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そよなかきむら
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3 . z =. z = z x x O C z w = z z +. w z x. z C w w = w w = (z z + )( z z + ) = z 4 z z + z z z + 4 z z z + z z + z = = 6 8z + z 8 z + 6 (z + z) + + z = 6 8(z + z) + z + z (z + z) = 0(z + z) + (z + z) z z z + z = x + yi + x yi = x, z z = z = 4 = 4x 0 x ( x )

4 . w = 4x 0 x = 4 ( x 5 ) ( 5, ) ⅱ w 0 z = x ⅰ 5 < < w -4-4 x x = 5 w = w -4-4 x -5 x = w 4( ) 0( ) = = (4 + 5) w 4 + 5

5 (ⅲ < <. w x x = w = = (4 5) -5 w 4 5 4

6 . f (x) = e x ( + f (t)sint e x dt = ) = e x [ t] + f (t)sintdt = e [] + dt + f (t)sintdt f (t)sintdt f (x) = e x + f (t)sintdt C = f (t)sintdt (C > 0 f (x) f (x) = ( e x + f (t)sint ) dt. f (x). 0 < π g() = f (t)sintdt f (x) = e x + C C = (e t + C)sintdt C = e t sintdt + C sintdt e t sintdt (e t sint) = e t sint + e t cost (e t cost) = e t cost e t sint 5

7 (e t sint) + (e t cost) = e t sint t e t sint + e t cost = e t sintdt f (x) = e x + C g () = (e e )sin e x > e x e t sintdt = [ e t ( sint + cost )] = sin ( e + e ) cos ( e e ) sintdt sintdt = [ cost ] = cos + cos = 0 C = sin ( e + e ) + cos ( e e ) f (x) = e x + sin ( e + e ) + cos ( e e ) < π (0) π π (0) 極小. g() = f (t)sintdt g () = f ()sin f ( )sin( )( ) = π g(π) = sinπ (eπ + e π ) + cosπ (eπ e π ) = (eπ e π ) 6

8 . A A A TypeⅠ A C = A={ } ( ) A B A B. A. A,B. A,B A Type Ⅰ = ( ) TypeⅡ A A={ } C ( ) C ( ( ) = C = ) = A A A ( ) = 8 = 7

9 . TypeⅢ B B={ }or{ } B={ } ( ) Cse : A A={ },{ } = ( ) 7 Cse : A A={ *}, * A={ * *}, * C ( ) ( ) + C ( )( ) B={ } ( ) ( = + = ) = 7 7 TypeⅢ 7 7 TypeⅣ B B={ } C ( )( Cse : A ) = A={ } { } = ( ) 7 Cse A A={ } { } C ( ) ( ) = TypeⅣ ( ) = = 7. X: A B={ } 7 7 Y; A,B P Y (X ) 8

10 Y P(Y ) TypeⅤ B={ } = ( ) 4 A A={ } { } { } Cse A={ } C ( )( ) = 9 Cse A= { } C ( ) C ( ) ( ) = 9 Cse A= { } C ( )( ) = 9 TypeⅤ 4 ( ) = 9 TypeⅤ TypeⅥ B={ } C ( )( A ) = Cse A={ } C ( ) ( ) = 9 Cse A={ } C ( ) ( ) = 9 TypeⅥ ( ) = 9 TypeⅥ TypeⅦ B={ } = 4 A A={ } ( ) TypeⅦ 4 TypeⅦ P(Y ) = = P(X Y ) Cse Cse Cse Cse P(X Y ) = 4 ( ) + ( ) = 8 5 A={ } { } 9

11 4. x + x x = 0 f (x) f (0) = x = 0 f (x) = x + x f (x) = 0 x + x = 0. x + x x = 0. s, t, u { n } n = s n (s t)(s u) + t n (t u)(t s) + u n n+ + n+ n+ n = 0 (n =,,,...) (u s)(u t) x = ± 7 α = 7 f (x) f (x), β = 極大極小 f (x) = f (x) ( x + 9 ) 4 9 x 7 9. n f (α) = f (α) ( α + 9 ) 4 9 α 7 9 = 4 9 α 7 9 = > 0 f (β) = f (β) ( β + 9 ) 4 9 β 7 9 = 4 9 β 7 9 = < 0 0

12 .5. n. α β n = s n (s t)(s u) + t n (t u)(t s) + u n n+ = n + n+ n+ (ⅰ) n = = (u s)(u t) (s t)(s u) + (t u)(t s) + (u s)(u t) -.5 = (t u) (s u) + (s t) (s t)(s u)(t u) = 0 f (α) > 0, f (β) < 0 x + x x = 0 0. s, t, u s + s s = 0 t + t t = 0 u + u u = 0 n+ + n+ n+ n = sn+ + s n+ s n s n (s t)(s u) = sn+ (s + s s ) (s t)(s u) = 0 + tn+ + t n+ t n s n (t u)(t s) + tn (t + t t ) (t u)(t s) + un+ + u n+ u n u n (u s)(u t) + un (u + u u ) (u s)(u t) (ⅱ) n = = = = s (s t)(s u) + t (t u)(t s) + u (u s)(u t) s(t u) t(s u) + u(s t) (s t)(s u)(t u) st su ts + tu + us ut (s t)(s u)(t u) (ⅲ) n = = = 0 s (s t)(s u) + t (t u)(t s) + u = s (t u) t (s u) + u (s t) (s t)(s u)(t u) (u s)(u t)

13 = (t u)s (t u )s + tu(t u) (s t)(s u)(t u) = (t u){s (t u)s + tu} (s t)(s u)(t u) = (t u)(s t)(s u) (s t)(s u)(t u) = = 0, = 0, = n = 4 = + = n = k k, k+, k+ k+ = k + k+ k+ k+ n n

14 5. A B Q l A(0,0,), B(0,,) l C(,0,0), D(,0,) m l m P(s, t, ). P l l Q P m C D P R m m R Q, R s, t, l AB. P l m s, t,. s, t (s, t) AB = OB l Q l OA = (0,,) (0,0,) = (0,0,) AQ AB AQ = q AB q OQ OA = q OQ = AB OA + q AB OQ = (0,0,) + q(0,,) = (0,q, + q)

15 m CD = OD m R m CD OC = (,0,) (,0,0) = (0,0,) CR CD CR = r CD r 0( s) + 0( t) + (r ) = 0 r = 0 r = OQ = 0, t + (, t + + ), OR = (,0,) OR OC = r CD OR = OC + r CD OP = (s, t, ) PQ = PR = OQ OR OR = (,0,0) + r(0,0,) = (,0,r) OP = ( s, q t, + q ) OP = ( s, t, r ) AB = (0,0,) CD = (0,0,) PQ AB PQ AB = 0 0( s) + (q t) + ( + q ) = 0 q = t + q = t + PR CD PR CD = 0 4

16 .. 4s = t + ( )t + 4 t (t + ) = 4 ( s + + ) (s, t) t = 4s (,0) s = s + t + ( +, + s = ) + 5. PQ = PQ = s, t ( PR = OR PR = ( s, t,0) OQ OP = ( s, q t, + q ), t + ) OP = ( s, t, r ) PQ, PR PQ = PR s + ( t) 4 + (t + ) 4 = ( s) + t t + ( )t + 4 4s = 0 5

17 6

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