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
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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 te t = n t (3) () f(x) f(x) dx (4) () f(x) log 3 log xf(x) dx 4 6
4 4. y = x P(p, p ), Q(q, q ) < p < q, POQ = π 4 O P Q R () p () q p (3) R x y (4) R (5) R 4 6
4 5. a, b, c, d (a d) + 4bc = ( ) ( ) a b E =, A =, B = A a + d c d A \= a + d () B () n A n = pa + qe p, q n a, b, c, d (3) A A ( ) A 5 = 5 9 E E 4 3 6
4 6. K O 3 A(a, b, ), B(r, s, t), C(3,, ) OA, AB, BC K OC K b >, t > () K l () A (3) B (4) AB P x H(x,, ) PH x (5) K x V 4 4 63
5 7. () x ŕ x > + log x () y = x log x (x > ) C a C (a, ) (3) () C x = e 5 64
5 8. f(x) =, g(x) = x + x + 3 x + () g(x) g (x) () g (f(g(x))) (3) c f(c) (4) {a n } a = g( ), a n+ = f(a n ) (n =,, 3, ) (5) (4) {a n } lim n a n 5 65
5 9. () 5 (()(iv) ) I = I 4 = x sin x dx, I = x cos x sin x dx, I 5 = x cos x dx, I 3 = sin x cos x dx sin x dx () y = sin x C C O(, ) l A(π, ) l l l B C P(t, sin t) ( ő t ő π) l PQ t = Q = P OQ = s OBA s t PQ t C l, l l V 5 3 66
5. ABCD ( ) BA = 66, BC = 7, BD = 65 ( ) BA #» BC #» #» = 8, BC BD #» #» = 35, BD BA #» = 4 A BCD AH () AC () BH #» BC, #» #» BD (3) CH (4) ABC AH ABC V 5 4 67
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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 te t = n t (3) () f(x) f(x) dx (4) () f(x) log 3 log xf(x) dx 4 () S n = r + r + 3r 3 + + nr n r rs n = r + r 3 + + (n )r n + nr n+ ( r)s n = r + r + r 3 + + r n nr n+ r \= ( r)s n = r( rn ) r nr n+ = r rn+ n( r)r n+ r S n = nrn+ (n + )r n+ + r ( r) () () r = e x f n (x) = n(e x ) n+ (n + )(e x ) n+ + e x ( e x ) = ne (n+)x (n + )e (n+)x + e x ( e x ) (n + )x = t n = t x ( ) ne (n+)x t = x e t = x t e t e t 88
4 (n + )x = t (n + )e (n+)x = x t e t n t, t ( lim n ne (n+)x = lim t x t e t e ) t = lim (n + n )e (n+)x = lim t x t e t = (3) lim f n(x) = n f(x) = ( e x ) = f(x) dx = e x ( e x ) e x ( e x ) e x ( e x ) e x + C (C ) (4) F (x) = e x = ex e x F (log ) =, F (log 3) = 3 log 3 log xf(x) dx = [ ] log 3 xf (x) log log 3 log F (x) dx = 3 log 3 + log [ log(e x ) = 3 log 3 + log + log = 3 log 3 log 3 ] log 3 log 89
4 4. y = x P(p, p ), Q(q, q ) < p < q, POQ = π 4 O P Q R () p () q p (3) R x y (4) R (5) R 4 () OP, OQ x α, β ( tan α = p, tan β = q < α < β < π ) POQ = π 4 y y = x Q β α = π 4 β = α + π 4 α >, < β < π < α < π 4 < tan α < tan π 4 < p < () tan(β α) = tan π 4 tan β tan α + tan α tan β = q p + pq = q p = + pq ( p)q = + q < p < p \= q = + p p O P p q x (3) y = x y = x P, Q y = px p, y = qx q, y (p q)x (p q ) = ( (p q) x p + q ) = 9
4 p q \= x = p + q y = p p + q p = pq ( ) p + q R, pq q = + p p = < p < q > p (4) x = p + q x >, y > (q p) = ( + pq) >, y = pq > (p + q) 4pq = + pq + (pq) (p + q) (pq) 6pq = p + q = x, pq = y (x) y 6y = 4x (y + 3) = 8 x (y + 3) 8 = (x >, y > ) (5) y x y 8 = y 3 x ± y = y = ±x x >, y > y = x 3 O x 9
4 6. K O 3 A(a, b, ), B(r, s, t), C(3,, ) OA, AB, BC K OC K b >, t > () K l () A (3) B (4) AB P x H(x,, ) PH x (5) K x V 4 4 () (i), (ii), (iii) K OC 3l = 3 C l = 3 H B () OA = 3 a + b = 3 AC = l = 6 (a 3) + b = 6 3(a 3) = 3 a 3 = a = + b = 3 b = b = ( b > ) A(,, ) O A P 9
4 (3) OB = l = 6 r + s + t = 6 AB = l = 3 (r ) + (s ) + t = 3 BC = l = 3 (r 3) + s + t = 3 3(r 3) = 3 r =, { s + t = (s ) + t = (s ) = s = + t = t = 3 6 t = ( t > ) ( ) 6 B,, (4) AB u : ( u) P(X, Y, Z) X = ( u) + u = u + Y = ( u) + u = ( u) Z = 6 u P x H(x,, ) X = x u + = x u = x, 8 y = { (x )} = (3 x) 6 z = (x ) 93
4 ( P x, ) 6 (3 x), (x ) PH = (3 x) + 3 (x ) PH = x 6x + 6 (5) A, B x H A, H B H A (,, ), H B (,, ) OC OC = 3 (i) (iii) 3 ő x ő OAH A OH AH A = OH A = V V = 3 π ( ) = 3 π ő x ő AB OC x x OC S(x) S(x) = π PH = π(x 6x + 6) V V = S(x) dx = π(x 6x + 6) dx ] [ = π 3 x3 3x + 6x { } (8 ) = π 3(4 ) + 6( ) 3 ( ) = π 4 3 9 + 6 = 5 3 π ő x ő 3 V 3 V (i) (iii) V 3 = V = 3 π V = V + V + V 3 = 3 π + 5 3 π + 3 π V = 3π 94
5 7. () x ŕ x > + log x () y = x log x (x > ) C a C (a, ) (3) () C x = e 5 () f(x) x ŕ a f(x) m f(x) ŕ m (x ŕ a) m > f(x) > (x ŕ a) f(x) > g(x) (x ŕ a) F (x) = f(x) g(x) x ŕ a F (x) f(x) = x ( + log x) x ŕ f(x) () x t y = mx x x t 95
5 C : y = f(x) (p, q) C (t, f(t)) y = f (t)(x t) + f(t) y = f (t)x tf (t) + f(t) (i) (p, q) q = f (t)p tf (t) + f(t) ( ) ( ) t ( ) 4 f(t) = a f(t) = a y = f(t) y = a a n ő b n lim n a n = lim n b n = () f(x) = x ( + log x) f (x) = x x = x x x ŕ f (x) ŕ x ŕ f(x) x ŕ f(x) ŕ f() f() = ( + ) = > x ŕ f(x) > x > + log x () g(x) = x log x g (x) = log x + C (t, t log t) y = (log t + )(x t) + t log t y = (log t + )x t (a, ) = (log t + )a t 96
5 log t + = t = e log t + \= t \= e a = t log t + ( ) g (x) = x x > g (x) > y = g(x) ( ) t (a, ) h(t) = t log t + h (t) = h (t) = log t = t = (log t + ) t t (log t + ) = log t (log t + ) t > h(t) t e h (t) + h(t) lim h(t) =, lim t + h(t) =, t e () t ŕ t > + log t h(t) = t log t + > t t = t t lim = lim t h(t) = t y = h(t) lim h(t) = t e + 97
5 y y = h(t) y = a O e t a < ő a < a = a > (3) g (x) = log x + = x = e C y = (x e) + e y = x e y O e e t 98
5 y = g(x) S e S = {x log x (x e)} dx e [ ] e e = x log x e e x = e [ ] e e 4 ( ) 4 x = e + e 4 4 e 4 + e 4 e = 4 e e + 9 4 e 4 [ x dx ] e x ex e e + (e 4 e ) 99
5 8. f(x) =, g(x) = x + x + 3 x + () g(x) g (x) () g (f(g(x))) (3) c f(c) (4) {a n } a = g( ), a n+ = f(a n ) (n =,, 3, ) (5) (4) {a n } lim n a n 5 () y = x + x + x x y () (3) (4) () g (f(g(x))) = 4 x a = g( ) x = g (f(g( ))) = 4 g (f(a )) = 4 g (a ) g (a ) = 4 g (a ) ( f(a ) = a ) g (a ) = b b = 4 b {b n } b n = g (a n )
5 p q (p, q ) p q (p, q ) #» a, #» b #» a \= #», #» b \=, #» a \// #» b independent depend in depend independent #» a // #» b () y = x + x + ( x + )y = x + (y + )x = y y = y \= () f(g(x)) = x = y y + g (x) = x x + g(x) + 3 = x + x + + 3 ( x + ) = (x + ) + 3( x + ) = x + 4 x + 8
5 g (f(g(x))) = f(g(x)) f(g(x)) + x + 4 = x + 8 x + 4 + x + 8 ( x + 4) (x + 8) = ( x + 4) + (x + 8) = 5x = 4 x (3) c f(c) c + 3 = n m (c + 3)n = m c = m 3n n (n, m ) f(c) (4) b n = g (a n ) a n = g(b n ) () b n+ = g (a n+ ) = g (f(a n )) = g (f(g(b n ))) = 4 b n a = g( ) b = g (a ) = {b n } 4 b n = ( ) n 4 a n = g(b n ) a n = b n + b n + ( ) n + a n = 4 ( ) n + 4 ( (5) lim ) n = (4) n 4 lim n a n =
5 9. () 5 (()(iv) ) I = I 4 = x sin x dx, I = x cos x sin x dx, I 5 = x cos x dx, I 3 = sin x cos x dx sin x dx () y = sin x C C O(, ) l A(π, ) l l l B C P(t, sin t) ( ő t ő π) l PQ t = Q = P OQ = s OBA s t PQ t C l, l l V 5 3 () I = I = = π + [ x sin x dx = [ ] π sin x = π x cos x dx = = I = π ] π x cos x [ ] π x sin x ( cos x) dx x sin x dx I 3 = = sin x dx = [ x 4 sin x ] π cos x = π dx 3
5 I 4 = x cos x sin x dx = [ = ] π 4 x cos x x sin x dx ( ) 4 cos x dx = π 4 + [ 8 sin x ] π = π 4 I 5 = sin x cos x dx = [ ] π 3 sin3 x = () y = sin x y = cos x (l ) = cos = (l ) = cos π = (l ) (l ) = l l OBA = π y l l Q B C s P O ( Q s, s t A π ) s OP #» OQ #» = OQ #» ( (t, sin t) s, s (t + sin t) = s s ) = s s \= s = t + sin t t = s = s = s = t + sin t PQ P(t, sin t) l : x y = PQ = t sin t P l sin t ő t t sin t ŕ PQ = t sin t x 4
5 Q = B s = π V = s = t + sin t π πpq ds = π π (t sin t) ds ds = + cos t dt, s π t π V = π = π π = 4 π = 4 π = 4 π = 4 = (t sin t) + cos t dt (t t sin t + sin t)( + cos t) dt (t + t cos t t sin t t sin t cos t + sin t + sin t cos t) dt ([ ] π ) 3 t3 + I I I 4 + I 3 + I 5 ( π 3 3 π π + π + π ) + ( ) π 3 3 3π π (π 9) V V = π (π 9) 5
5. ABCD ( ) BA = 66, BC = 7, BD = 65 ( ) BA #» BC #» #» = 8, BC BD #» #» = 35, BD BA #» = 4 A BCD AH () AC () BH #» BC, #» #» BD (3) CH (4) ABC AH ABC V 5 4 () BA #» = #» #» a, BC = #» c, #» AC = #» c #» a #» BD = #» d = #» c #» c #» a + #» a = 49 8 + 66 = 59 #» AC ŕ AC #» = 59 () #» BH = s #» c + t #» d #» AH = BH #» BA #» AH #» = s #» c + t #» d #» a #» AH BC #» #» AH BC #» = B 66 65 7 A H C D (s #» c + t #» d #» a ) #» c = s #» c + t #» c #» d #» a #» c = 49s + 35t 8 = 7s + 5t 4 = AH #» BD #» #» AH BD #» = (s #» c + t #» d #» a ) #» d = s #» c #» d + t #» d #» a #» d = 35s + 65t 4 = 7s + 3t 8 = 6
5, s = 3 4, t = #» BH = 3 #» BC + #» BD 4 (3) CH #» = BH #» BC #» = 3 4 #» c + #» #» d c = 4 ( #» c + 7 #» d ) #» CH = 4 #» c + 7 #» d = 4 ( #» c 4 #» c #» d + 49 #» d ) = ( 49 4 35 + 49 65) 4 = 49 ( + 65) 4 = 4 76 = 9 #» CH ŕ #» CH = 9 (4) BC H () #» BH = 4 3 #» c + 7 #» d = 4 (9 #» c + 4 #» c #» d + 49 #» d ) = (9 49 + 4 35 + 49 65) 4 = 49 (9 + 3 + 65) 4 = 4 4 = 6 #» BH ŕ BH #» = 6 BCH CH < BH < BC BC H BC K K BC 7
5 A H B K H C B K C BC H B K V = 3 πbh AH 3 πkh AH = πah 3 = πah 3 (BH KH ) BK BK = x BKH CKH 6 x = 9 (7 x) x (7 x) = 7 7(x 7) = 7 x 7 = x = 4 B H 6 9 x K 7 x C BK = 6 ABH AH = AB BH = 66 6 = 4 AH = V = π 3 6 V = 3 3 π 8
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