(1) (2) (3) (4) HB B ( ) (5) (6) (7) 40 (8) (9) (10)

2017 12 9 4 1 30 4 10 3 1 30 3 30 2 1 30 2 50 1 1 30 2 10 (1) (2) (3) (4) HB B ( ) (5) (6) (7) 40 (8) (9) (10)

(1) i 23 c 23 0 1 2 3 4 5 6 7 8 9 a b d e f g h i (2) 23 23 (3) 23 ( 23 ) 23 x 1 x 2 23 x 2 ( x 1) (4) R (5) log x x 1

1 3 2 15 3 25 4 35 2

1 1 6 1 17 ( ) sin 1 x, cos 1 x, tan 1 x sin x, cos x, tan x, arcsin x, arccos x, arctan x. π 2 sin 1 x π 2, 0 cos 1 x π, π 2 < tan 1 x < π 2 1 2 lim x 2 x + 7 (x + 1) x 2 1 sin x lim x x = 2 = 1 1 2 0 0 1 1 2 2 3 1 2 4 3 4 5 5 6 6 7 1 8 2 9 1 2 a 3 4 b 5 6 c 3

4

2 I = tan x 1 + cos 2 dx x tan x = t t 3 dx dt = 1 1 cos 2 x = 1 + tan2 x = 1 + t 2 dx dt = 4, 1 + cos2 x = 5 t I = 6 3 0 0 1 1 2 tan x 3 1 tan 2 x 4 cos 1 x 5 cos 2 x 6 1 cos x 7 1 cos 2 x 4 5 0 t 1 1 + t 2 1 1 + t 2 3 1 + t 2 4 2 1 + t 2 5 2 + t2 1 + t 2 6 1 t 7 t2 2 + t 2 6 0 1 2 x 1 1 2 tan 1 x 2 1 2 x + 1 4 log(1 + tan2 x) 3 1 2 tan2 x 4 1 1 + x 2 5 1 2 log(1 + tan2 x) 6 1 2 log(2 + x2 ) 7 1 2 log(2 + tan2 x) 5

6

3 sinh x, cosh x sinh x = ex e x, cosh x = ex + e x 2 2 (1) d sinh x = 7 dx 7 0 cos 1 x 1 sinh x 2 sinh x 3 cosh x + sinh x 4 sin 1 x 5 cosh x 6 cosh x 7 cosh x sinh x (2) sinh x y = sinh 1 x x = sinh y x = ey e y 2 t = e y t 2 t 2 2xt 1 = 0 t > 0 t = 8 y = 9 8 9 0 1 + x 2 + 1 1 log(1 + x 2 + 1) 2 e 1+ x 2 +1 3 x + x 2 + 1 4 log(x + x 2 + 1) 5 e x+ x 2 +1 (3) (2) d dx sinh 1 x = d dx 9 = 10 10 0 x x 2 + 1 + x 2 + 1 1 1 x2 + 1 2 1 + x x2 + 1 3 x + x 2 + 1 e x+ x 2 +1 x2 + 1 4 1 x + x 2 + 1 5 x x2 + 1 7

8

4 1 1 t t = 0 1 1 t = 1 + t + t2 + t 3 + ( 1 < t < 1) 1 1 + x 2 1 = 11 ( 1 < x < 1) 1 + x2 tan 1 x = x 0 1 ds 1 + s2 tan 1 x = 12 ( 1 < x < 1) tan 1 x 11 0 1 + x + x 2 + x 3 + x 4 + 1 1 x 2 x 4 x 6 x 8 2 1 x + x 2 x 3 + x 4 3 1 x 2 + x 4 x 6 + x 8 4 1 + x + 2x 2 + 3x 3 + 4x 4 + 5 1 x 3 + x 5 x 7 + x 9 6 1 + x 2 + x 4 + x 6 + x 8 + 7 1 + x + x2 2 + x3 3 + x4 4 + 12 0 x + 2x 2 + 3x 3 + 4x 4 + 1 x 2x 3 + 4x 5 6x 7 + 2 x x2 2 + x3 3 x4 4 + 3 x x3 3 x5 5 x7 7 4 x x3 3 + x5 5 x7 7 + 5 3x + 6x2 + 12x 3 + 24x 4 + 6 x + 3x 3 + 5x 5 + 7x 7 + 7 x + x2 2 + x3 3 + x4 4 + 9

10

5 1 f(x), g(x) f (x), g (x) 2 h(t, s) h t h (t, s), (t, s) s z = f(h(t, s)) t z t = 13 w = h(f(x), g(x)) dw dx = 14 13 14 0 f (0) h (t, s) t 1 2 f (h(t, s)) h (t, s) t 4 f (h(t, s)) h (t, s) s 6 h t (f(x), g(x))(f (x) + g (x)) f (0) h (t, s) s h 3 t (f(x), g(x))f (x) h 5 s (f(x), g(x))(f (x) + g (x)) 7 h s (f(x), g(x))g (x) 8 h t (f(x), g(x))(f (x) + g (x)) + h s (f(x), g(x))(f (x) + g (x)) 9 h t (f(x), g(x))f (x) + h s (f(x), g(x))g (x) a h t (f(x), g(x))f (0) + h s (f(x), g(x))g (0) 11

12

6 xy D D = { (x, y) 1 } 2 y 1, 0 x y y 3 I = x 2 + y 2 dxdy D (1) I I = 1 1 2 ( 15 0 ) y 3 x 2 + y 2 dx dy 15 0 1 2 4 x y 1 1 2 y x 3 y 5 x 6 x y 7 1 y 15 (2) I 1 (y) = 0 1 x 2 + y 2 dx I 1(y) = 16 y 16 0 1 4 4 log 2 2 1 π 2 π 4 5 2π 6 π 2 3 1 π 4 7 π 2 1 13

(3) (1), (2) I = 1 1 2 y 3 I 1 (y) dy = 17 17 0 1 1 7 24 4 π ( π ) 7 5 12 2 1 24 8 7π ( 9 1 π ) 7 48 4 24 2 7π 12 6 7π 96 a 7 96 3 7 log 2 48 7 log 2 3 14

2 1 5 18 34 1 A (1, 1, 1), B (5, 2, 1), C (1, 2, 1), D (5, 5, 2) (1) 2 18 4 AB AC (2) AB, AC, AD 3 19 18 19 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 a 10 b 11 c 12 d 13 e 14 f 15 15

16

2 1 x y 3z = 2 ( ) x + 2y + 4z = 4 3x 2y 8z = a a (1) a = 20 ( ) (2) a = 20 x = 2 y = 21, z = 22 20 22 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 1 a 2 b 3 c 4 d 5 e 6 f 7 17

18

3 a, b, c a + b + 2c a b c b + c + 2a b c a c + a + 2b 23 (a + b + c) 24 23 24 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 1 a 2 b 3 c 4 d 5 e 6 f 7 19

20

1 2 2 4 A = 1 1 1 1 2 4 A 1 x 2 2 1 1 x 1 1 2 4 x = (1 x)(2 x)( 25 x) = 0 A 1 2 25 1 1 c 1 1 c 1 0 26 2, 25 c 2 0 1 27, c 3 1 28 1 c 2, c 3 0 A 1 0 1 P = 1 1 28 26 27 1 P 1 AP = 1 0 0 0 2 0 0 0 25 25 28 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 1 a 2 b 3 c 4 d 5 e 6 f 7 21

22

a 1 0 5 a 0 A = 0 a 1 3 n 0 0 a A n 1 0 0 0 1 0 S = a 0 1 0, T = 0 0 1 0 0 1 0 0 0 A = S + T ST = T S A n = (S + T ) n = S n + 29 S n 1 T + 30 S n 2 T 2 + + T n T 2 = 0 0 31 0 0 0 0 0 0, T 3 = 0 0 0 0 0 0 0 0 0 A n = 32 33 34 0 32 33 0 0 32 29 31 0 0 1 1 2 2 3 n 4 n + 1 5 n 1 2 6 n(n 1) 2 7 n(n + 1) 2 32 34 0 a n 1 1 a n 2 a n+1 3 (n 1)a n 4 na n 1 5 (n + 1)a n 1 6 n(n 1)an+1 2 7 n(n 1)an 2 2 8 n(n + 1)an 1 2 23

24

3 1 5 35 51 ( ) y x y, y y dy dx, d 2 y dx 2. 1 y = 2 y y 2 = 35 + C (C ) y(1) = 0 C = 36 35 0 x 1 x 2 2x 3 2x 4 4x 5 4x 6 x 2 7 x 2 8 2x 2 9 2x 2 a 4x 2 b 4x 2 c 1 x d 1 x e 1 x 2 f 1 x 2 36 0 0 1 1 2 2 3 3 4 4 5 5 6 1 7 2 8 3 9 4 a 5 25

26

2 (a) x 2 + y 2 = 2xyy (1) y = x u(x) (a) u(x) (b) u = 37 37 0 1 + u2 2u 1 1 + u2 2xu 2 1 + 3u2 2u 3 1 + 3u2 2xu 4 1 u2 2u 5 1 u2 2xu 6 u 3 2(1 + u 2 ) 7 u 3 2(1 u 2 ) (2) (b) (c) 38 = C (C ) 38 0 x(1 + u 2 ) 4 x(1 u 2 ) 1 1 + u2 x 5 1 u2 x 2 1 + u2 e x 3 x 3 (1 + 3u 2 ) 6 1 u2 e x 7 x 3 (1 3u 2 ) (3) (c) u = y x (a) C 0 xy 39 39 0 1 2 3 4 5 27

28

3 xy K P ( t, f(t) ) A ( 0, 2t{f(t)} 2) f(t) t > 0 f(t) 0 (1) P y = f (t) x + 40 f (t) df dt 40 0 0 1 t 2 f(t) 3 tf(t) 6 tf (t) 9 f (t) tf(t) 4 f(t) t 7 f (t) t a f(t) f (t) t 5 f (t) 8 f(t) tf (t) b f (t) f(t) t (2) (1) A 40 = 2t{f(t)} 2 z = 1 f(t) dz dt + 41 z = 42 41 42 0 0 1 1 2 2 3 1 4 2 5 t 6 t 7 t 2 8 t 2 9 1 t a 1 t b 1 t 2 c 1 t 2 d t e t 29

(3) (2) z f(t) = 1 z = 43 (C ) 43 0 1 t + C 3 1 t 2 + C 6 t t 2 + C 9 1 t + t C 1 t C 4 1 t 2 + Ct 7 1 t 3 + Ct a 1 2t log t + Ct 2 C t 5 8 b 1 t 2 + t + C 1 t 3 + Ct 2 1 2t log t + Ct 30

4 a y + ay = 0 y (0) = y (1) = 0 y(0) > 0 y (1) a = 44 y (2) a 44 < a < 45 y a = 45 y A y = A cos 46 x 44 46 0 0 1 1 2 2 3 π 4 π 2 5 π 6 π 2 7 2π 2 8 π2 4 2 a π b 2 π π c d 2π 2 9 π2 4 π e 2 31

32

5 k > 2 k y + 2ky + 4y = e 4x C 1, C 2 (1) k 47 y = e kx ( C 1 e 48 x + C 2 e 48 x ) + 1 49 e 4x 47 0 3 1 4 2 5 3 6 4 8 5 10 6 5 2 7 7 2 8 9 2 9 9 4 a 11 4 b 13 4 48 49 0 k 1 k 2 2 k 2 3 k 4 k 2 4 5 4 k 2 6 k 2 16 7 16 k 2 8 2k + 20 9 2k + 20 a 2k 12 b 2k 12 c 8k + 20 d 8k + 20 e 8k 12 f 8k 12 (2) k = 47 y = C 1 e 4x + C 2 e 50 x + 51 xe 4x 50 51 0 0 1 1 2 2 3 3 4 1 2 5 1 3 6 2 3 7 1 4 8 3 4 9 1 a 2 b 3 c 1 2 d 1 3 e 2 3 f 1 4 g 3 4 33

34

4 1 5 52 70 ( ) A P (A) A A B P (B A) X E(X), V (X) ( ) 1 X, Y X 1 0 1 1 6 a b Y 1 0 1 1 2 c d a, b, c, d E(X + Y ) = 0 E(X Y ) = 2 3 E(X) = 52, V (X) = 53, c = 54 52 54 0 0 1 1 2 1 2 3 1 3 4 2 3 5 1 4 6 3 4 7 1 6 8 5 6 9 1 9 a 2 9 b 4 9 c 5 9 d 7 9 e 8 9 35

36

2 96% 6% 1 A, B P (A) = P (B) = 1 2 C (1) P (A C) = 55, P (B C) = 56 P (C) = 55 + 56 (2) P (A C) 57 55 57 0 1 2 1 2 3 2 3 5 3 4 5 4 15 17 5 16 17 6 6 25 7 12 25 8 24 25 9 3 50 a 47 50 b 2 100 c 3 100 d 47 100 e 49 100 37

38

3 1 : 1 1 5 5 X Y Y = 2X 5 ( (1) X B 5, 1 ) V (X) = 58 V ( Y ) V (X) 2 59 58 59 0 0 1 1 2 2 3 3 4 4 5 5 6 10 7 20 8 1 2 9 5 2 a 1 4 b 5 4 (2) X Y 60 P ( Y = 1) = 61, P ( Y = 1 3 X 5) = 62 (3) 3 X 5 Y = 1 63 P ( Y = 1) = 61 2, P ( Y = 1 3 X 5) = 62 60 63 0 1 2 39

61 62 0 0 1 1 2 1 2 3 1 4 4 3 4 5 1 8 6 3 8 7 5 8 8 7 8 9 1 16 a 3 16 b 5 16 c 7 16 d 9 16 e 15 16 f 1 32 g 5 32 h 15 32 40

4 a, b X f(x) ( b a x x b ) a f(x) = ( 0 x > b ) a b a b = 64 E(X) = 65 V (X) = 1 a = 66 64 65 0 0 1 1 2 2 3 a 4 2a 5 a 2 6 2a 2 7 a 8 2a 9 1 a a 1 2a b 1 a 2 c 1 2a 2 d 1 a e 1 2a 66 0 0 1 1 2 2 3 3 4 4 5 1 2 6 1 3 7 2 3 8 1 4 9 3 4 a 1 6 b 5 6 41

42

5 U 100 1 1151 g µ g µ 95% 75 g n (n = 1, 2,..., 100) X n 100 N(µ, 75 2 ) X = X 1 + X 2 + + X 100 100 ( ) N µ, 67 Z = X µ 68 Z N(0, 1) P ( 1.96 Z 1.96) 0.95 P ( X 69 µ X + 69 ) 0.95 [ x 69, x + 69 ] x = 1151 100 70 67 68 0 0.75 1 7.5 2 75 3 750 4 7500 5 0.75 2 6 7.5 2 7 75 2 8 750 2 9 7500 2 a 0.75 b 7.5 c 75 d 750 e 7500 43

69 0 0.0110 1 0.110 2 1.10 3 11.0 4 110 5 0.0147 6 0.147 7 1.47 8 14.7 9 147 a 0.0196 b 0.196 c 1.96 d 19.6 e 196 70 0 25 1 20 2 10 3 5 4 4 5 2 6 1 25 7 1 20 8 1 10 9 1 5 a 1 4 b 1 2 44