06 5.. ( y = x x y 5 y 5 = (x y = x + ( y = x + y = x y.. ( Y = C + I = 50 + 0.5Y + 50 r r = 00 0.5Y ( L = M Y r = 00 r = 0.5Y 50 (3 00 0.5Y = 0.5Y 50 Y = 50, r = 5
.3. (x, x = (, u = = 4 (, x x = 4 x, x 0 x = 0 x = 4 x.4. ( z + z = 8 z, z 0 (z, z = (0, 8, (,, (8, 0 3 (0, 8, (,, (8, 0 z = z 4 z + 8..5. (g f(x = g(f(x = (x + = 4x + 4x + (f g(x = f(g(x = x +.6. 5 (i 5 5 (ii 5 5 (i x = 5, x 5 (ii x 5, x = 5 x 5 min{x, x } = 5 0 5 x
.7. f(tx, tx = (α(tx ρ + β(tx ρ ρ = {t ρ (αx ρ + βx ρ } ρ = t(αx ρ + βx ρ ρ = tf(x, x.8. ( ( (3 (4 (5.9. ( (tk(tl = t KL = t KL. ( y = Y L = K L KL = L L L = k... ( ( e x = e x, ( e x = 4e x ( ( e log x ( =, e log x = 0 e log x = x (3 (x log x = log x +, (x log x = /x (4 cos x = cos x sin x, sin x = sin x cos x (sin x cos x sin x = cos x, (sin x cos x sin x = sin x (5 ( / x = x 3/, ( / x = 3 4 x 5/ (6 (x /x = + x, (x /x = x 3.. ( y a = (a + b(x a ( x = (a + b(x a + k x = (a + b/ (a + b = x (3 3
.3. cos h = cos(h/ + h/ = cos h sin h = sin h cos h lim h 0 h = lim h 0 sin h/ h ( = lim sin h ( sin h/ h 0 h/ = 0 cos h h = cos(0 + h cos 0 h cos x = sin x cos h lim h 0 h = lim h 0 cos h h = cos 0 = sin 0 = 0.4. x + y 3 = 0000.5. x = a, a x = a f a a > h > 0 f a (a + h f a (a = 0 h 0 (f a (a + h f a (a/h 0 sin ( π a (a h sin π h = π a cos π a h π a h.3 (.8 h 0 x = a f a(a = 0 x = a f a 0 > h > a ( π ( sin a ( a h sin π = cos π a h + (.8 f a( a = 0 4
a 0 f a (x x 0 x 0.6. ( ϕ(x = f( z g (x ( ϕ (x = f ( z g (x/g (g (x 3 3.. 3.5 f(x f (x = a < 0 3.. ( f (x = x < 0 ( f (x = e x > 0 (3 f (x = x 0 3.3. f (x = e x > 0 f(x f (x = e x e = 0 x = 3.6 f( = 0 3.4. π(x p π(x = p log(x + wx π (x = (x+ < 0 π(x π (x = p p w x+ w = 0, x = w x = p w w 3.5. 3.6 ( f (x = 3 x < 0 ( f (x = 3x + 5 = 0 x = 5 5
4 4.. ( α + xα+ log(x + ex n ( /(x + dx = lim /(x + dx = lim log(n + =. 0 n 0 n (3 y = /(x + + + + n n n 0 dx = log(n + x + y y = n 0 x+ dx = n k= k O y = x+ x n O y = x+ n n x (4 + 4 + + n + n x dx dx = x x + C, n x dx = n (5 (3 4.. 6
y y = + n x dx = n k= k (, (, O y = x x n O y = x n n x ( y = 3x + x = y/3 /3 ( y x(3x + 6 = 3 y 6 dx, 3 dy = 3 ( y 3 y 6 3 3 dy = (y 7 y 6 dy 9 = ( y 8 9 8 y7 + C 7 x(3x + 6 dx = ( 3x 9 (3x + 7 8 + C 56 ( y = 3x + x = y/3 /3 y x (3x + = 3 3 y = 3y 3y, dx dy = 3 ( 3y 3y 3 dy = ( 9 y y dy = ( log y + + C 9 y x (3x + dx = ( log 3x + + + C 9 3x + 7
4.3. ( x sin x + cos x + C (x sin x = sin x + x cos x x cos x dx = x sin x sin xdx + C = x sin x + cos x + C ( x sin x + x cos x sin x + C ( (x cos x = cos x x sin x x sin x dx = cos x dx x cos x + C = sin x x cos x + C (x sin x = x sin x + x cos x x cos x dx = x sin x x sin x dx + C = x sin x (sin x x cos x + C = x sin x + x cos x sin x + C (3 x cos x + x sin x + cos x + C (x cos x = x cos x x sin x ( x sin x dx = x cos x dx x cos x + C = (x sin x + cos x x cos x + C = x cos x + x sin x + cos x + C 4.4. x 0 f(x = 0 < x f(x = 0 [, ] 8
4.5. ( p = c (q ( p q q p = c (q p (3 c q c (q c c c (q (4 x(p (3 c (q = x (q = px(p x(p x (qdq = px(p x(p c (qdq = px(p c(x(p+c(0 = + 0 0 5 5.. ( (ˆx ŷ, ˆx ŷ (ŷ ˆx, ŷ ˆx ( x = t(ˆx ŷ, ˆx ŷ + (ŷ, ŷ x = tˆx + ( tŷ x = tŷ + ( tˆx (3 x = t(ˆx ŷ 5.. ( ( (3, 3, 3 3 3 (6, 6, 6, 0 9
(3 det a b c = det 0 b a c a a b c 0 b a c a = det 0 b a c a 0 0 c a (c a(b + a = det 0 b a c a 0 0 (c a(c b = (a b(b c(c a (4 a b c a b c det c a b = det c a b b c a a + b + c a + b + c a + b + c a b c = (a + b + c det c a b 0 b a c a = (a + b + c det 0 a c b c = (a + b + c(a + c + b ab bc ca = a 3 + b 3 + c 3 3abc (, (, (3, (4 5.3. ( ( ( adj a b d = = c d c b a 0
( 0 0 ( d b ad bc c a (3 a 0 0, a = 0 0 /a 0 5.4. ( 3 4 = 0 0 = 0 3 4 3 4 0 4 3 (- x y z w xz yw xw yz = = z w x y y x w z yz + xw xz yw w z y x (-, (-3, (-4 (-5 x, y, x, y (- (-3 i i = (x + yi(x + y i = xx yy + (xy + x yi i = α ᾱ (-4 C C 4 5.5. ( F (x = det A F (x = f(x + g(x +h(x 3 ij (i, j,, 3 x F (x = f (x +g (x +h (x 3 A (f (x, g (x, h (x (
( F (a F (b 3 0 5.6. ( a b a b det A = 0 a, a 5.5 (ii k, k k a + k a = (0, 0 T k, k a = ka k x = (x, x T Ax = b (x + kx a = b a b a b y a + y b = (0, 0 T y, y 0 a b 0 y 0 y 0 ya = b, y 0 y x = 0, x = y/k x a + x a = x ka = ya = b x = 0, x = y/k a b ( Ax = b ˆx Ax = 0 S 0 S 0 + {ˆx} = {x + ˆx x S 0 } Ax = b ˆx ˆx ( a b ( ˆx = (0, y/k Ax = 0 S 0 = {(x, x x a + x a = (0, 0 T } S = S 0 + {ˆx} S Ax = b Ax = b S (3 (adj Aa = (0, 0 T adj A 5.5 (ii det(adja = 0
6 6.. A B P (A B = P (A B P (B = 57 30 57 = 30 9 + = 30 30 6.. ( P (A = 4 6 = 3 X X X, X P (B = = 4 X X (X, X = (3,, (4,, (5,, (6,, (4,, (5,, (6,, (5, 3, (6, 3, (6, 4 0 P (C = 0 36 = 5 8 ( X 4 X X X = 3 X (X, X = (,, (, 3, (, 5, (3,, (3, 3, (3, 5 6 P (A B = 6 36 = 6 = P (AP (B A B (3 X 4 X X (X, X = (3,, (4,, (4, 3 P (A C = 3 36 = 3 5 7 = P (AP (C 3
A C 6.3. ( E[X] = 4400 + 4 0000 + 4 3600 = 0600. ( E[u(X] = 4400 + 4 0000 + 4 3600 = 00. 0000 = 00 (3 E[X] = 3 4400 + 6 0000 + 6 (4 3600 = 35600 3. E[u(X] = 3 4400 + 6 0000 + 6 3600 = 30 3. 00 6.4. ( ( (3 (4 7 7.. ( α 5α + 6 = (α (α 3 = 0 α =, 3 3 ( α 8α + 5 = (α 3(α 5 = 0 α = 3, 5 3 5 7.. α (a + cα + ac = (α a(α c = 0 α = a, c a c 4
7.3. A(ax + by = aax + bay = a(αx + b(αy = α(ax + by 7.4. ( x, x 0 x + 3x = 0 x = x = 0 ( (x, x = (, 0, (x, x = (0, (3 x + 4x x + 4x = (x + x 0 x = x 0 (4 (x x 0 x = x 0 7.4 ( A = ( 0 0 3 > 0, det A = 6 > 0 ( A = ( 0 0 > 0, det A = < 0 (3 A = ( 4 > 0, 4 > 0, det A = 0 (4 A = ( > 0, > 0, det A = 0 7.5. ( α 4α + 3 = (α (α 3 = 0 α =, 3 3 ( Ã = ( 0 0 3. (3 A, Ã 7.6. 5
( y = 3 y + 4 y = 3 ( ( 7.6 y t = t (y 0 3 + 3 3 (3 t ( 3 t 0 ( yt 3 7.7. ( 5 4 y 3 4 z = y y = z = 0 3 4 y + 5 4 z = z ( A = ( 5 4 3 4 3 4 5 4 α 5α + = 0 α =, A (3 y 0 = z 0 = a y = 5 4 y 0 3 4 z 0 = a z = 3 4 y 0 + 5 4 z 0 = a y t = z t = a t lim y t = lim z t = 0 t t 6
7.8. ( y = 5 6 y 6 y + y = 3 ( A = ( 5 6 6 0 α 5 6 α + 6 = 0 α =, 3 A (3 y t = ( y t+ y t y t+ = Ay t 7.8 y t = A t y 0 y 0 = ( y y 0 (,, (, 3, 3 P = ( 3 7
A t = P ( t 0 0 3 t P ( ( ( = 0 t 3 0 3 ( t 3 t 3 t + t 3 t = 6 t 6 3 t t + 3 3 t 3 ( y t+ y t ( 3 = t 3 t + t 3 t 6 6 t 3 t + 3 t 3 t ( y y 0. y t = (y 0 3y t + (y 0 y 3 t y t = (y 0 3y t + (y 0 y + 3. 3t 8 8.. a a a n b n b b (, ( (3 b n a n = b a n (4 (i (a n (b n (ii a n, n =,,... (iii b n a n, n =,,... a n, n =,,... 8
8.. ( (i, (ii C C ]0, [ p > 0 p = C (q q ( 8.3. K L α, α ρ CES F (K, L, ρ = (αk ρ + ( αl ρ /ρ log F (K, L, ρ = log ( αk ρ + ( αl ρ (K, L ρ 0 α(log KK ρ ( α(log LL ρ lim log F (K, L, ρ = lim ρ 0 ρ 0 αk ρ + ( αl ρ = α log K + ( α log L = log K α L α ρ lim ρ 0 F (K, L, ρ = K α L α ρ K L ( log α ( K ρ L + ( α log F (K, L, ρ = log L ρ K > L ρ (K/L ρ 0 K = L log(α + α = 0 ρ 0 lim log F (K, L, ρ = log L ρ 9
L > K lim log F (K, L, ρ = log K ρ lim F (K, L, ρ = min(k, L ρ ρ ρ K, L K L L ρ K ρ L ρ αk ρ + ( αl ρ ( αl ρ L ( αk ρ + ( αl ρ /ρ = F (K, L, ρ ( α /ρ L ρ ( α /ρ lim F (K, L, ρ = L ρ L K lim F (K, L, ρ = min(k, L ρ 8.4. ( x = c/β x α/β ( ( u / u (x, x (x, x = α x x x β x ( c /β ( = c /β α/β x x α/β = c /β α β x α/β = α β 0 c /β x α/β = α x x β x
8.5. A A( + r n = A n log( + r = log lim r 0 rn = r log( + r log r log( + r = lim r 0 /( + r = r log = 0.69 rn log 0.70 f(r = log( + r f(0 = 0, d dr f(0 r=0 = f(r r = 0 r r/log( + r 8.6. rn = r log rn = log log( + r ( lim x x = lim eax x a e ax = 0 ( n = n = k x k+ lim x e ax = lim (k + x k x ae ax n = k + k + x k = lim x a e ax = 0
9 9.. ( x + 3x = 5, x 0, x 0 (x, x x x 3 (x, x = (0, 5/ (x, x = (5/3, 0 x x 3 = 0 (x, x h, h u(x + h, x + h = u(x, x + h u x (x, x + h u x (x, x + o(h, h (h, h o(h, h / (h, h h, h h + 3h = 0 (x + h + 3(x + h = 5, x + h > 0, x + h > 0 u(x + h, x + h u(x, x h u x (x, x + h u x (x, x + o(h, h 0, h + 3h = 0 h, h h > 0 u (x x, x + h u (x h x, x + o(h, h 0, h u (x x, x u (x 3 x, x + o(h, h 0, h h o(h, h = o(h, h (h, h = o(h, h 3 h (h, h h (h, h 3 (h, h = 3h /3 u x (x, x 3 u x (x, x 0 0
h < 0 u x (x, x = 3 ( 5 (3 ( (x, x u x (x, x = 3 λ u x (x, x = λ u x (x, x = 3λ x + 3x = 5 (4 x = x =. u x (x, x u x (x, x 9.. ( g(x = x x x = g(/g ( = + / =.5 x x = x g(x /g (x =.5 0.5/3 =.466 ( ˆx ˆx > 0 n x n x n = x n x n x n, x 0 = ˆx, n =,, x 0 > 0 x n = x n + x n = x n + x n > 0, n =,,... 3
II x n = x n + x n = x n x n x 0 x n, n =,,... n x n x n = x n x n < 0 x, x,... x n 0 8 x ( x /x = 0 x = ˆx 9.3. ( u(x, x x, x ( x, x u ( x, x x u(x, x = u( x, x x x (i x = ϕ(x (ii ϕ(x u( x, x = c x u( x, x = c x = x u x ( x, x 0 x S ϕ(x x S u(x, ϕ(x = c ϕ( x = x 4
x u (x, ϕ(x + u (x, ϕ(x ϕ (x = 0 x x ϕ (x = u x (x, ϕ(x u x (x, ϕ(x ϕ (x ( ( 9.4. ( x = y = z = a f (x, y, a = y x y f/ y(a, a, a = a 0 ( y y (a, a =, (a, a = x z ( y(x, z (x, z = (a, a f(x, y, z = 0 y (x, z x x(y(x, z z = 0 x = a y y y x (y z x x x = 0 y(a, a = a y y x (a, a = z (a, a (3 x = y = 0 f(x, y, z = 0 f y (0, 0, z = 0 ( 9.5. 5
( y = (y(0 + /α e αt t/α /α y = z + k t + k ẏ = αy + t ż + k = αz + (αk + t + αk αk + = 0, k = αk k k k = /α, k = /α ż = αz z = z(0e αt z(0 = y(0 k y = z(0e αt +k t+k ( y = y(0e t / y(t = 0 t = t y(t = 0, t t y(t 0 ẏ/y = t (log y = t log y = ẏ y dt = t dt = t + C y = e C e t / e C = y(0 y = y(0e t / ẏ = ty + α, α 0 y = αe t / dt e t / (elementary function f(x x = 0 (3 y = t α e C, C y(0 t > 0 ẏ y = α t log y = α log t + C = log ( t α e C C = log e C y = t α e C y(0 y = y(0t α C (4 y = (y(0 + β/αc αt β/α 6
y = z + k tẏ = αy + β tż = αz + αk + β αk + β = 0 k = β/α tż = αz (3 z = z(0c αt y = z(0c αt + k (5 y = re 3 cos (θ + t r θ y(0 = ẏ(0 = λ λ + = 0 λ = + 3 i λ = 3 i (9.3 y = Ae λ t + Be λ t e λt = e t 3 e ti (8.3 ( e λt = e 3 3 cos t + i sin t e λ t = e ( cos t 3 + i sin t 3 θ a, θ b r a, r b, A = r a (cos θ a + i sin θ a, B = r b (cos θ b + i sin θ b 7
0 < θ a, θ b < π ( y =r a (cos θ a + i sin θ a e 3 3 cos t + i sin t ( + r b (cos θ b + i sin θ b e cos t 3 + i sin t 3 ( ( ( =r a e 3 3 cos θ a + t + i sin θ a + t ( ( ( + r b e cos θ b + t 3 + i sin θ b + t 3 ( =r a e 3 cos (θ a + t + r b e cos θ b + t 3 { ( ( } + ie 3 r a sin θ a + t + r b sin θ b + t 3 y(0 = ẏ(0 = t > 0 ( ( 3 r a sin θ a + t + r b sin θ b + t 3 = 0 θ a + t 3 = π/ t r a r b θ b + t 3 = π/ r a r b r a = r b ( ( 3 sin θ a + t + sin θ b + t 3 = 0 t θ a + t 3 = (θ b + t 3 θ a + θ b = 0 { ( ( } y = r a e 3 3 cos θ a + t + cos θ a t = r a e 3 cos (θ a + t 8
r a θ a y(0, ẏ(0 y(0 = r a e cos θa, ẏ(0 = 3r a e sin θa 0 0.. ( π(x, x π(x, x = x + x (x + x. π (x, x = = 0 x x π (x, x = = 0 x x x = 4, x = y = 4 + = 3 ( π(x, x D π(x, x D π(x, x = ( π(x,x π(x,x x x x π(x,x π(x,x x x x (x, x = ( 4, D π ( 4, = ( = 4 0 0 ( x x 0 0 x x 9
0.. ( L(x, x, λ = x x λ(p x + p x w. L x (x, x, λ = L x (x, x, λ = x λp = 0 x x λp = 0 x L λ (x, x, λ = (p x + p x w = 0 ( λ x x = p p p x = p x 3 x p x + p x = w x x = w p x = w p (3 L(x, x, λ = x a x b λ(p x + p x w. 30
L x (x, x, λ = ax a x b λp = 0 L x (x, x, λ = bx a x b λp = 0 L λ (x, x, λ = (p x + p x w = 0 λ ax bx = p p x = bp ap x 3 x x p x + bp a x = w x = aw (a + bp x = bw (a + bp 0.3. ( u(x, x = x x x = p p x + m p ( U(x = x p x + m = p x 3 + m x. p p p p ( U (x = 3p p x + m p x 3
0.4. U(x x = m 3p x 0 m 3p U (x 0 + 0 U(x 0 ( L(x, x, λ = x + x λ(x + x. L x (x, x, λ = λ = 0 L x (x, x, λ = λ = 0 L λ (x, x, λ = (x + x = 0 λ ( L(x, x, λ = x + x λ(x + x. L x (x, x, λ = x λ = 0 L x (x, x, λ = x λ = 0 L λ (x, x, λ = (x + x = 0 x = x =, λ =. 3
(x, x = (, (, ( + ( = (, 0 + 0 = 0.5. ( π(x, x π(x, x = p ( x + x π x (x, x = π x (x, x = p x 4 = 0 p x 6 = 0 ( 4 x + 6 x. (x, x = (4p, 9p π(x, x D π(x, x D π(x, x = ( π(x,x π(x,x x x x π(x,x π(x,x x x x = ( 4x x 0 0 4x x x, x > 0 0.9 π(x, x 0.0 x = 4p, x = 9p y = 4p + 9p = 5p 33
( L(x, x, λ 4 x + 6 x λ( x + x y. L (x, x, λ = x 4 λ = 0 x L (x, x, λ = x 6 λ = 0 x L λ (x, x, λ = ( x + x y = 0 λ x x = 3 x = 3 x 3 x x + 3 x = y x x = 4 5 y. x = 9 5 y. C(y C(y = 4 4 5 y + 6 9 5 y = 0 y. (3 C (y = 0 > 0 C(y 34
(4 π(y π(y = py 0 y. π(y p 5 y = 0 y = 5p.. E(X = 6 + 6 + 6 3 + 6 4 + 6 5 + 6 6 = 7 V (X = ( 6 7 + ( 6 7 + ( 6 3 7 + 6 ( 5 7 + 6 ( 6 7 = 35 + 6 ( 4 7.. X x,, x n p,, p n E(aX + b = p (ax + b + + p n (ax n + b = a(p x + + p n x n + b(p + + p n = ae(x + b V (ax + b = p {ax + b (ae(x + b} + + p n {ax n + b (ae(x + b} = p a (x E(X + + p n a (x n E(X = a V (X.3. X.4 a = 0 Prob{ X 60 0} 0 0 = 4. 40 80 75% 35
.4. X \ Y 6 4 4 3 4 6 3 3 4 X, Y X 4 Y 6 3 3 3 4 4 ( E(X = 3 + 3 4 = 3 E(Y = 3 4 + 4 6 = 3 V (X = 3 ( 3 + 3 (4 3 = V (Y = 3 4 ( 3 + 4 (6 3 = 3 ( Cov(X, Y = 4 ( 3( 3 + ( 3(6 3 + (4 3( 3 + (4 3(6 3 = 0 6 ρ(x, Y = 0 = 0 3 (3 Prob{X = }Prob{Y = } = 3 3 4 = 4 = Prob{X =, Y = } Prob{X = }Prob{Y = 6} = 3 4 = = Prob{X =, Y = 6} Prob{X = 4}Prob{Y = } = 3 3 4 = Prob{X = 4}Prob{Y = 6} = 3 4 = 6 = Prob{X = 4, Y = } = Prob{X = 4, Y = 6} 36
.5.. Cov(X, Y = E[(X E(X(Y E(Y ] = E[XY E(Y X E(XY + E(XE(Y ] = E(XY E[E(Y X] E[E(XY ] + E[E(XE(Y ] = E(XY E(Y E(X E(XE(Y + E(XE(Y = E(XY E(XE(Y. X = Y Cov(X, X = E(X E(X = V (X.6. 00 x θ V [(θx + ( θy 00 ] = 00 V (θx + ( θy V [θx + ( θy ] θ V [θx + ( θy ] = V [θx] + V [( θy ] + Cov(θX, ( θy = θ V (X + ( θ V (Y + θ( θcov(x, Y. V (X = 4, V (Y =, Cov(X, Y = 3 V [θx + ( θy ] θ = 5 9 V [θx + ( θy ] = 9 3 θ 0 3 θ +. 3 57 00 6 x 74 y 37
.7. (.,.4,.7 [ ] X + X n E(Y n = E n = n E[X + + X n ] = n [E(X + + E(X n ] = (m n = m. n [ ] X + X n V (Y n = V n = n V [X + + X n ] = n [V (X + + V (X n ] = n (n v = v n. ( ( E(Y n = m.3 Prob { Y n m ε} V (Y n ε = v nε. (3 ( { X + + X n Prob n } m ε v nε { lim Prob X + + X n n n } m ε lim n v nε = 0 { } lim Prob X + + X n m n n < ε ( { } X + + X n = lim Prob m n n ε { } = lim Prob X + + X n m n n ε = 0 = 38
39