = tan 37

Size: px
Start display at page:

Download "(1) θ a = 5(cm) θ c = 4(cm) b = 3(cm) (2) ABC A A BC AD 10cm BC B D C 99 (1) A B 10m O AOB 37 sin 37 = cos 37 = tan 37"

Transcription

1 4. 98 () θ a = 5(cm) θ c = 4(cm) b = (cm) () D 0cm 0 60 D 99 () 0m O O 7 sin 7 = 0.60 cos 7 = tan 7 = () xkm km R km 00 () θ cos θ = sin θ = () θ sin θ = 4 tan θ = () 0 < x < 90 tan x = 4 sin x = cos x =

2 4 98 () (sine) (cosine) (tangent) c b c b c b θ a θ a θ a sin θ = b c cos θ = a c tan θ = b a () = D + D D D D D 99 () O () 00 a + b = c ( ) a ( ) + b = c c b b a = c a c ( ) + b ( ) = c a a cos θ + sin θ = tan θ = sin θ cos θ + tan θ = cos θ θ c a b

3 4 98 () sin θ = 5 cos θ = 4 5 tan θ = 4 () tan 0 = D D D = D tan 0 = 0 = 0 tan 60 = D D D = D tan 60 = 0 = 0 = D + D = = 40 (cm) 99 () tan 7 = O = O tan 7 = = 7.54 (m) () O O D O D H O = θ H = r tan θ = O = R (x + R) R = R x + xr OH O OH = O = θ tan θ = OH H = R r r R x + xr = R r r r R O θ x H R O D 7 0 m D

4 4 R x + xr = R r r R r = (R r )(x + xr) (x + xr + R )r = R (x + xr) r = R x + xr x + R (km) 00 () sin θ = ( ) sin θ = cos θ = = 8 9 θ sin θ > 0 sin θ = () tan θ = 5 = 5 5 cos θ = sin θ = 6 = 5 6 θ cos θ > 0 cos θ = 5 4 tan θ = sin θ cos θ = θ tan θ > 0 tan θ = cos θ = 6 5 = 5 () θ θ 5 = 5 = tan θ = sin x = 5, cos x = 4 5 cos x = + tan x = = 5 6 cos x = < x < 90 cos x > 0 cos x = 4 5 x 5 4 sin x cos x = tan x sin x = tan x cos x = = 5

5 4. 0 () θ sin θ = 4 5 cos θ = tan θ = () 0 θ 80 cos θ = 5 sin θ tan θ () 0 θ 80 tan θ = sin θ = cos θ = 0 POE = θ (0 θ 80 ) cos θ = x = x, sin θ = y = y, tan θ = y x θ 90 < θ < 80 cos θ < 0, sin θ > 0, tan θ < 0 P x y y θ O E x

6 4 0 () sin θ = 4 5 ( ) cos θ = sin θ = 4 = θ cos θ < 0 cos θ = 9 5 = 5 4 tan θ = sin θ cos θ = 5 = 4 5 y θ O 4 5 x () cos θ = 5 ( sin θ = cos θ = ) = θ 80 sin θ 0 sin θ = 5 = 5 5 y θ O x tan θ = sin θ cos θ = 5 5 = () cos θ = + tan θ = + ( ) = 9 cos θ = 9 0 θ 80 tan θ < 0 θ cos θ < 0 cos θ = 9 = tan θ = sin θ cos θ sin θ = tan θ cos θ = ( ) =

7 tan (90 θ) = 7 tan θ = 0 (sin 0 cos 0 ) + (sin 70 + cos 70 ) = 04 sin 60 cos 70 + cos 0 sin 70 0 sin (90 θ) = cos θ cos (90 θ) = sin θ tan (90 θ) = sin (90 θ) cos (90 θ) = cos θ sin θ = tan θ 0 0 sin 70 = sin (90 0 ) = cos 0 cos 70 = cos (90 0 ) = sin 0 04 sin (80 θ) = sin θ cos (80 θ) = cos θ tan (80 θ) = sin (80 θ) cos (80 θ) = sin θ cos θ = tan θ

8 4 0 tan (90 θ) = 7 tan θ = 7 tan θ = 7 0 sin 70 = sin (90 0 ) = cos 0 cos 70 = cos (90 0 ) = sin 0 sin 70 y 70 sin 0 0 (sin 0 cos 0 ) + (sin 70 + cos 70 ) = (sin 0 cos 0 ) + (cos 0 + sin 0 ) O cos 70 x = (sin 0 sin 0 cos 0 + cos 0 ) cos 0 + (cos 0 + sin 0 cos 0 + sin 0 ) = (sin 0 + cos 0 ) = 04 sin 60 = sin (80 0 ) = sin 0 cos 70 = cos (90 0 ) = sin 0 sin 70 = sin (90 0 ) = cos 0 sin 60 cos 70 + cos 0 sin 70 = sin 0 sin 0 + cos 0 cos 0 = sin 0 + cos 0 = 60 y sin O x cos 0 sin 60 = sin( ) = cos 70 cos 0 = cos(90 70 ) = sin 70 70

9 tan 0 sin 0 cos 40 sin 5 06 cos 5 cos 50 sin tan 0 sin 0 sin 5 cos 40 cos 0 > cos 40 > cos cos 5 cos 50 sin 50 cos

10 4 05 tan 0 = sin 0 = sin 5 = sin 0 < tan 0 < sin 5 cos 0 > cos 40 > cos 45 > cos 40 > sin 0 < tan 0 < sin 5 < cos sin 50 = sin (90 40 ) = cos 40 0 < θ < 90 θ cos θ cos 50 < cos 40 < cos 5 cos 5 cos 50 y O x cos 45 cos 0 cos 40

11 sin θ + cos θ = sin θ cos θ =, sin 4 θ + cos 4 θ = 08 sin θ + cos θ = 0 θ 80 sin θ = 09 sin θ + cos θ = sin θ cos θ = tan θ + tan θ = tan + tan = 0 sin + cos sin cos

12 4 07 sin θ + cos θ = sin θ cos θ sin 4 θ + cos 4 θ sin θ cos θ sin θ + cos θ sin θ cos θ 08 sin θ + cos θ = sin θ cos θ sin θ cos θ II 09 sin θ + cos θ = sin θ cos θ tan θ + tan θ = ( sin θ cos θ ) + ( cos θ sin θ sin θ cos θ 0 tan + tan ) sin + cos sin cos sin cos tan + tan = 0 sin cos sin + cos = (sin ± cos ) sin ± cos

13 4 07 sin θ + cos θ = sin θ + sin θ cos θ + cos θ = sin θ + cos θ = + sin θ cos θ = sin θ cos θ = sin 4 θ + cos 4 θ = (sin θ + cos θ) sin θ cos θ ( ) = = 08 sin θ + cos θ = cos θ = sin θ sin θ + cos θ = cos θ = sin θ cos θ sin θ + ( sin θ ) = sin θ sin θ 8 9 = 0 sin θ = t 9t t 4 = 0 t = ± θ 80 0 t sin θ = t = = ± 7 6 sin θ + cos θ = sin θ + sin θ cos θ + cos θ = 9 sin θ + cos = + sin θ cos θ = sin θ cos θ = II sin θ cos θ t t 4 9 = 0 9t t 4 = 0

14 4 09 sin θ + cos θ = sin θ + sin θ cos θ + cos θ = 4 sin θ + cos θ = + sin θ cos θ = 4 sin θ cos θ = 8 tan θ + ( tan θ = tan θ + tan θ ( = sin θ cos θ + cos θ sin θ ( = sin θ cos θ ) tan θ tan θ ) ( ) = sin θ + cos θ sin θ cos θ ) ( = 8 ) = tan + tan = sin cos + cos sin = sin + cos cos sin = sin cos tan + tan = 0 sin cos = 0 sin cos = 0 (sin + cos ) = sin + sin cos + cos = + 0 = 8 5 (sin cos ) = sin sin cos + cos = 0 = sin cos = 0 > 0 sin > 0 cos > 0 sin + cos > 0 sin + cos = 8 5 = 0 5 sin cos = ± 0 5 = ± 5

15 4.6 x 90 x 80 6 cos x + cos x = 0 cos θ + sin θ = 0 θ 0 < θ < 80 0 θ < 80 θ cos θ + sin θ cos θ = cos x + sin y = 4 sin x cos y = 4 0 x 80, 0 y 80 () sin x + cos y () x () y cos x cos x = t t t cos θ sin θ cos θ = sin θ sin θ sin θ cos θ sin θ cos θ cos θ tan θ \= 0 4 cos x + sin y sin x cos y sin x cos y

16 4 90 x 80 cos x 0 cos x = t 6t + t = 0 (t )(t + ) = 0 t 0 t = cos x = cos θ + sin θ = 0 x = 0 ( sin θ) + sin θ = 0 sin θ sin θ + = 0 sin θ = t 0 < θ < 80 0 < sin θ 0 < t t t + = 0 (t )(t ) = 0 t = sin θ =, 0 < θ < 80 θ = 0, 90, 50 cos θ = 0 cos θ \= 0 cos θ + sin θ cos θ = cos θ tan θ = sin θ cos θ + tan θ = cos θ + tan θ = + tan θ tan θ = 0, 0 θ < 80 θ = 0, 60

17 4 4 () cos x + sin y = ( sin x) + ( cos y) = (sin x + cos y) cos x + sin y = sin x + cos y = (sin x + cos y) = sin x + cos y + sin x cos y = + 4 = 0 x 80 sin x 0 sin x cos y > 0 sin x > 0 cos y > 0 sin x + cos y > 0 sin x + cos y = () sin x + cos y = sin x cos y = 4 4 = sin x( sin x) sin x sin x + 4 = 0 sin x = 0 x 80 x = 0, 50 () () cos y = sin x = = 0 y 80 y = 60

18 < θ < 5 sin θ cos θ > 0 6 cos θ + sin θ < 0 0 < θ < θ < 90 cos θ = θ tan θ > θ < θ < 5 sin θ = cos θ cos θ 6 cos θ = sin θ sin θ 7 0 θ < 90 tan θ = θ 0 tan θ > θ

19 4 5 ( cos θ) cos θ > 0 cos θ = t t + t < 0 (t )(t + ) < 0 y 5 < t < < cos θ < 0 < θ < 5 60 < θ < 5 6 ( sin θ) + sin θ < 0 sin θ sin θ > 0 sin θ( sin θ ) > 0 60 O y 50 x sin θ < 0, sin θ > 0 < θ < 80 0 < θ < 50 0 O x 7 0 θ < 90 cos θ = θ = 0 y tan θ = θ θ = 0 0 θ < 90 tan θ > θ 0 < θ < 90 O 0 x

20 θ 80 cos θ + sin θ + sin θ 9 0 θ < 90 f(θ) = tan θ + sin θ + cos θ + = cos θ + = cos θ f(θ) = tan θ + sin θ + cos θ f(θ) = + + tan θ f(θ) = + cos θ + = ( cos θ ) + = θ = f(θ) 0 f(x) = sin x + a cos x + 0 x 80 () a = f(x) x () a 0 < a < f(x) 5 a () a a f(x) x f(x) x

21 4 8 cos θ = sin θ sin θ 0 θ 80 sin θ 0 sin θ 9 f(θ) cos θ 0 sin x = cos x f(x) cos x cos x = t t 0 x 80 t

22 4 8 sin x = t 0 θ 80 0 t cos θ + sin θ + sin θ = ( sin θ) + sin θ + sin θ = t + t + = (t ) + 0 t t = x = 90 t = 0 x = 0, 80 y max min O t 9 0 θ < 90 f(θ) = tan θ + sin θ + cos θ sin θ + cos θ = cos θ + tan θ = cos θ f(θ) = tan θ + sin θ + cos θ f(θ) = tan θ + ( cos θ) + cos θ = + cos θ + tan θ f(θ) = + cos θ + ( + tan θ) ( ) = + cos θ + cos θ = cos θ + cos θ cos θ cos θ = 0 f(θ) cos θ cos θ = 0 cos θ = 0 θ < 90 cos θ = θ = 0 f(θ)

23 4 0 f(x) = sin x + a cos x + = ( cos x) + a cos x + = cos x + a cos x + cos x = t 0 x 80 t ( f(x) = t + at + = t a ) + a 4 + = g(t) () a = ( g(t) = t ) t g(t) f(x) ( ) g = θ 80 t = cos x = x = y O max t () 0 < a < t = a 0 < a < t ( ) g(t) g a 5 y max a 4 + = 5 a = ( 0 < a < ) () a y = g(t) t = a O a t a t g(t) g() = a + g( ) = a + y a + max 0 θ 80 t = cos x = x = 0 t = cos x = x = 80 f(x) min O a + a t a + x 0 a + x 80

24 4. a b c H H H = H = H = H a = b a H c () = = = 60 = () = 0 = = = () = 5 = 8 = 7 cos

25 4 H sin = H, cos = H b b () x = b + c bc cos () () cos = c + a b b x c b b c ca b H b c () x x () a ()

26 4 H H = sin = b sin H = cos = b cos H = c b cos H a = H + H = b sin + (c b cos ) = b sin + c bc cos + b cos = b (sin + cos ) + c bc cos = b + c bc cos () = + cos 60 = = 7 > 0 = 7 () = x = + x x cos 0 x x 0 = 0 (x 5)(x + ) = 0 x > 0 = x () cos = = < < 80 = 60 = 5 8 sin 60 = 0 5 b 8 c H 7 a

27 a < a < a < a < 4 = x, =, = 5 x () x () x () a b c a b < c < a + b a b b + c < a c 4 S S = = ab sin (= bc sin = ) ca sin c b S S = s(s a)(s b)(s c) s = a + b + c a

28 4 8 4 < a < < a < < a a > < a < 4 a 8 4 () (x ) (5 x) < < (x ) + (5 x) x < < x < < x < 4 () = θ cos θ = + (5 x) (x ) (5 x) = (5 x) sin θ = 8 8x 4(5 x) = 7 x 5 x = (5 x) cos θ ( = (5 x) 7 x 5 x = (5 x) (7 x) = x + 8x 4 = (x ) + () < x < 4 x = ) x θ 5 x (x ) + + (5 x) s = = = s(s x + )(s )(s 5 + x) = (4 x) (x ) = x + 8x 4

29 4. 5 a = R sin R 6 () = 60 = () 4 4 sin ( + ) sin = = = () = 0 = sin : sin : sin = 5 : 6 : 7 θ cos θ = 5 6 () () () 7 sin : sin : sin = a R : b R : c R = a : b : c

30 4 5 D D = D a = R sin D = R sin R O D 6 () R sin = R R = sin 60 = a 60 () + + = 80 sin ( + ) = sin (80 ) = sin 4 sin ( + ) sin = sin = 4 0 < < 80 0 < sin sin = = 60 0 R R = 4 sin = R = R sin = 4 = 4 () = 80 ( ) = 45 sin 0 = sin 45 = sin 45 sin 0 = 0 = 5 7 R a sin = b sin = c = R sin sin = a R, sin = b R, sin = c R sin : sin : sin = 5 : 6 : 7 a : b : c = 5 : 6 : 7 t > 0 a = 5t b = 6t c = 7t c θ cos θ = a + b c = 5t + 6t 49t = ab 5t 6t

31 = 8 = 5 = 7 r 0 a, b, c, a b c () a () b c 8 9 I r = I + I + I = r + r + r = r ( + + ) 0 R r a = E + E = D + F = (c r) + (b r) r = b + c a r I r r b a c a E r r r c D b F

32 4 8 4 θ cos θ = < θ < 80 sin θ > 0 sin θ = cos θ = ( 4 = 4 ) = 5 4 R R = 4 R = sin θ sin θ = 4 = cos = = 8 5 S = 60 S = sin 60 = 8 5 = 0 r S = r ( + + ) r ( ) = 0 r = 0 () a b c a a = = () b + c = r r = b + c = c = 4 b b + (4 b) = 9 b = 4 ± c = 4 b = 4 4 ± b c b = 4 + c = 4 = 4 b b a r c c

33 4.5 = 4 = 5 = 6 D D D = D = D = 4 = 4 D = 5 D = 9 O D = = D = = D = = 0 D > D = D = O sin D = D = + D = D

34 4 80 D D D D D 80 D D = 80 cos = cos (80 ) = cos D + D = D D D D D D 0

35 4 = θ cos θ = = D = x D = 80 θ D 6 = + x x cos(80 θ) 6 = 4 + x 4x( cos θ) x + x = 0 x = ± 57 4 x = D > 0 D = θ 5 6 x D D = α D D = cos α = 97 7 cos α D = 80 α D D = cos (80 α) = cos α 97 7 cos α = cos α cos α = D = 97 7 = 6 D = 6 α α D = β D = 80 β D = cos β = cos β = cos (80 β) = cos β 4 cos β = 74 cos β = 7 6 = ( + 7 ) 6 =

36 4 = + cos = cos 0 = 4 > 0 = 6 D = = 80 0 = 0 0 x R sin 0 = R R = sin 0 = 6 = D R = D sin D sin D = D R = = D = = 0 D = x D ( ) = ( ) + x x cos 0 x 6x + 4 = 0 x = ± 5 D D D D = D = 0 = = D D D > D D = + 5, D = 5 D + D = sin + D D sin D = sin 0 + ( + 5)( 5) sin D = (8 + 4) =

37 4.6 4 = a = b = c () a = b cos () sin = sin + sin () a = b + c + bc 5 a b c () a cos = b cos () a cos b cos = c 6 a = b = c = tan a = tan b

38 4 4 a b c () cos = a + b c ab a b c () sin = a R, sin = b R, sin = c R () a 5 cos a b c 6 tan cos sin a b c

39 4 4 () cos = a + b c ab a = b cos a = b a + b c ab a = a + b c b = c b > 0 c > 0 b = c b = c c a b () R a sin = b sin = c sin = R sin = a R, sin = b R, sin = c R sin = sin + sin ( ) a ( ) = b ( ) + c a = b + c R R R = 90 () a = b + c bc cos a = b + c + bc b + c bc cos = b + c + bc bc( + cos ) = 0 b \= 0 c \= 0 + cos = 0 cos = = 0

40 4 5 () a cos = b cos a cos = b cos a a + c b = b b + c a ac bc a + c b = b + c a c c a = b a > 0 b > 0 a = b a = b c a b () a cos b cos = c a a + c b b b + c a = c ac bc a + c b c b + c a c = c a + c b (b + c a ) = c a = b + c = 90 6 R tan a = sin a cos = a a R bc b + c a = bc a(b + c a )R tan b = b b R tan a = tan b bc a(b + c a )R = b a(b + c a ) = ac a + c b = ca b(a + c b )R ca b(a + c b )R a b(a + c b ) b (a + c b ) = a (b + c a ) c (b a )c (b 4 a 4 ) = 0 (b a )c (b + a )(b a ) = 0 (b + a)(b a){c (b + a )} = 0 b + a > 0 b = a c = a + b = = 90 c a b

41 4.7 7 () : = : = 7 = 60 5 () : : = : : 4 8 = 60 = c = b () () P P 9 () D = D = 5 = D = 7 D = 8 D () D = 60, = 75, D =, =, = 40 x cm cm cm

42 4 7 () sin = 60 : = : = x 60 = x = 7 x () : : = : : 4 = x = x = 4x 8 () b c () = P + P P 9 () D = D () 40 8

43 4 7 () : = : = x = x ( 7) = (x) + (x) x x cos 60 8 = 4x + 9x x x = 4 x = ( x > 0) x 7 60 x x sin 60 = 4 6 = 6 x () : : = : : 4 = x = x = 4x cos = (x) + (4x) (x) = 7 x 4x 8 x x 0 < < 80 sin > 0 4x sin = ( ) cos = 7 5 = x x 8 = = = () = bc sin 60 = x = bc () P = x () = P + P 4 bc = cx sin 0 + bx sin 0 bc bc = (b + c)x P = x = b + c P P x = c P : P = : = c : b cx sin 0 = c b + c bc sin 60 x = bc b + c ( x > 0) 0 0 x P b

44 4 9 () D cos = = < < 80 sin > 0 sin = ( ) cos = 4 = 7 7 D D = = 0 7 () D = D = D = D = 75 0 = 45 S S = D + D = + sin 45 = + = 40 x 6 6 x x sin 60 = x (cm ) r O = O = r = x O = 45 O x = r + r r r cos 45 x = ( )r r = x = + 8 O = 8 r sin 45 = + = ( + )x (cm ) ( x + x + x x ) 4 ( ) x = ( + ) x x = ( + )x (cm ) x D D x x r r 45 O x O x

45 4.8 4 P Q R S QR PQS : : Q P R S 4 D E D EF DEF EF S D S : = : F S : S = : 4 S T S : T 44 DE F G H I J () H H I H : H = : J () IH I IH : I = : F () FGHIJ DE FGHIJ : DE = : H G D E

46 4 4 a : b a : b 4 D EF : = E : DEF = E E = = a E = x 4 O 6 ( ) 44 () D ED I H E = 80 5 = 6 = 08 6 H () IH H I D E

47 4 4 QR PS : : QR : PS : = : : R S QR : QRS = : 9 QR = PQS = PS QR QRS = 4 = QR : PQS : = : : 9 P Q R S 4 = = a E = x D EF a : = : x ax = F + F = x + = a x a(a ) = a a = 0 a > 0 a = + 5 : = : a = : + 5 D EF S : S = : a + 5 = : x E D F a

48 4 4 O D O a : D = : D = a : a tan 0 = : ( = : ) S : T = ( ) : = : 4 O D 0 44 DE D ED 80 E = 80 5 = 08 I = I = IH = HI = 7 () H : H = 6 : 7 = : () IH = I = x IH H H = I = I IH : H = H : H : x = x : (x + ) = 6 x x = 0 x = IH : I = : I H ( x > 0) () FGHIJ DE IH = = H = + x = + 5 ( ) + 5 : = : J I F H G D E

49 m 57 m m 67 m tan 57 =.599 tan 67 = m 0 m m 47 cm 4cm 4cm = α sin α =, cos α = D EFGH = D = E = D DE = θ () D DE E H G () cos θ E F () DE (4) DE

50 (4) DE E D h DE

51 4 45 m x m tan 57 = x 0 x = 0 tan 57 = = x + = (m) m 57 0m xm 67 m y m y = 0 tan 67 = = y + = (m) 46 H = x H = x tan 60 = x = H tan 0 = H = x + = x + 00 = 9x x > 0 x = 00 8 = 00 = 5 = x = 5 (m), H = 5 6 (m) H 60 0 xm 00m 47 = 4 = 4 + = = 0 (cm ) = = 4 sin α = = 5 4, cos α = 4 = α 5

52 4 48 () D = + = DE = + = 5 E = + = 0 () DE cos θ = ( 5) + ( 0) ( ) 5 0 = 5 E D D 5 F G = 0 () 0 < θ < 80 sin θ > 0 sin θ = ( cos θ = 0 ) = 7 0 E θ 0 DE = = 7 (4) DE V V = E D D = = h H G DE h V = h DE E F = h 7 = 7 6 h = 7 6 h h = 6 7

53 = = D = = D = D = 5 D D D 50 OD O = O = O = OD = = = D = O D = 4 cm M OD M D OD cm M 5 = cm D = N D 6 cm E = cm D R M EFGH D E : M N FG H Q EH P Q S MN R F P G EG PQ S () () () MR () MR PGS

54 4 49 D = = D D 50 a : b a : b V ( ) V = ( OD ) ( ) = ( OD ) 5 () MR : () G SR PM

55 4 49 D = θ D cos θ = + ( 5) = 0 < θ < 80 sin θ > 0 sin θ = ( ) cos θ = 5 = D R D R = D 5 R = = sin θ 5 D H H = H = 9 H H = H = 9 H DH = D H = 9 H H = H = DH H D H = R = H = H = = D D H = 5 = H D D D = D + D D = 90 D R R = =

56 4 50 OD O V ( ) V = OD = 8 OD O D H H = = = OH = O H = 4 ( ) M 4 H O D = V = 6 4 = (cm ) H 5 () MR MR : = M : = : = : MR = = cm () G SR PM T TMR TPGS M : PG = : : = 8 : 7 ( V V = 8 ) TPGS 7 PGS = PG PS = = 9 4 TG = h T : TG = : (h ) : h = : h = (h ) h = 9 TPGS = PGS h = = 7 4 V = = 9 4 (cm ) P M S R T G

57 4. 5 D M MD = θ cos θ = MD θ D M 5 O O P P = θ cos θ = 5 OP < P OP 54 O O : P O P = θ P S () P () cos θ = S = () O V P θ h V = (4) O P h

58 4 5 4 M M DM D MD M = DM MD cos θ cos θ sin θ 5 P P = P cos θ 54 () O H O = O = O H H = H sin 60 = H H = (4) OP h

59 4 5 M DM M = DM = sin 60 = MD cos θ = ( ) + ( ) = 0 < θ < 80 sin θ > 0 sin θ = ( ) cos θ = = MD = = M θ D 5 P = x P = P P cos θ = x + x x x cos θ = 5 x x = 5 6x = 0x x = 6 = x x P = x = ( x > 0) 4 OP = 60 OP = y OP ) = + y y cos 60 ( 4 y y + 6 = 0 (4y )(4y ) = 0 OP = y = ( y < ) 4 P y θ x O x

60 4 54 () P ( ) P = cos 60 = 6 P = ( P > 0) 4 () P = P = 4 cos θ = P + P P P P = 5 0 < θ < 80 sin θ > 0 ( ) sin θ = 5 = P S S = 4 4 = 8 P θ 4 O () O OH H H M H M O H = M = sin 60 = ( ) 6 OH = = P M O V H V = OH = 6 sin 60 = (4) OP V OP : P = : V = V 4 = 4 = 6 V = S h = h 8 h 8 = 6 h = P O h

61 4. 55 a D 56 DE P Q P DE Q DE P Q DE () () P () Q E Q P D 57 = 6 = 6 O O 58

62 4 r V = 4 πr, S = 4πr r 55 D

63 4 55 D H = = D H D D H D D M H M H = a sin 60 = a ( ) 6 H = a a = a O R 6 OH = a R H + HO = O ( ) ( ) 6 a + a R = R 6 R = 4 a M MO M O : OH = M : MH = : R = O = 4 H = 6 4 a 56 () F DE F = F = = 4 = F + F = 4 + = 7 () ED G F G P Q FH = FG = = R H R H E O M M G D D H = F FH = 4 = 5 F

64 4 P r FH PI F : P = FH : PI 4 : ( 5 r) = : r 4r = 5 r r = 5 5 () J = H r = 5 5 = J : H = 5 : 5 = : 5 5 Q P QJ : PH = : 5 ( Q = 5 F H ) ( ) P = 4 5 π r ( ) 5 = π = π = 4π 5 I Q J P G 57 6 G M GM = M tan 0 = = O = 4 π( ) = 4 π G G M 0 M 58 O r OH = r O = r OH r = (r ) + r = 5 4 π r = 5 6 π H r O r

さくらの個別指導 ( さくら教育研究所 ) A 2 2 Q ABC 2 1 BC AB, AC AB, BC AC 1 B BC AB = QR PQ = 1 2 AC AB = PR 3 PQ = 2 BC AC = QR PR = 1

さくらの個別指導 ( さくら教育研究所 ) A 2 2 Q ABC 2 1 BC AB, AC AB, BC AC 1 B BC AB = QR PQ = 1 2 AC AB = PR 3 PQ = 2 BC AC = QR PR = 1 ... 0 60 Q,, = QR PQ = = PR PQ = = QR PR = P 0 0 R 5 6 θ r xy r y y r, x r, y x θ x θ θ (sine) (cosine) (tangent) sin θ, cos θ, tan θ. θ sin θ = = 5 cos θ = = 4 5 tan θ = = 4 θ 5 4 sin θ = y r cos θ =

More information

5. F(, 0) = = 4 = 4 O = 4 =. ( = = 4 ) = 4 ( 4 ), 0 = 4 4 O 4 = 4. () = 8 () = 4

5. F(, 0) = = 4 = 4 O = 4 =. ( = = 4 ) = 4 ( 4 ), 0 = 4 4 O 4 = 4. () = 8 () = 4 ... A F F l F l F(p, 0) = p p > 0 l p 0 P(, ) H P(, ) P l PH F PF = PH PF = PH p O p ( p) + = { ( p)} = 4p l = 4p (p 0) F(p, 0) = p O 3 5 5. F(, 0) = = 4 = 4 O = 4 =. ( = = 4 ) = 4 ( 4 ), 0 = 4 4 O 4 =

More information

1 29 ( ) I II III A B (120 ) 2 5 I II III A B (120 ) 1, 6 8 I II A B (120 ) 1, 6, 7 I II A B (100 ) 1 OAB A B OA = 2 OA OB = 3 OB A B 2 :

1 29 ( ) I II III A B (120 ) 2 5 I II III A B (120 ) 1, 6 8 I II A B (120 ) 1, 6, 7 I II A B (100 ) 1 OAB A B OA = 2 OA OB = 3 OB A B 2 : 9 ( ) 9 5 I II III A B (0 ) 5 I II III A B (0 ), 6 8 I II A B (0 ), 6, 7 I II A B (00 ) OAB A B OA = OA OB = OB A B : P OP AB Q OA = a OB = b () OP a b () OP OQ () a = 5 b = OP AB OAB PAB a f(x) = (log

More information

70 : 20 : A B (20 ) (30 ) 50 1

70 : 20 : A B (20 ) (30 ) 50 1 70 : 0 : A B (0 ) (30 ) 50 1 1 4 1.1................................................ 5 1. A............................................... 6 1.3 B............................................... 7 8.1 A...............................................

More information

A(6, 13) B(1, 1) 65 y C 2 A(2, 1) B( 3, 2) C 66 x + 2y 1 = 0 2 A(1, 1) B(3, 0) P 67 3 A(3, 3) B(1, 2) C(4, 0) (1) ABC G (2) 3 A B C P 6

A(6, 13) B(1, 1) 65 y C 2 A(2, 1) B( 3, 2) C 66 x + 2y 1 = 0 2 A(1, 1) B(3, 0) P 67 3 A(3, 3) B(1, 2) C(4, 0) (1) ABC G (2) 3 A B C P 6 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

More information

4 4 θ X θ P θ 4. 0, 405 P 0 X 405 X P 4. () 60 () 45 () 40 (4) 765 (5) 40 B 60 0 P = 90, = ( ) = X

4 4 θ X θ P θ 4. 0, 405 P 0 X 405 X P 4. () 60 () 45 () 40 (4) 765 (5) 40 B 60 0 P = 90, = ( ) = X 4 4. 4.. 5 5 0 A P P P X X X X +45 45 0 45 60 70 X 60 X 0 P P 4 4 θ X θ P θ 4. 0, 405 P 0 X 405 X P 4. () 60 () 45 () 40 (4) 765 (5) 40 B 60 0 P 0 0 + 60 = 90, 0 + 60 = 750 0 + 60 ( ) = 0 90 750 0 90 0

More information

( )

( ) 18 10 01 ( ) 1 2018 4 1.1 2018............................... 4 1.2 2018......................... 5 2 2017 7 2.1 2017............................... 7 2.2 2017......................... 8 3 2016 9 3.1 2016...............................

More information

2 (1) a = ( 2, 2), b = (1, 2), c = (4, 4) c = l a + k b l, k (2) a = (3, 5) (1) (4, 4) = l( 2, 2) + k(1, 2), (4, 4) = ( 2l + k, 2l 2k) 2l + k = 4, 2l

2 (1) a = ( 2, 2), b = (1, 2), c = (4, 4) c = l a + k b l, k (2) a = (3, 5) (1) (4, 4) = l( 2, 2) + k(1, 2), (4, 4) = ( 2l + k, 2l 2k) 2l + k = 4, 2l ABCDEF a = AB, b = a b (1) AC (3) CD (2) AD (4) CE AF B C a A D b F E (1) AC = AB + BC = AB + AO = AB + ( AB + AF) = a + ( a + b) = 2 a + b (2) AD = 2 AO = 2( AB + AF) = 2( a + b) (3) CD = AF = b (4) CE

More information

熊本県数学問題正解

熊本県数学問題正解 00 y O x Typed by L A TEX ε ( ) (00 ) 5 4 4 ( ) http://www.ocn.ne.jp/ oboetene/plan/. ( ) (009 ) ( ).. http://www.ocn.ne.jp/ oboetene/plan/eng.html 8 i i..................................... ( )0... (

More information

(4) P θ P 3 P O O = θ OP = a n P n OP n = a n {a n } a = θ, a n = a n (n ) {a n } θ a n = ( ) n θ P n O = a a + a 3 + ( ) n a n a a + a 3 + ( ) n a n

(4) P θ P 3 P O O = θ OP = a n P n OP n = a n {a n } a = θ, a n = a n (n ) {a n } θ a n = ( ) n θ P n O = a a + a 3 + ( ) n a n a a + a 3 + ( ) n a n 3 () 3,,C = a, C = a, C = b, C = θ(0 < θ < π) cos θ = a + (a) b (a) = 5a b 4a b = 5a 4a cos θ b = a 5 4 cos θ a ( b > 0) C C l = a + a + a 5 4 cos θ = a(3 + 5 4 cos θ) C a l = 3 + 5 4 cos θ < cos θ < 4

More information

入試の軌跡

入試の軌跡 4 y O x 4 Typed by L A TEX ε ) ) ) 6 4 ) 4 75 ) http://kumamoto.s.xrea.com/plan/.. PDF) Ctrl +L) Ctrl +) Ctrl + Ctrl + ) ) Alt + ) Alt + ) ESC. http://kumamoto.s.xrea.com/nyusi/kumadai kiseki ri i.pdf

More information

17 ( ) II III A B C(100 ) 1, 2, 6, 7 II A B (100 ) 2, 5, 6 II A B (80 ) 8 10 I II III A B C(80 ) 1 a 1 = 1 2 a n+1 = a n + 2n + 1 (n = 1,

17 ( ) II III A B C(100 ) 1, 2, 6, 7 II A B (100 ) 2, 5, 6 II A B (80 ) 8 10 I II III A B C(80 ) 1 a 1 = 1 2 a n+1 = a n + 2n + 1 (n = 1, 17 ( ) 17 5 1 4 II III A B C(1 ) 1,, 6, 7 II A B (1 ), 5, 6 II A B (8 ) 8 1 I II III A B C(8 ) 1 a 1 1 a n+1 a n + n + 1 (n 1,,, ) {a n+1 n } (1) a 4 () a n OA OB AOB 6 OAB AB : 1 P OB Q OP AQ R (1) PQ

More information

A (1) = 4 A( 1, 4) 1 A 4 () = tan A(0, 0) π A π

A (1) = 4 A( 1, 4) 1 A 4 () = tan A(0, 0) π A π 4 4.1 4.1.1 A = f() = f() = a f (a) = f() (a, f(a)) = f() (a, f(a)) f(a) = f 0 (a)( a) 4.1 (4, ) = f() = f () = 1 = f (4) = 1 4 4 (4, ) = 1 ( 4) 4 = 1 4 + 1 17 18 4 4.1 A (1) = 4 A( 1, 4) 1 A 4 () = tan

More information

1 (1) ( i ) 60 (ii) 75 (iii) 315 (2) π ( i ) (ii) π (iii) 7 12 π ( (3) r, AOB = θ 0 < θ < π ) OAB A 2 OB P ( AB ) < ( AP ) (4) 0 < θ < π 2 sin θ

1 (1) ( i ) 60 (ii) 75 (iii) 315 (2) π ( i ) (ii) π (iii) 7 12 π ( (3) r, AOB = θ 0 < θ < π ) OAB A 2 OB P ( AB ) < ( AP ) (4) 0 < θ < π 2 sin θ 1 (1) ( i ) 60 (ii) 75 (iii) 15 () ( i ) (ii) 4 (iii) 7 1 ( () r, AOB = θ 0 < θ < ) OAB A OB P ( AB ) < ( AP ) (4) 0 < θ < sin θ < θ < tan θ 0 x, 0 y (1) sin x = sin y (x, y) () cos x cos y (x, y) 1 c

More information

さくらの個別指導 ( さくら教育研究所 ) A 2 P Q 3 R S T R S T P Q ( ) ( ) m n m n m n n n

さくらの個別指導 ( さくら教育研究所 ) A 2 P Q 3 R S T R S T P Q ( ) ( ) m n m n m n n n 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

漸化式のすべてのパターンを解説しましたー高校数学の達人・河見賢司のサイト

漸化式のすべてのパターンを解説しましたー高校数学の達人・河見賢司のサイト https://www.hmg-gen.com/tuusin.html https://www.hmg-gen.com/tuusin1.html 1 2 OK 3 4 {a n } (1) a 1 = 1, a n+1 a n = 2 (2) a 1 = 3, a n+1 a n = 2n a n a n+1 a n = ( ) a n+1 a n = ( ) a n+1 a n {a n } 1,

More information

6 2 2 x y x y t P P = P t P = I P P P ( ) ( ) ,, ( ) ( ) cos θ sin θ cos θ sin θ, sin θ cos θ sin θ cos θ y x θ x θ P

6 2 2 x y x y t P P = P t P = I P P P ( ) ( ) ,, ( ) ( ) cos θ sin θ cos θ sin θ, sin θ cos θ sin θ cos θ y x θ x θ P 6 x x 6.1 t P P = P t P = I P P P 1 0 1 0,, 0 1 0 1 cos θ sin θ cos θ sin θ, sin θ cos θ sin θ cos θ x θ x θ P x P x, P ) = t P x)p ) = t x t P P ) = t x = x, ) 6.1) x = Figure 6.1 Px = x, P=, θ = θ P

More information

(1) PQ (2) () 2 PR = PR P : P = R : R (2) () = P = P R M = XM : = M : M (1) (2) = N = N X M 161 (1) (2) F F = F F F EF = F E

(1) PQ (2) () 2 PR = PR P : P = R : R (2) () = P = P R M = XM : = M : M (1) (2) = N = N X M 161 (1) (2) F F = F F F EF = F E 5 1 1 1.1 2 159 O O PQ RS OR P = PQ P O M MQ O (1) M P (2) P : P R : R () PR P 160 > M : = M : M X (1) N = N M // N X M (2) M 161 (1) E = 8 = 4 = = E = (2) : = 2 : = E = E F 5 F EF F E 5 1 159 (1) PQ (2)

More information

高等学校学習指導要領解説 数学編

高等学校学習指導要領解説 数学編 5 10 15 20 25 30 35 5 1 1 10 1 1 2 4 16 15 18 18 18 19 19 20 19 19 20 1 20 2 22 25 3 23 4 24 5 26 28 28 30 28 28 1 28 2 30 3 31 35 4 33 5 34 36 36 36 40 36 1 36 2 39 3 41 4 42 45 45 45 46 5 1 46 2 48 3

More information

IMO 1 n, 21n n (x + 2x 1) + (x 2x 1) = A, x, (a) A = 2, (b) A = 1, (c) A = 2?, 3 a, b, c cos x a cos 2 x + b cos x + c = 0 cos 2x a

IMO 1 n, 21n n (x + 2x 1) + (x 2x 1) = A, x, (a) A = 2, (b) A = 1, (c) A = 2?, 3 a, b, c cos x a cos 2 x + b cos x + c = 0 cos 2x a 1 40 (1959 1999 ) (IMO) 41 (2000 ) WEB 1 1959 1 IMO 1 n, 21n + 4 13n + 3 2 (x + 2x 1) + (x 2x 1) = A, x, (a) A = 2, (b) A = 1, (c) A = 2?, 3 a, b, c cos x a cos 2 x + b cos x + c = 0 cos 2x a = 4, b =

More information

1 3 1.1.......................... 3 1............................... 3 1.3....................... 5 1.4.......................... 6 1.5........................ 7 8.1......................... 8..............................

More information

18 ( ) ( ) [ ] [ ) II III A B (120 ) 1, 2, 3, 5, 6 II III A B (120 ) ( ) 1, 2, 3, 7, 8 II III A B (120 ) ( [ ]) 1, 2, 3, 5, 7 II III A B (

18 ( ) ( ) [ ] [ ) II III A B (120 ) 1, 2, 3, 5, 6 II III A B (120 ) ( ) 1, 2, 3, 7, 8 II III A B (120 ) ( [ ]) 1, 2, 3, 5, 7 II III A B ( 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

More information

) 9 81

) 9 81 4 4.0 2000 ) 9 81 10 4.1 natural numbers 1, 2, 3, 4, 4.2, 3, 2, 1, 0, 1, 2, 3, integral numbers integers 1, 2, 3,, 3, 2, 1 1 4.3 4.3.1 ( ) m, n m 0 n m 82 rational numbers m 1 ( ) 3 = 3 1 4.3.2 3 5 = 2

More information

高校生の就職への数学II

高校生の就職への数学II II O Tped b L A TEX ε . II. 3. 4. 5. http://www.ocn.ne.jp/ oboetene/plan/ 7 9 i .......................................................................................... 3..3...............................

More information

Part y mx + n mt + n m 1 mt n + n t m 2 t + mn 0 t m 0 n 18 y n n a 7 3 ; x α α 1 7α +t t 3 4α + 3t t x α x α y mx + n

Part y mx + n mt + n m 1 mt n + n t m 2 t + mn 0 t m 0 n 18 y n n a 7 3 ; x α α 1 7α +t t 3 4α + 3t t x α x α y mx + n Part2 47 Example 161 93 1 T a a 2 M 1 a 1 T a 2 a Point 1 T L L L T T L L T L L L T T L L T detm a 1 aa 2 a 1 2 + 1 > 0 11 T T x x M λ 12 y y x y λ 2 a + 1λ + a 2 2a + 2 0 13 D D a + 1 2 4a 2 2a + 2 a

More information

() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi)

() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi) 0. A A = 4 IC () det A () A () x + y + z = x y z X Y Z = A x y z ( 5) ( s5590) 0. a + b + c b c () a a + b + c c a b a + b + c 0 a b c () a 0 c b b c 0 a c b a 0 0. A A = 7 5 4 5 0 ( 5) ( s5590) () A ()

More information

1990 IMO 1990/1/15 1:00-4:00 1 N N N 1, N 1 N 2, N 2 N 3 N 3 2 x x + 52 = 3 x x , A, B, C 3,, A B, C 2,,,, 7, A, B, C

1990 IMO 1990/1/15 1:00-4:00 1 N N N 1, N 1 N 2, N 2 N 3 N 3 2 x x + 52 = 3 x x , A, B, C 3,, A B, C 2,,,, 7, A, B, C 0 9 (1990 1999 ) 10 (2000 ) 1900 1994 1995 1999 2 SAT ACT 1 1990 IMO 1990/1/15 1:00-4:00 1 N 1990 9 N N 1, N 1 N 2, N 2 N 3 N 3 2 x 2 + 25x + 52 = 3 x 2 + 25x + 80 3 2, 3 0 4 A, B, C 3,, A B, C 2,,,, 7,

More information

1 12 ( )150 ( ( ) ) x M x 0 1 M 2 5x 2 + 4x + 3 x 2 1 M x M 2 1 M x (x + 1) 2 (1) x 2 + x + 1 M (2) 1 3 M (3) x 4 +

1 12 ( )150 ( ( ) ) x M x 0 1 M 2 5x 2 + 4x + 3 x 2 1 M x M 2 1 M x (x + 1) 2 (1) x 2 + x + 1 M (2) 1 3 M (3) x 4 + ( )5 ( ( ) ) 4 6 7 9 M M 5 + 4 + M + M M + ( + ) () + + M () M () 4 + + M a b y = a + b a > () a b () y V a () V a b V n f() = n k= k k () < f() = log( ) t dt log () n+ (i) dt t (n + ) (ii) < t dt n+ n

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 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

More information

1 θ i (1) A B θ ( ) A = B = sin 3θ = sin θ (A B sin 2 θ) ( ) 1 2 π 3 < = θ < = 2 π 3 Ax Bx3 = 1 2 θ = π sin θ (2) a b c θ sin 5θ = sin θ f(sin 2 θ) 2

1 θ i (1) A B θ ( ) A = B = sin 3θ = sin θ (A B sin 2 θ) ( ) 1 2 π 3 < = θ < = 2 π 3 Ax Bx3 = 1 2 θ = π sin θ (2) a b c θ sin 5θ = sin θ f(sin 2 θ) 2 θ i ) AB θ ) A = B = sin θ = sin θ A B sin θ) ) < = θ < = Ax Bx = θ = sin θ ) abc θ sin 5θ = sin θ fsin θ) fx) = ax bx c ) cos 5 i sin 5 ) 5 ) αβ α iβ) 5 α 4 β α β β 5 ) a = b = c = ) fx) = 0 x x = x =

More information

> > <., vs. > x 2 x y = ax 2 + bx + c y = 0 2 ax 2 + bx + c = 0 y = 0 x ( x ) y = ax 2 + bx + c D = b 2 4ac (1) D > 0 x (2) D = 0 x (3

> > <., vs. > x 2 x y = ax 2 + bx + c y = 0 2 ax 2 + bx + c = 0 y = 0 x ( x ) y = ax 2 + bx + c D = b 2 4ac (1) D > 0 x (2) D = 0 x (3 13 2 13.0 2 ( ) ( ) 2 13.1 ( ) ax 2 + bx + c > 0 ( a, b, c ) ( ) 275 > > 2 2 13.3 x 2 x y = ax 2 + bx + c y = 0 2 ax 2 + bx + c = 0 y = 0 x ( x ) y = ax 2 + bx + c D = b 2 4ac (1) D >

More information

18 ( ) I II III A B C(100 ) 1, 2, 3, 5 I II A B (100 ) 1, 2, 3 I II A B (80 ) 6 8 I II III A B C(80 ) 1 n (1 + x) n (1) n C 1 + n C

18 ( ) I II III A B C(100 ) 1, 2, 3, 5 I II A B (100 ) 1, 2, 3 I II A B (80 ) 6 8 I II III A B C(80 ) 1 n (1 + x) n (1) n C 1 + n C 8 ( ) 8 5 4 I II III A B C( ),,, 5 I II A B ( ),, I II A B (8 ) 6 8 I II III A B C(8 ) n ( + x) n () n C + n C + + n C n = 7 n () 7 9 C : y = x x A(, 6) () A C () C P AP Q () () () 4 A(,, ) B(,, ) C(,,

More information

No2 4 y =sinx (5) y = p sin(2x +3) (6) y = 1 tan(3x 2) (7) y =cos 2 (4x +5) (8) y = cos x 1+sinx 5 (1) y =sinx cos x 6 f(x) = sin(sin x) f 0 (π) (2) y

No2 4 y =sinx (5) y = p sin(2x +3) (6) y = 1 tan(3x 2) (7) y =cos 2 (4x +5) (8) y = cos x 1+sinx 5 (1) y =sinx cos x 6 f(x) = sin(sin x) f 0 (π) (2) y No1 1 (1) 2 f(x) =1+x + x 2 + + x n, g(x) = 1 (n +1)xn + nx n+1 (1 x) 2 x 6= 1 f 0 (x) =g(x) y = f(x)g(x) y 0 = f 0 (x)g(x)+f(x)g 0 (x) 3 (1) y = x2 x +1 x (2) y = 1 g(x) y0 = g0 (x) {g(x)} 2 (2) y = µ

More information

x () g(x) = f(t) dt f(x), F (x) 3x () g(x) g (x) f(x), F (x) (3) h(x) = x 3x tf(t) dt.9 = {(x, y) ; x, y, x + y } f(x, y) = xy( x y). h (x) f(x), F (x

x () g(x) = f(t) dt f(x), F (x) 3x () g(x) g (x) f(x), F (x) (3) h(x) = x 3x tf(t) dt.9 = {(x, y) ; x, y, x + y } f(x, y) = xy( x y). h (x) f(x), F (x [ ] IC. f(x) = e x () f(x) f (x) () lim f(x) lim f(x) x + x (3) lim f(x) lim f(x) x + x (4) y = f(x) ( ) ( s46). < a < () a () lim a log xdx a log xdx ( ) n (3) lim log k log n n n k=.3 z = log(x + y ),

More information

211 kotaro@math.titech.ac.jp 1 R *1 n n R n *2 R n = {(x 1,..., x n ) x 1,..., x n R}. R R 2 R 3 R n R n R n D D R n *3 ) (x 1,..., x n ) f(x 1,..., x n ) f D *4 n 2 n = 1 ( ) 1 f D R n f : D R 1.1. (x,

More information

#A A A F, F d F P + F P = d P F, F y P F F x A.1 ( α, 0), (α, 0) α > 0) (x, y) (x + α) 2 + y 2, (x α) 2 + y 2 d (x + α)2 + y 2 + (x α) 2 + y 2 =

#A A A F, F d F P + F P = d P F, F y P F F x A.1 ( α, 0), (α, 0) α > 0) (x, y) (x + α) 2 + y 2, (x α) 2 + y 2 d (x + α)2 + y 2 + (x α) 2 + y 2 = #A A A. F, F d F P + F P = d P F, F P F F A. α, 0, α, 0 α > 0, + α +, α + d + α + + α + = d d F, F 0 < α < d + α + = d α + + α + = d d α + + α + d α + = d 4 4d α + = d 4 8d + 6 http://mth.cs.kitmi-it.c.jp/

More information

29

29 9 .,,, 3 () C k k C k C + C + C + + C 8 + C 9 + C k C + C + C + C 3 + C 4 + C 5 + + 45 + + + 5 + + 9 + 4 + 4 + 5 4 C k k k ( + ) 4 C k k ( k) 3 n( ) n n n ( ) n ( ) n 3 ( ) 3 3 3 n 4 ( ) 4 4 4 ( ) n n

More information

4 4 4 a b c d a b A c d A a da ad bce O E O n A n O ad bc a d n A n O 5 {a n } S n a k n a n + k S n a a n+ S n n S n n log x x {xy } x, y x + y 7 fx

4 4 4 a b c d a b A c d A a da ad bce O E O n A n O ad bc a d n A n O 5 {a n } S n a k n a n + k S n a a n+ S n n S n n log x x {xy } x, y x + y 7 fx 4 4 5 4 I II III A B C, 5 7 I II A B,, 8, 9 I II A B O A,, Bb, b, Cc, c, c b c b b c c c OA BC P BC OP BC P AP BC n f n x xn e x! e n! n f n x f n x f n x f k x k 4 e > f n x dx k k! fx sin x cos x tan

More information

A S- hara/lectures/lectures-j.html r A = A 5 : 5 = max{ A, } A A A A B A, B A A A %

A S-   hara/lectures/lectures-j.html r A = A 5 : 5 = max{ A, } A A A A B A, B A A A % A S- http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html r A S- 3.4.5. 9 phone: 9-8-444, e-mail: hara@math.kyushu-u.ac.jp, http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html Office

More information

OABC OA OC 4, OB, AOB BOC COA 60 OA a OB b OC c () AB AC () ABC D OD ABC OD OA + p AB + q AC p q () OABC 4 f(x) + x ( ), () y f(x) P l 4 () y f(x) l P

OABC OA OC 4, OB, AOB BOC COA 60 OA a OB b OC c () AB AC () ABC D OD ABC OD OA + p AB + q AC p q () OABC 4 f(x) + x ( ), () y f(x) P l 4 () y f(x) l P 4 ( ) ( ) ( ) ( ) 4 5 5 II III A B (0 ) 4, 6, 7 II III A B (0 ) ( ),, 6, 8, 9 II III A B (0 ) ( [ ] ) 5, 0, II A B (90 ) log x x () (a) y x + x (b) y sin (x + ) () (a) (b) (c) (d) 0 e π 0 x x x + dx e

More information

B. 41 II: 2 ;; 4 B [ ] S 1 S 2 S 1 S O S 1 S P 2 3 P P : 2.13:

B. 41 II: 2 ;; 4 B [ ] S 1 S 2 S 1 S O S 1 S P 2 3 P P : 2.13: B. 41 II: ;; 4 B [] S 1 S S 1 S.1 O S 1 S 1.13 P 3 P 5 7 P.1:.13: 4 4.14 C d A B x l l d C B 1 l.14: AB A 1 B 0 AB 0 O OP = x P l AP BP AB AP BP 1 (.4)(.5) x l x sin = p l + x x l (.4)(.5) m d A x P O

More information

Note.tex 2008/09/19( )

Note.tex 2008/09/19( ) 1 20 9 19 2 1 5 1.1........................ 5 1.2............................. 8 2 9 2.1............................. 9 2.2.............................. 10 3 13 3.1.............................. 13 3.2..................................

More information

9 5 ( α+ ) = (α + ) α (log ) = α d = α C d = log + C C 5. () d = 4 d = C = C = 3 + C 3 () d = d = C = C = 3 + C 3 =

9 5 ( α+ ) = (α + ) α (log ) = α d = α C d = log + C C 5. () d = 4 d = C = C = 3 + C 3 () d = d = C = C = 3 + C 3 = 5 5. 5.. A II f() f() F () f() F () = f() C (F () + C) = F () = f() F () + C f() F () G() f() G () = F () 39 G() = F () + C C f() F () f() F () + C C f() f() d f() f() C f() f() F () = f() f() f() d =

More information

function2.pdf

function2.pdf 2... 1 2009, http://c-faculty.chuo-u.ac.jp/ nishioka/ 2 11 38 : 5) i) [], : 84 85 86 87 88 89 1000 ) 13 22 33 56 92 147 140 120 100 80 60 40 20 1 2 3 4 5 7.1 7 7.1 1. *1 e = 2.7182 ) fx) e x, x R : 7.1)

More information

1 http://www.manabino-academy.com 1 1 1.1............................................... 1 1.2............................................. 4 1.3............................................. 6 2 8 2.1.............................................

More information

untitled

untitled 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/

More information

さくらの個別指導 ( さくら教育研究所 ) A a 1 a 2 a 3 a n {a n } a 1 a n n n 1 n n 0 a n = 1 n 1 n n O n {a n } n a n α {a n } α {a

さくらの個別指導 ( さくら教育研究所 ) A a 1 a 2 a 3 a n {a n } a 1 a n n n 1 n n 0 a n = 1 n 1 n n O n {a n } n a n α {a n } α {a ... A a a a 3 a n {a n } a a n n 3 n n n 0 a n = n n n O 3 4 5 6 n {a n } n a n α {a n } α {a n } α α {a n } a n n a n α a n = α n n 0 n = 0 3 4. ()..0.00 + (0.) n () 0. 0.0 0.00 ( 0.) n 0 0 c c c c c

More information

1 26 ( ) ( ) 1 4 I II III A B C (120 ) ( ) 1, 5 7 I II III A B C (120 ) 1 (1) 0 x π 0 y π 3 sin x sin y = 3, 3 cos x + cos y = 1 (2) a b c a +

1 26 ( ) ( ) 1 4 I II III A B C (120 ) ( ) 1, 5 7 I II III A B C (120 ) 1 (1) 0 x π 0 y π 3 sin x sin y = 3, 3 cos x + cos y = 1 (2) a b c a + 6 ( ) 6 5 ( ) 4 I II III A B C ( ) ( ), 5 7 I II III A B C ( ) () x π y π sin x sin y =, cos x + cos y = () b c + b + c = + b + = b c c () 4 5 6 n ( ) ( ) ( ) n ( ) n m n + m = 555 n OAB P k m n k PO +

More information

さくらの個別指導 ( さくら教育研究所 ) A AB A B A B A AB AB AB B

さくらの個別指導 ( さくら教育研究所 ) A AB A B A B A AB AB AB B 1 1.1 1.1.1 1 1 1 1 a a a a C a a = = CD CD a a a a a a = a = = D 1.1 CD D= C = DC C D 1.1 (1) 1 3 4 5 8 7 () 6 (3) 1.1. 3 1.1. a = C = C C C a a + a + + C = a C 1. a a + (1) () (3) b a a a b CD D = D

More information

知能科学:ニューラルネットワーク

知能科学:ニューラルネットワーク 2 3 4 (Neural Network) (Deep Learning) (Deep Learning) ( x x = ax + b x x x ? x x x w σ b = σ(wx + b) x w b w b .2.8.6 σ(x) = + e x.4.2 -.2 - -5 5 x w x2 w2 σ x3 w3 b = σ(w x + w 2 x 2 + w 3 x 3 + b) x,

More information

知能科学:ニューラルネットワーク

知能科学:ニューラルネットワーク 2 3 4 (Neural Network) (Deep Learning) (Deep Learning) ( x x = ax + b x x x ? x x x w σ b = σ(wx + b) x w b w b .2.8.6 σ(x) = + e x.4.2 -.2 - -5 5 x w x2 w2 σ x3 w3 b = σ(w x + w 2 x 2 + w 3 x 3 + b) x,

More information

さくらの個別指導 ( さくら教育研究所 ) 1 φ = φ 1 : φ [ ] a [ ] 1 a : b a b b(a + b) b a 2 a 2 = b(a + b). b 2 ( a b ) 2 = a b a/b X 2 X 1 = 0 a/b > 0 2 a

さくらの個別指導 ( さくら教育研究所 ) 1 φ = φ 1 : φ [ ] a [ ] 1 a : b a b b(a + b) b a 2 a 2 = b(a + b). b 2 ( a b ) 2 = a b a/b X 2 X 1 = 0 a/b > 0 2 a φ + 5 2 φ : φ [ ] a [ ] a : b a b b(a + b) b a 2 a 2 b(a + b). b 2 ( a b ) 2 a b + a/b X 2 X 0 a/b > 0 2 a b + 5 2 φ φ : 2 5 5 [ ] [ ] x x x : x : x x : x x : x x 2 x 2 x 0 x ± 5 2 x x φ : φ 2 : φ ( )

More information

2 2 MATHEMATICS.PDF 200-2-0 3 2 (p n ), ( ) 7 3 4 6 5 20 6 GL 2 (Z) SL 2 (Z) 27 7 29 8 SL 2 (Z) 35 9 2 40 0 2 46 48 2 2 5 3 2 2 58 4 2 6 5 2 65 6 2 67 7 2 69 2 , a 0 + a + a 2 +... b b 2 b 3 () + b n a

More information

1 n A a 11 a 1n A =.. a m1 a mn Ax = λx (1) x n λ (eigenvalue problem) x = 0 ( x 0 ) λ A ( ) λ Ax = λx x Ax = λx y T A = λy T x Ax = λx cx ( 1) 1.1 Th

1 n A a 11 a 1n A =.. a m1 a mn Ax = λx (1) x n λ (eigenvalue problem) x = 0 ( x 0 ) λ A ( ) λ Ax = λx x Ax = λx y T A = λy T x Ax = λx cx ( 1) 1.1 Th 1 n A a 11 a 1n A = a m1 a mn Ax = λx (1) x n λ (eigenvalue problem) x = ( x ) λ A ( ) λ Ax = λx x Ax = λx y T A = λy T x Ax = λx cx ( 1) 11 Th9-1 Ax = λx λe n A = λ a 11 a 12 a 1n a 21 λ a 22 a n1 a n2

More information

DVIOUT-HYOU

DVIOUT-HYOU () P. () AB () AB ³ ³, BA, BA ³ ³ P. A B B A IA (B B)A B (BA) B A ³, A ³ ³ B ³ ³ x z ³ A AA w ³ AA ³ x z ³ x + z +w ³ w x + z +w ½ x + ½ z +w x + z +w x,,z,w ³ A ³ AA I x,, z, w ³ A ³ ³ + + A ³ A A P.

More information

ma22-9 u ( v w) = u v w sin θê = v w sin θ u cos φ = = 2.3 ( a b) ( c d) = ( a c)( b d) ( a d)( b c) ( a b) ( c d) = (a 2 b 3 a 3 b 2 )(c 2 d 3 c 3 d

ma22-9 u ( v w) = u v w sin θê = v w sin θ u cos φ = = 2.3 ( a b) ( c d) = ( a c)( b d) ( a d)( b c) ( a b) ( c d) = (a 2 b 3 a 3 b 2 )(c 2 d 3 c 3 d A 2. x F (t) =f sin ωt x(0) = ẋ(0) = 0 ω θ sin θ θ 3! θ3 v = f mω cos ωt x = f mω (t sin ωt) ω t 0 = f ( cos ωt) mω x ma2-2 t ω x f (t mω ω (ωt ) 6 (ωt)3 = f 6m ωt3 2.2 u ( v w) = v ( w u) = w ( u v) ma22-9

More information

x = a 1 f (a r, a + r) f(a) r a f f(a) 2 2. (a, b) 2 f (a, b) r f(a, b) r (a, b) f f(a, b)

x = a 1 f (a r, a + r) f(a) r a f f(a) 2 2. (a, b) 2 f (a, b) r f(a, b) r (a, b) f f(a, b) 2011 I 2 II III 17, 18, 19 7 7 1 2 2 2 1 2 1 1 1.1.............................. 2 1.2 : 1.................... 4 1.2.1 2............................... 5 1.3 : 2.................... 5 1.3.1 2.....................................

More information

() n C + n C + n C + + n C n n (3) n C + n C + n C 4 + n C + n C 3 + n C 5 + (5) (6 ) n C + nc + 3 nc n nc n (7 ) n C + nc + 3 nc n nc n (

() n C + n C + n C + + n C n n (3) n C + n C + n C 4 + n C + n C 3 + n C 5 + (5) (6 ) n C + nc + 3 nc n nc n (7 ) n C + nc + 3 nc n nc n ( 3 n nc k+ k + 3 () n C r n C n r nc r C r + C r ( r n ) () n C + n C + n C + + n C n n (3) n C + n C + n C 4 + n C + n C 3 + n C 5 + (4) n C n n C + n C + n C + + n C n (5) k k n C k n C k (6) n C + nc

More information

PSCHG000.PS

PSCHG000.PS a b c a ac bc ab bc a b c a c a b bc a b c a ac bc ab bc a b c a ac bc ab bc a b c a ac bc ab bc de df d d d d df d d d d d d d a a b c a b b a b c a b c b a a a a b a b a

More information

0.6 A = ( 0 ),. () A. () x n+ = x n+ + x n (n ) {x n }, x, x., (x, x ) = (0, ) e, (x, x ) = (, 0) e, {x n }, T, e, e T A. (3) A n {x n }, (x, x ) = (,

0.6 A = ( 0 ),. () A. () x n+ = x n+ + x n (n ) {x n }, x, x., (x, x ) = (0, ) e, (x, x ) = (, 0) e, {x n }, T, e, e T A. (3) A n {x n }, (x, x ) = (, [ ], IC 0. A, B, C (, 0, 0), (0,, 0), (,, ) () CA CB ACBD D () ACB θ cos θ (3) ABC (4) ABC ( 9) ( s090304) 0. 3, O(0, 0, 0), A(,, 3), B( 3,, ),. () AOB () AOB ( 8) ( s8066) 0.3 O xyz, P x Q, OP = P Q =

More information

1. 4cm 16 cm 4cm 20cm 18 cm L λ(x)=ax [kg/m] A x 4cm A 4cm 12 cm h h Y 0 a G 0.38h a b x r(x) x y = 1 h 0.38h G b h X x r(x) 1 S(x) = πr(x) 2 a,b, h,π

1. 4cm 16 cm 4cm 20cm 18 cm L λ(x)=ax [kg/m] A x 4cm A 4cm 12 cm h h Y 0 a G 0.38h a b x r(x) x y = 1 h 0.38h G b h X x r(x) 1 S(x) = πr(x) 2 a,b, h,π . 4cm 6 cm 4cm cm 8 cm λ()=a [kg/m] A 4cm A 4cm cm h h Y a G.38h a b () y = h.38h G b h X () S() = π() a,b, h,π V = ρ M = ρv G = M h S() 3 d a,b, h 4 G = 5 h a b a b = 6 ω() s v m θ() m v () θ() ω() dθ()

More information

x, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y)

x, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y) x, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 1 1977 x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y) ( x 2 y + xy 2 x 2 2xy y 2) = 15 (x y) (x + y) (xy

More information

極限

極限 si θ = ) θ 0 θ cos θ θ 0 θ = ) P T θ H A, 0) θ, 0 < θ < π ) AP, P H A P T PH < AP < AT si θ < θ < ta θ si θ < θ < si θ cos θ θ cos θ < si θ θ < θ < 0 θ = h θ 0 cos θ =, θ 0 si θ θ =. θ 0 cos θ θ θ 0 cos

More information

ORIGINAL TEXT I II A B 1 4 13 21 27 44 54 64 84 98 113 126 138 146 165 175 181 188 198 213 225 234 244 261 268 273 2 281 I II A B 292 3 I II A B c 1 1 (1) x 2 + 4xy + 4y 2 x 2y 2 (2) 8x 2 + 16xy + 6y 2

More information

直交座標系の回転

直交座標系の回転 b T.Koama x l x, Lx i ij j j xi i i i, x L T L L, L ± x L T xax axx, ( a a ) i, j ij i j ij ji λ λ + λ + + λ i i i x L T T T x ( L) L T xax T ( T L T ) A( L) T ( LAL T ) T ( L AL) λ ii L AL Λ λi i axx

More information

koji07-01.dvi

koji07-01.dvi 2007 I II III 1, 2, 3, 4, 5, 6, 7 5 10 19 (!) 1938 70 21? 1 1 2 1 2 2 1! 4, 5 1? 50 1 2 1 1 2 2 1?? 2 1 1, 2 1, 2 1, 2, 3,... 3 1, 2 1, 3? 2 1 3 1 2 1 1, 2 2, 3? 2 1 3 2 3 2 k,l m, n k,l m, n kn > ml...?

More information

( ) 2.1. C. (1) x 4 dx = 1 5 x5 + C 1 (2) x dx = x 2 dx = x 1 + C = 1 2 x + C xdx (3) = x dx = 3 x C (4) (x + 1) 3 dx = (x 3 + 3x 2 + 3x +

( ) 2.1. C. (1) x 4 dx = 1 5 x5 + C 1 (2) x dx = x 2 dx = x 1 + C = 1 2 x + C xdx (3) = x dx = 3 x C (4) (x + 1) 3 dx = (x 3 + 3x 2 + 3x + (.. C. ( d 5 5 + C ( d d + C + C d ( d + C ( ( + d ( + + + d + + + + C (5 9 + d + d tan + C cos (sin (6 sin d d log sin + C sin + (7 + + d ( + + + + d log( + + + C ( (8 d 7 6 d + 6 + C ( (9 ( d 6 + 8 d

More information

7 27 7.1........................................ 27 7.2.......................................... 28 1 ( a 3 = 3 = 3 a a > 0(a a a a < 0(a a a -1 1 6

7 27 7.1........................................ 27 7.2.......................................... 28 1 ( a 3 = 3 = 3 a a > 0(a a a a < 0(a a a -1 1 6 26 11 5 1 ( 2 2 2 3 5 3.1...................................... 5 3.2....................................... 5 3.3....................................... 6 3.4....................................... 7

More information

.1 A cos 2π 3 sin 2π 3 sin 2π 3 cos 2π 3 T ra 2 deta T ra 2 deta T ra 2 deta a + d 2 ad bc a 2 + d 2 + ad + bc A 3 a b a 2 + bc ba + d c d ca + d bc +

.1 A cos 2π 3 sin 2π 3 sin 2π 3 cos 2π 3 T ra 2 deta T ra 2 deta T ra 2 deta a + d 2 ad bc a 2 + d 2 + ad + bc A 3 a b a 2 + bc ba + d c d ca + d bc + .1 n.1 1 A T ra A A a b c d A 2 a b a b c d c d a 2 + bc ab + bd ac + cd bc + d 2 a 2 + bc ba + d ca + d bc + d 2 A a + d b c T ra A T ra A 2 A 2 A A 2 A 2 A n A A n cos 2π sin 2π n n A k sin 2π cos 2π

More information

zz + 3i(z z) + 5 = 0 + i z + i = z 2i z z z y zz + 3i (z z) + 5 = 0 (z 3i) (z + 3i) = 9 5 = 4 z 3i = 2 (3i) zz i (z z) + 1 = a 2 {

zz + 3i(z z) + 5 = 0 + i z + i = z 2i z z z y zz + 3i (z z) + 5 = 0 (z 3i) (z + 3i) = 9 5 = 4 z 3i = 2 (3i) zz i (z z) + 1 = a 2 { 04 zz + iz z) + 5 = 0 + i z + i = z i z z z 970 0 y zz + i z z) + 5 = 0 z i) z + i) = 9 5 = 4 z i = i) zz i z z) + = a {zz + i z z) + 4} a ) zz + a + ) z z) + 4a = 0 4a a = 5 a = x i) i) : c Darumafactory

More information

2000年度『数学展望 I』講義録

2000年度『数学展望 I』講義録 2000 I I IV I II 2000 I I IV I-IV. i ii 3.10 (http://www.math.nagoya-u.ac.jp/ kanai/) 2000 A....1 B....4 C....10 D....13 E....17 Brouwer A....21 B....26 C....33 D....39 E. Sperner...45 F....48 A....53

More information

1. (8) (1) (x + y) + (x + y) = 0 () (x + y ) 5xy = 0 (3) (x y + 3y 3 ) (x 3 + xy ) = 0 (4) x tan y x y + x = 0 (5) x = y + x + y (6) = x + y 1 x y 3 (

1. (8) (1) (x + y) + (x + y) = 0 () (x + y ) 5xy = 0 (3) (x y + 3y 3 ) (x 3 + xy ) = 0 (4) x tan y x y + x = 0 (5) x = y + x + y (6) = x + y 1 x y 3 ( 1 1.1 (1) (1 + x) + (1 + y) = 0 () x + y = 0 (3) xy = x (4) x(y + 3) + y(y + 3) = 0 (5) (a + y ) = x ax a (6) x y 1 + y x 1 = 0 (7) cos x + sin x cos y = 0 (8) = tan y tan x (9) = (y 1) tan x (10) (1 +

More information

1 1 3 ABCD ABD AC BD E E BD 1 : 2 (1) AB = AD =, AB AD = (2) AE = AB + (3) A F AD AE 2 = AF = AB + AD AF AE = t AC = t AE AC FC = t = (4) ABD ABCD 1 1

1 1 3 ABCD ABD AC BD E E BD 1 : 2 (1) AB = AD =, AB AD = (2) AE = AB + (3) A F AD AE 2 = AF = AB + AD AF AE = t AC = t AE AC FC = t = (4) ABD ABCD 1 1 ABCD ABD AC BD E E BD : () AB = AD =, AB AD = () AE = AB + () A F AD AE = AF = AB + AD AF AE = t AC = t AE AC FC = t = (4) ABD ABCD AB + AD AB + 7 9 AD AB + AD AB + 9 7 4 9 AD () AB sin π = AB = ABD AD

More information

limit&derivative

limit&derivative - - 7 )................................................................................ 5.................................. 7.. e ).......................... 9 )..........................................

More information

1. 2 P 2 (x, y) 2 x y (0, 0) R 2 = {(x, y) x, y R} x, y R P = (x, y) O = (0, 0) OP ( ) OP x x, y y ( ) x v = y ( ) x 2 1 v = P = (x, y) y ( x y ) 2 (x

1. 2 P 2 (x, y) 2 x y (0, 0) R 2 = {(x, y) x, y R} x, y R P = (x, y) O = (0, 0) OP ( ) OP x x, y y ( ) x v = y ( ) x 2 1 v = P = (x, y) y ( x y ) 2 (x . P (, (0, 0 R {(,, R}, R P (, O (0, 0 OP OP, v v P (, ( (, (, { R, R} v (, (, (,, z 3 w z R 3,, z R z n R n.,..., n R n n w, t w ( z z Ke Words:. A P 3 0 B P 0 a. A P b B P 3. A π/90 B a + b c π/ 3. +

More information

2016 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 1 16 2 1 () X O 3 (O1) X O, O (O2) O O (O3) O O O X (X, O) O X X (O1), (O2), (O3) (O2) (O3) n (O2) U 1,..., U n O U k O k=1 (O3) U λ O( λ Λ) λ Λ U λ O 0 X 0 (O2) n =

More information

IA 2013 : :10722 : 2 : :2 :761 :1 (23-27) : : ( / ) (1 /, ) / e.g. (Taylar ) e x = 1 + x + x xn n! +... sin x = x x3 6 + x5 x2n+1 + (

IA 2013 : :10722 : 2 : :2 :761 :1 (23-27) : : ( / ) (1 /, ) / e.g. (Taylar ) e x = 1 + x + x xn n! +... sin x = x x3 6 + x5 x2n+1 + ( IA 2013 : :10722 : 2 : :2 :761 :1 23-27) : : 1 1.1 / ) 1 /, ) / e.g. Taylar ) e x = 1 + x + x2 2 +... + xn n! +... sin x = x x3 6 + x5 x2n+1 + 1)n 5! 2n + 1)! 2 2.1 = 1 e.g. 0 = 0.00..., π = 3.14..., 1

More information

85 4

85 4 85 4 86 Copright c 005 Kumanekosha 4.1 ( ) ( t ) t, t 4.1.1 t Step! (Step 1) (, 0) (Step ) ±V t (, t) I Check! P P V t π 54 t = 0 + V (, t) π θ : = θ : π ) θ = π ± sin ± cos t = 0 (, 0) = sin π V + t +V

More information

.5 z = a + b + c n.6 = a sin t y = b cos t dy d a e e b e + e c e e e + e 3 s36 3 a + y = a, b > b 3 s363.7 y = + 3 y = + 3 s364.8 cos a 3 s365.9 y =,

.5 z = a + b + c n.6 = a sin t y = b cos t dy d a e e b e + e c e e e + e 3 s36 3 a + y = a, b > b 3 s363.7 y = + 3 y = + 3 s364.8 cos a 3 s365.9 y =, [ ] IC. r, θ r, θ π, y y = 3 3 = r cos θ r sin θ D D = {, y ; y }, y D r, θ ep y yddy D D 9 s96. d y dt + 3dy + y = cos t dt t = y = e π + e π +. t = π y =.9 s6.3 d y d + dy d + y = y =, dy d = 3 a, b

More information

Microsoft Word - 11問題表紙(選択).docx

Microsoft Word - 11問題表紙(選択).docx A B A.70g/cm 3 B.74g/cm 3 B C 70at% %A C B at% 80at% %B 350 C γ δ y=00 x-y ρ l S ρ C p k C p ρ C p T ρ l t l S S ξ S t = ( k T ) ξ ( ) S = ( k T) ( ) t y ξ S ξ / t S v T T / t = v T / y 00 x v S dy dx

More information

Part () () Γ Part ,

Part () () Γ Part , Contents a 6 6 6 6 6 6 6 7 7. 8.. 8.. 8.3. 8 Part. 9. 9.. 9.. 3. 3.. 3.. 3 4. 5 4.. 5 4.. 9 4.3. 3 Part. 6 5. () 6 5.. () 7 5.. 9 5.3. Γ 3 6. 3 6.. 3 6.. 3 6.3. 33 Part 3. 34 7. 34 7.. 34 7.. 34 8. 35

More information

(1.2) T D = 0 T = D = 30 kn 1.2 (1.4) 2F W = 0 F = W/2 = 300 kn/2 = 150 kn 1.3 (1.9) R = W 1 + W 2 = = 1100 N. (1.9) W 2 b W 1 a = 0

(1.2) T D = 0 T = D = 30 kn 1.2 (1.4) 2F W = 0 F = W/2 = 300 kn/2 = 150 kn 1.3 (1.9) R = W 1 + W 2 = = 1100 N. (1.9) W 2 b W 1 a = 0 1 1 1.1 1.) T D = T = D = kn 1. 1.4) F W = F = W/ = kn/ = 15 kn 1. 1.9) R = W 1 + W = 6 + 5 = 11 N. 1.9) W b W 1 a = a = W /W 1 )b = 5/6) = 5 cm 1.4 AB AC P 1, P x, y x, y y x 1.4.) P sin 6 + P 1 sin 45

More information

n ( (

n ( ( 1 2 27 6 1 1 m-mat@mathscihiroshima-uacjp 2 http://wwwmathscihiroshima-uacjp/~m-mat/teach/teachhtml 2 1 3 11 3 111 3 112 4 113 n 4 114 5 115 5 12 7 121 7 122 9 123 11 124 11 125 12 126 2 2 13 127 15 128

More information

1

1 1 1 7 1.1.................................. 11 2 13 2.1............................ 13 2.2............................ 17 2.3.................................. 19 3 21 3.1.............................

More information

) a + b = i + 6 b c = 6i j ) a = 0 b = c = 0 ) â = i + j 0 ˆb = 4) a b = b c = j + ) cos α = cos β = 6) a ˆb = b ĉ = 0 7) a b = 6i j b c = i + 6j + 8)

) a + b = i + 6 b c = 6i j ) a = 0 b = c = 0 ) â = i + j 0 ˆb = 4) a b = b c = j + ) cos α = cos β = 6) a ˆb = b ĉ = 0 7) a b = 6i j b c = i + 6j + 8) 4 4 ) a + b = i + 6 b c = 6i j ) a = 0 b = c = 0 ) â = i + j 0 ˆb = 4) a b = b c = j + ) cos α = cos β = 6) a ˆb = b ĉ = 0 7) a b = 6i j b c = i + 6j + 8) a b a b = 6i j 4 b c b c 9) a b = 4 a b) c = 7

More information

[ ] 0.1 lim x 0 e 3x 1 x IC ( 11) ( s114901) 0.2 (1) y = e 2x (x 2 + 1) (2) y = x/(x 2 + 1) 0.3 dx (1) 1 4x 2 (2) e x sin 2xdx (3) sin 2 xdx ( 11) ( s

[ ] 0.1 lim x 0 e 3x 1 x IC ( 11) ( s114901) 0.2 (1) y = e 2x (x 2 + 1) (2) y = x/(x 2 + 1) 0.3 dx (1) 1 4x 2 (2) e x sin 2xdx (3) sin 2 xdx ( 11) ( s [ ]. lim e 3 IC ) s49). y = e + ) ) y = / + ).3 d 4 ) e sin d 3) sin d ) s49) s493).4 z = y z z y s494).5 + y = 4 =.6 s495) dy = 3e ) d dy d = y s496).7 lim ) lim e s49).8 y = e sin ) y = sin e 3) y =

More information

3.300 m m m m m m 0 m m m 0 m 0 m m m he m T m 1.50 m N/ N

3.300 m m m m m m 0 m m m 0 m 0 m m m he m T m 1.50 m N/ N 3.300 m 0.500 m 0.300 m 0.300 m 0.300 m 0.500 m 0 m 1.000 m 2.000 m 0 m 0 m 0.300 m 0.300 m -0.200 he 0.400 m T 0.200 m 1.50 m 0.16 2 24.5 N/ 3 18.0 N/ 3 28.0 18.7 18.7 14.0 14.0 X(m) 1.000 2.000 20 Y(m)

More information

t θ, τ, α, β S(, 0 P sin(θ P θ S x cos(θ SP = θ P (cos(θ, sin(θ sin(θ P t tan(θ θ 0 cos(θ tan(θ = sin(θ cos(θ ( 0t tan(θ

t θ, τ, α, β S(, 0 P sin(θ P θ S x cos(θ SP = θ P (cos(θ, sin(θ sin(θ P t tan(θ θ 0 cos(θ tan(θ = sin(θ cos(θ ( 0t tan(θ 4 5 ( 5 3 9 4 0 5 ( 4 6 7 7 ( 0 8 3 9 ( 8 t θ, τ, α, β S(, 0 P sin(θ P θ S x cos(θ SP = θ P (cos(θ, sin(θ sin(θ P t tan(θ θ 0 cos(θ tan(θ = sin(θ cos(θ ( 0t tan(θ S θ > 0 θ < 0 ( P S(, 0 θ > 0 ( 60 θ

More information

e a b a b b a a a 1 a a 1 = a 1 a = e G G G : x ( x =, 8, 1 ) x 1,, 60 θ, ϕ ψ θ G G H H G x. n n 1 n 1 n σ = (σ 1, σ,..., σ N ) i σ i i n S n n = 1,,

e a b a b b a a a 1 a a 1 = a 1 a = e G G G : x ( x =, 8, 1 ) x 1,, 60 θ, ϕ ψ θ G G H H G x. n n 1 n 1 n σ = (σ 1, σ,..., σ N ) i σ i i n S n n = 1,, 01 10 18 ( ) 1 6 6 1 8 8 1 6 1 0 0 0 0 1 Table 1: 10 0 8 180 1 1 1. ( : 60 60 ) : 1. 1 e a b a b b a a a 1 a a 1 = a 1 a = e G G G : x ( x =, 8, 1 ) x 1,, 60 θ, ϕ ψ θ G G H H G x. n n 1 n 1 n σ = (σ 1,

More information

mugensho.dvi

mugensho.dvi 1 1 f (t) lim t a f (t) = 0 f (t) t a 1.1 (1) lim(t 1) 2 = 0 t 1 (t 1) 2 t 1 (2) lim(t 1) 3 = 0 t 1 (t 1) 3 t 1 2 f (t), g(t) t a lim t a f (t) g(t) g(t) f (t) = o(g(t)) (t a) = 0 f (t) (t 1) 3 1.2 lim

More information

D xy D (x, y) z = f(x, y) f D (2 ) (x, y, z) f R z = 1 x 2 y 2 {(x, y); x 2 +y 2 1} x 2 +y 2 +z 2 = 1 1 z (x, y) R 2 z = x 2 y

D xy D (x, y) z = f(x, y) f D (2 ) (x, y, z) f R z = 1 x 2 y 2 {(x, y); x 2 +y 2 1} x 2 +y 2 +z 2 = 1 1 z (x, y) R 2 z = x 2 y 5 5. 2 D xy D (x, y z = f(x, y f D (2 (x, y, z f R 2 5.. z = x 2 y 2 {(x, y; x 2 +y 2 } x 2 +y 2 +z 2 = z 5.2. (x, y R 2 z = x 2 y + 3 (2,,, (, 3,, 3 (,, 5.3 (. (3 ( (a, b, c A : (x, y, z P : (x, y, x

More information

1W II K =25 A (1) office(a439) (2) A4 etc. 12:00-13:30 Cafe David 1 2 TA appointment Cafe D

1W II K =25 A (1) office(a439) (2) A4 etc. 12:00-13:30 Cafe David 1 2 TA  appointment Cafe D 1W II K200 : October 6, 2004 Version : 1.2, kawahira@math.nagoa-u.ac.jp, http://www.math.nagoa-u.ac.jp/~kawahira/courses.htm TA M1, m0418c@math.nagoa-u.ac.jp TA Talor Jacobian 4 45 25 30 20 K2-1W04-00

More information

1 I

1 I 1 I 3 1 1.1 R x, y R x + y R x y R x, y, z, a, b R (1.1) (x + y) + z = x + (y + z) (1.2) x + y = y + x (1.3) 0 R : 0 + x = x x R (1.4) x R, 1 ( x) R : x + ( x) = 0 (1.5) (x y) z = x (y z) (1.6) x y =

More information

arctan 1 arctan arctan arctan π = = ( ) π = 4 = π = π = π = =

arctan 1 arctan arctan arctan π = = ( ) π = 4 = π = π = π = = arctan arctan arctan arctan 2 2000 π = 3 + 8 = 3.25 ( ) 2 8 650 π = 4 = 3.6049 9 550 π = 3 3 30 π = 3.622 264 π = 3.459 3 + 0 7 = 3.4085 < π < 3 + 7 = 3.4286 380 π = 3 + 77 250 = 3.46 5 3.45926 < π < 3.45927

More information

keisoku01.dvi

keisoku01.dvi 2.,, Mon, 2006, 401, SAGA, JAPAN Dept. of Mechanical Engineering, Saga Univ., JAPAN 4 Mon, 2006, 401, SAGA, JAPAN Dept. of Mechanical Engineering, Saga Univ., JAPAN 5 Mon, 2006, 401, SAGA, JAPAN Dept.

More information

4.6: 3 sin 5 sin θ θ t θ 2t θ 4t : sin ωt ω sin θ θ ωt sin ωt 1 ω ω [rad/sec] 1 [sec] ω[rad] [rad/sec] 5.3 ω [rad/sec] 5.7: 2t 4t sin 2t sin 4t

4.6: 3 sin 5 sin θ θ t θ 2t θ 4t : sin ωt ω sin θ θ ωt sin ωt 1 ω ω [rad/sec] 1 [sec] ω[rad] [rad/sec] 5.3 ω [rad/sec] 5.7: 2t 4t sin 2t sin 4t 1 1.1 sin 2π [rad] 3 ft 3 sin 2t π 4 3.1 2 1.1: sin θ 2.2 sin θ ft t t [sec] t sin 2t π 4 [rad] sin 3.1 3 sin θ θ t θ 2t π 4 3.2 3.1 3.4 3.4: 2.2: sin θ θ θ [rad] 2.3 0 [rad] 4 sin θ sin 2t π 4 sin 1 1

More information