1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0

1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0 0 < t < τ I II 0 No.2 2 C x y x y > 0 x 0 x > b a dx = axy < 0 dy = ay(x b a ) 1 = 100 dy > 0 y x < b a = 100

dy < 0 y x x = 50 y 0 x = 1000 x = 100 y 0 dy = 0 x = y = 0 x = 0 dx = 0 9 10 No.3 4 B 1 x (1 + x) n. = 1 + nx T dt m dm k = 4π 2 m T 2 k + dk = 4π 2 m + dm (T + dt ) 2 = 4π 2 m T 2 1 + dm m (1 + dt T )2. = 4π 2 m T 2 (1 + dm m )(1 2dT T ). = 4π 2 m T 2 (1 + dm m 2dT T ) dm m = ±0.1% dt T = ±0.2% k 0.1 + 2 0.2 = 0.5% 2 k = 4π 2 m T 2 1 log k = log(4π 2 ) + log m 2 log T dk k = dm m 2 log dt T 3 0.1 + 2 0.2 = 0.5% 2

3 1 1 + x 1 2 3 No.4 5 B 1 K dk = 0 W Q(K K 0 ) = 0 K = K 0 + W Q = 0.04 + 4 0.02 60 100 = 0.173% 2 dk = W V Q(K K 0) = Q V V ( K K 0 W ) Q K K 0 W Q = F K 0 W Q dk = df df = Q V F F (t) = A exp ( QV ) t K(t) = A exp ( QV ) t + K 0 + W Q 0 0 No.5 2 C t = 0 T dx dx x = ax bxy = (a by) T 0 dx x = log x(t ) log x(0) = T 0 T (a by) = at b y(t) 0 3

x(t ) = x(0) T 0 y(t) = T Y at bt Y = 0 Y = a b 2 dy y = ( c + dx) log y(t ) log y(0) = 0 = ct + dxt = 0 X = c d x y dx = ax bxy rx = (a r)x bxy dy = cy + dxy ry = (c + r)y + dxy X = c + r d > X Y = a r b No.1 x y < Y No.6 3 C y T dy = ky y(0) = y y(t) = A exp( kt) y(t) = y exp( kt) y(t ) = y exp( kt ) y y exp( kt ) + y 0 = y y = y 0 1 exp( kt ) 4

No.7 5 A 1 2 ẍ = rω 2 sin(ωt) ÿ = rω 2 cos(ωt) x sin ωt = ±1, cos ωt = 0 y = r B y cos ωt = ±1, sin ωt = 0 y = 0, 2r A C ẍ 2 + ÿ 2 = r 2 ω 4 2 0 rω 2 x x B y y A C 2 2 No.8 1 B Taylor y(x + h) = y(x) + hy (x) + h2 2 y (ξ) h 2 Y i+1 = Y i + h(x i + Y i ) = 1.1Y i + 0.1 0.1i = 1.1Y i + 0.01i 5

x i = 0.1iY 0 = 1 Y 1 = 1.1 + 0 = 1.1 Y 2 = 1.21 + 0.01 = 1.22 Y 3 = 1.1 1.22 + 0.02 = 1.362 Y 4 = 1.1 1.362 + 0.03 = 1.5282 y = 2e 0.4 1.4 2e 0.4 1.4 1.5282 = 2e 0.4 2.9282 Taylor No.9 2 B dx = f(x) > 0 x f(x) f(x) = 0 0 < x < x 0 x = x 0 f(x) No.10 1 B 1 dp 0 (t) = ap 0 (t) < 0 (P 0 (t) P 0 (t) dp 1 (t) = a(p 0 (t) P 1 (t)) P 0 > P 1 P 0 P 1 dp 1(t) = 0 2 dp 0 (t) = ap 0 P 0 (0) = 1 P 0 (t) = e at 6

dp 1 (t) = ap 1 (t) + ae at dp 1 (t) = ap 1 (t) P 1 (t) = Ce at C C(t) P 1 (t) = C(t)e at dp 1 (t) dp 1 (t) = ap 1 (t) + ae at = C (t)e at ac(t)e at = ac(t)e at + ae at C (t) = a C(t) = at + C 1 P 1 (0) = 0 C 1 = 0 P 1 (t) = ate at No.11 2 A x x = 0 d 2 y dx 2 = e ax dy dx = 1 a e ax + C 1 2 = 1 a + C 1 C 1 = 2 + 1 a x dy dx = 1 a e ax 2 + 1 a y = 1 ( a 2 e ax + 2 + 1 ) x + C 2 a 7

x 2 + 1 a = 0 a = 1 2 y(x) 2 e ax x dy = 0 a = 2 dx No.12 4 B 1 F df dx df = f = f 0 + av 3 df dx = df d dv 1 dx = f 0 + av 3 V = f 0 V + av 2 ( ) df = f 0 dx V 2 + 2aV = 0 V = 3 f0 2a 2 t F F = (f 0 + av 3 )t L L = V t Lagrange g g(t, V ) = V t + k((f 0 + av 3 )t F ) g V = t + 3kaV 2 t = 0 g t = V + k(f 0 + av 3 ) = 0 2 t V = 3kaV 2 t k(f 0 + av 3 ) f 0 + av 3 = 3aV 3 8

V = 3 f0 2a Lagrange H.7 No.13 3 A x y Lagrange f(x, y) = x 3 + y 3 + k(x 2 + y 2 3) f x = 3x2 + 2kx = 0 f y = 3y2 + 2ky = 0 x = 0 y = 0 x = y 2 x = 0 x = y x = 0 y = 3 x = y = x 3 + y 3 = 3 3 3 2 x 3 + y 3 = 3 6 2 No.14 1 A 1 xy P (r cos θ, r sin θ) = (aωt cos ωt, aωt sin ωt) v x = dx = aω cos ωt aω2 t sin ωt v y = dy = aω sin ωt + aω2 t cos ωt 9

v = v 2 x + v 2 y = a 2 ω 2 + a 2 ω 4 t 2 = aω 1 + ω 2 t 2 2 v 2 = v r = dr = aω v θ = r dθ = aω2 t v 2 r + v 2 θ = aω 1 + ω 2 t 2 IV No.15 2 C g(x) = sin ωx d 2 g(x) dx 2 + g(x) = 0 ω 2 + 1 = 0 ω = 1 g(x) = A sin x + B cos x f(x) = A sin x + B cos x 1 sin 2x 3 f(0) = 0 B = 0 f (x) = A cos x 2 cos 2x 3 f (0) = A 2 3 = 1 3 A = 1 f(x) = sin x 1 3 sin 2x = sin x 2 3 sin x cos x = sin x ( 1 2 3 cos x ) f(x) = 0 x x = 0, ±π, ±2π 2 10

2 2 1 f(x) = 1 sin 2x 3 f(x) = C sin 2x g(x) g(x) = e kx k g(x) = sin ωx 2 No.16 5 B 1 2 x 3 (4m + 1)x 2 + 2(m + 3)x + 2(m 3) = x 3 x 2 + 6x 6 2m(2x 2 x 1) = (x 1)(x 2 + 6 2m(2x + 1)) x = 1 2 x 2 4mx + 6 2m = 0 x = 1 2 f(x) = x 2 4mx + 6 2m 2 D/4 = (2m) 2 (6 2m) = 4m 2 + 2m 6 = (2m + 3)(m 1) > 0 m < 3 2, m > 1 4m > 0, 6 2m > 0 0 < m < 3 x = 1 f(1) = 7 6m 0 1 < m < 7 6, 7 6 < m < 3 2 11

x = 1 m = x3 x 2 + 6x 6 4x 2 2x + 6 = (x2 + 6)(x 1) 2(2x + 1)(x 1) = x2 + 6 2(2x + 1) y = g(x) = x2 + 6 2(2x + 1) y = m x = 1 x > 0 2 y = g(x) x = 1 2 x = 1 0 1 No.17 3 A ϕ x = 1 1 2 x 2 + y 2 (2x) = x x 2 + y 2 1 x y 2 ϕ x 2 = x2 + y 2 x 2x (x 2 + y 2 ) 2 = x2 + y 2 (x 2 + y 2 ) 2 2 ϕ y 2 = y2 + x 2 (x 2 + y 2 ) 2 2 ϕ x 2 + 2 ϕ y 2 = 0 i z = x + iy z = x iy f(z) f(z) 2 ϕ x 2 + 2 ϕ y 2 = 0 ϕ(x, y) = 1 2 ln zz = 1 2 ln z + 1 2 ln z No.18 5 A 1 r V V = 4 3 πr3 12

r = 4 dr = 3 dv = 4πr2 dr dv = 4π 42 3 = 192π 2 r V r = 1 + 3t V = 4 3 π(1 + 3t)3 = 4π 3 (27t3 + 27t 2 + 9t + 1) dv = 4π 3 (81t2 + 54t + 9) = 4π(27t 2 + 18t + 3) t = 1 dv = 192π 1 2 II I No.19 5 A 1 (x a) 2 + (y a) 2 = r 2 x 2(x a) + 2(y a) dy dx = 0 x = 1 y = 3 dy dx = 3 2(1 a) + 2(3 a) 3 = 0 a = 5 2 2 (x, y) = (1, 3) (1 a) 2 + (3 a) 2 = r 2 y y > 0 ) y a = r 2 (x a) 2 x dy dx = (x a) r2 (x a) 2 13

(x, y) = (1, 3) dy dx = 3 3 = (1 a) (1 a) (1 a) = = r2 (1 a) 2 3 a 3 a a < 3 1 a = 5 2 3 x 2 + y 2 = r 2 3 y = x ±3r r (x, y) = (, ) 3 10 10 x y a (1, 3) (y > x ) 3r r + a = 1, + a = 3 10 10 r 4a = 10 a = 5 2 1 2 2 No.20 1 A t > 0 r t + r2 t 2 t r2 t = 2r e x + 1 e x 2 e x 1 e x = 2 e y + 9 e y 2 e y 9 e y = 6 ( e x + 1 ) (e y e x + 9e ) y 12 2 x y No.21 3 A 14

h V = πr 2 h h = V πr 2 S = 2πrh + 2πr 2 = 2V r + 2πr2 ds dr = 2V r 2 + 4πr = 0 r = ( ) 1 V 3 2π II 1 No.22 3 A 1 f(a) = g(a) = 0 g(x) lim x a f(x) = lim g (x) x a f (x) d dx ( 1 9 + x 3) = 2 9 + x d dx ( 1 9 x 3) = 2 9 x 9 + x 3 9 x lim = lim = 1 x 0 9 x 3 x 0 9 + x 2 x (1 + x) n. = 1 + nx 9 + x 3 = 3 1 + x 9 3 =. ( 3 1 + x ) 3 = x 9 3 9 x 3 = 3 1 x 9 3 =. ( 3 1 x ) 3 = x 9 3 9 + x 3 lim = 1 x 0 9 x 3 15

2 1 No.23 4 A 1 y = 3 4 x + 5 2 ( x 2 + y 2 = x 2 + 3 4 x + 5 ) 2 2 = 25 16 x2 15 4 x + 25 4 = 25 16 ( x 6 5) 2 + 4 4 2 f(x, y) = x 2 + y 2 + k(3x + 4y 10) f = 2x + 3k = 0 x f = 2y + 4k = 0 y x : y = 3 : 4 x = 3t, y = 4t 9t + 16t = 25t = 10 t = 2 5 x 2 + y 2 = 25t 2 = 4 3 x 2 + y 2 2 3x + 4y 10 = 0 10 32 + 4 2 = 2 x 2 + y 2 2 2 = 4 I 1 2 2 3 16