入試の軌跡

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i.................................. 7.................................................................... 5.4 4................................. 8.5 5..................................6 6..................................7 7................................. 4.8 8................................. 6.9 9................................. 8...................................................................................................... 4.................................. 6.4 4................................. 8 4.................................. 4.................................. 54.................................. 64.4 4................................. 77.5 5................................. 8.6 6................................. 88.7 7................................. 9.8 8..................................9 9................................. 7.................................. 6.................................. 4.................................. 9.................................. 4.4 4................................. 47 iii

55.................................. 56.................................. 6.................................. 69.4 4................................. 75.5 5................................. 8.6 6................................. 8.7 7................................. 8.8 8................................. 84.9 9................................. 85.................................. 9.................................. 94.................................. 98.................................. 4.4 4................................. 8 iv

5 4 5 5 9 5 5 5 4 4 6 7 9 II III III 4 III 5 C 6 B 7 III 8 III 9 C

4 6 7 9 III I III 4 B 5 6 A 7 III 8 III 9 C 4 6 7 9 III II III 4 B 5 6 C 7 C 8 II 9 III 4 III C n III 4 B 5 A C B

5 III C n 4 II 5 III II 6 III B B II 4 III 5 III 7 III C B 4 A 5 II 8 III A B 4 III 5 II

4 9 B A n III 4 C 5 III I A III 4 III 5 C III III B 4 B 5 A III C A B I III 4 III 5 A

5 III B A 4 III 5 C 4 III A II C 4 A 5 III

6 4-4 4 5 6 7 8 9 4 I II 4 4 5 5 4 5 4 III 5 5 4 5 4 4 4 5 4 5 5 4 A B 4 4 4 5 5 C 5 5 4 5

.. 7. 4 6 7 9 fx = ax + a + bx + b + x fx a b ab a = fx x > b fx x > a b ab y = x + ax + bx + c G xy p, q X, Y G mx + ny = X, Y m n 4 G x r z xy r x y < r K z = t r t r xy K K V r < r V r

8 4 z a c b b ac > azz + bz + bz + c = z a z z d p q dz pz q = dz qz p z 5 n i X i i =,, n n = 7 5 X 4 n X = P X = n X EX 4 lim n logex log 5 n EX + X n n 6 m n a i = m j = n k = b i = Ans = k + j c i k = k + j d i = [i/] e j = j f b [x] x m = 4 b i j k i = j = n k = m b i j k i j + k m Ans 4 l m = l d

.. 9 7 fx y = fx G Pt, ft x θt θt π < θt < π P G P Qαt, βt G θt αt βt t a b a < b P Q L L L L θa θb 8 e fx = e x + x + x < x < < fx < x lim n n f = n e < gx = e x x n gx K n n n k gx dx K n k { 4x + 9y = p 9 p q x y x + 6y = q x y p, q p 5 q 5 d d p q x y 4 d

. 4 6 7 9 Px, y t x y x = et e t y = et + e t e O, M t t P C t C OM OP At C MP St Px, y x y y x t At St At St t a a fa a f = a = 5 5 5 5 f5 = 4 a b m a = m b fa = m+ fb a p q a = pq fa p + q q = p a b m n r s a = m r b = n s a b { fa = b fb = a r s r = n+ s = m+

.. x y x log x x log y x + y log a b a < b fx gx [a, b] b a b a fx dx = b fx log fx dx a gx dx b a fx log gx dx a b a < b [a, b] fx M M = b fx dx b a b a b a a fx log fx dx M log M 4 ABC AB AC AB AC AB AC AB AC ABCD-EFGH AB = AD = AE = FBC = BCD = π EAB = θ θ < θ < π EFGH P P EF PI P EH PJ x = EI y = EJ ACP θ x y P EFGH ACP E J H I A P D F G B ABCD-EFGH C

5 C z z z w = z + z + z w z z z w = z z z w z z z z C w z z z z z w w w = w w w w 6 x y 45 45 y O O 9, n k k n O n, k n + k O n, k n k k n n + k O n, k, k, k n, k O n, k + 9 i T i i, T i i =,,,, 9 T = T 9 = T i i =,,, 7 x

.. θ π 7 P x y θ fθ = + { π x = fθ cos θ θ θ π y = fθ sin θ π P C P Pθ P = P P = P P = Pπ x α y β + = a >, b > P a b P α P C l l i ii iii D i D x ii D P P iii P D l P D 8 a a p fx gx fx = x p ax p gx = x fx f x f x fx gx a gx a = x a p = gx a gx a = x a x x x x a = 9 p = < 9 <. < 9 g < 9 4

4 9 A A = A a b A = ad bc c d A a b A = ad bc = A c d a d a b e f A = B = A + B c d g h A + B A B

.. 5. 4 6 7 9 xy x = rt cos t, y = rt sin t t π C rt = e t x y C t rt π {rt} r t sin t cos t dt = π {rt} sin t sin t dt r t rt C x x π V = π e t sin t dt x 4 + y 5, x 4 + y 5 A B A B B x, y x y log x y x y p n D n = {x, y x, x p y n} S n L n M n D n y = x p x n p n M n < S n < M n + n S n lim n p+ p Ln n

6 4 OABC AOB BOC COA 9 OA OB OC a b c ABC G OGA OGB OGC 9 a b c BC : D P AD A Q APQ G OQ 5 < a < a z = t + ai t l i z l z l θ m m sin θ 6,,,, y xa x < a a = a = a 9 7 a

.. 7 7 Pa, b P a b Qx, y AX = B A = x + a X = B = y + b P O Q R P O RQ OP P OQ P 8 θ < θ < π a > 4b Pa, b y = 4 x Q R PQ PR m m P QPR = θ P G m < < m tan θ m m G θ = π G 4 9 n {S k } S k = + + + + k logn + < S n < + log n {c k } c k = c k+ c k k =,, {a k } {b k } n n a k b k = a n b n a b b k+ a k k= n ks k = pn k= k= n ks k log nn n < k= S n pn n

8.4 4 n =,, I n = n n!! = x n e x dx I n =,, I n I n I x e x e lim n n! x n e x dx e n n =,, n k k k! k= A = a b b a a b a + b 4 a b 4 A 4 a + b 4 a b 4 n A n 4 < a < b = a A n, x n lim n x n Pxt, yt t { xt = cos t + π 4 t < π yt = cost C P C x y C 4 C

.4. 4 9 4 x, y, x y, z xy S xz T Aa, b, S Bc,, d T C,, A B O OA OC OB OABC O A B C A S B T OABC A B 5 n n n m m n 4

.5 5 l : y = x + a C : y = sin x π x π a a C l y x A a b A =, a = p n n =,,, b = p = a, pn+ = A p n + b n =,, p = A p + b p q n = p n p n =,, q n+ q n n =,, A n 4 p n n =,, t z + tz + t = t zt t zt C z C w = iz z + w

.5. 5 4 x [x] x [ ] = [] = < θ < π sin α = α < α < π log [ 5 + cos θ ] θ [ ] + log sin θ θ [ ] [ ] 5 log + cos θ + log sin θ θ 5 t t xy Pt, e t C a C x y x = a Sa Sa a > Sa lim a ae a = Sa =.5 a < a <.5 < e <

.6 6 log x lim = e x x e < n log x x = x > n α n β n α n < β n < α n < e n, ne < βn lim n α n OAB OB M AB α : α P < α < OP AM Q Q AM OA R OA = a OB = b OP OQ a b α a = b = AOB = θ cos θ = 6 OR a α R OA α {a n } {b n } a = b = a n+ = a n b n b n+ = a n + b n n a n b n n a n b n

4 fx = sin x.6. 6 π x π a π F a = a fxf fx = x y = fx F a x π dx 5 [a, b] fx a x b a fx b fx = 4x x [, ] fx < a < b < [a, b] fx

4.7 7 fx = xe x p p y = fx Pp, fp y = gx e x fx gx L y = fx y = gx x = x = L Sp p Sp p p < p < A B C A =, B =, C = + p p A n n =,,, A = A, A n+ = A n B BA n + C n =,,, A A A n = a n c n b n d n n = a n d n b n c n lim n n a b Aa, a, b B a, a, b Ca, a, b A B C α O A B C S AB D DC AB DO AB ABC DC DO θ sin θ α O α H OH P S ABCP P α

.7. 7 5 4 a b c a b c ax + bx + c = ac ac a c a, c n n 5 fx p fx + p = fx fx p sin x π y = sin x sin x m n fx fx = sin mx sin nx p p fx f = f p = mp π np π m n sin mx sin nx

6.8 8 fx = ex e e x + y = fx fx f x { } lim n f f n n + n + k 9 k k 9 9 n n OAB AB Q OQ P P Q O OAP OBP ABP a b c OQ OA OB a b OP OA OB a b c OA OB AB 5 6 P OA OB AB OP OA OB

.8. 8 7 4 a > fx = a + log x x > gx = x x y = fx y = gx P l a P l P x 5 Q R Q R θ < θ < π sin θ π < θ < π

8.9 9 O, A, 6 B, 4 O AB OC s t P OP = s OA + t OB C CP s t s t t CP k k k M N M + N n n p n p p M N p n+ p n M N M N k M + N 4 p n n k C : y = x P a b a, Q b, R b a b a R A A a P C A C C C C C C C

.9. 9 9 4 X Y Z X = Y X x y x + y A AX = Y, AY = Z, AZ = X Y X Z Z = sx + ty s t X + Y + Z = 4 A 5 y = e x P t xt, yt P dx v = dt, dy d x α = dt dt, d y dt t v = dx dt > P s, e s v s P s, e s α s P α

. ABC a = BC b = CA c = AB t A B AP = tc P CP a b c t P CP = a t P AB A B 7 6 xy y = x C C C P a, C C Q Q C C P P C Q C P n C Q n Q P Q P S P n Q n P n+ n =,,, S n 4 S n n= a

.. 4, a a xy x x P, t P t π P C C x C a 5 A = c QX, Y X Y = a c b d b Px, y d P y = x Q 9X = Y A P y = x Q X + Y = A P y = x Q L a b c d L x y

. y = x Pt, t y = x H t > H t x y = x y = x PH S S t y = x y = x S S = S t a fx = x + x + a e x x x e x 7e lim x x e x k y = x + x + y = ke x + a a k a, a,, a n, a n+ = a n, n =,,, a n a = a n tan π a = tan π a n+k = a n, n =, 4, 5, k

.. 4 4 O,, A,, B,, C,, 4 O A B C D A B C D F DF ABCD 5 4 4 4 4 4 i j i j 4 n n = 4 n = 4 n = 4 n =

4. x + y = 4 x A B A =, A 5 B = 5 6, B 7 9 8 = 8 E A B A B AB = E A B A A, AB, ABA, ABAB,, B A B B, BA, BAB, BABA, a n x ax + x + + n = nx + a n n a = 7,

.. 5 4 p q x + px + q = α β α β {a n } a n = α n β n n =,, a a a α β > lim a n+ n a n lim a n+ n a n = + 5 p q 5 5 A B T T A B B A A B T A n p n n =,, T A n q n n =,, T A

6. a > y = x x, y = a x x > C C C C P C l C y l a P C l θa < θa < π lim a a sin θa OABC P POABC P OA OP = OC OP = 4 AP : M CP N P BC Q PQ OMN BQ : QC OP 6 L R L: R: L 6 L L R L R L

.. 7 4 O A, S B, S T C y T D D x l S AB T BC AC l 5 x y t x y A =, B = t x x AB α β 5 4 6 5 x y t = A = tb 4 A E A 4 = E E x y t = x y t 4 x y t A

8.4 4 fx = x sin x x π y = fx l l a, b a y = fx x = a x x V a a a b c a + b = c a b c a + b = c a b c x + 6 + y 4 = y = x + a a x + y = x, y x, y x + y x, y

.4. 4 9 4 A 5 B 5 A B p q A E 5 n f n x f n x = x x nx k =,,, n f n x k + < x < k

B5 7 5 B4 5 4 5. 5 5 5 5. 4 5. 4

4. fx f x = ax + a + bx + b + a x f x f x = D a >, D D/4 = a + b ab + = a + b a a >, a + b b a = fx = bx + b + x b = fx = x O a a > a + b a = b = a + b a + b = a = fx = bx + b + x b x > fx fx f x = bx + b + b > f b >, b + b + < b b = fx = x b a x > fx a = a > D a > D > f x = a x + a + b a + b a a a a + b a f = b + D > a + b a > b O a a > b a b a + b >

x, y x + X.. 4 = p y + Y p X, q Y fx = x + ax + bx + c p = q fx = fp + f px p + f px p + x p G p, fp y = fx fp y = fp x y = fp fp x { y = fp fp + f pp x p + f pp x p + p x p } = fp + f px p f px p + x p f p = p = a G mx + ny = n d m n =, n d = n m s n + t d s, t s n + t d A As n + t d = s n + t d sa n + ta d = s n + t d s t A n = n, A d = m d A n n m = m n n m

44 m + n A = m + n m + n mn mn m n A X, Y X m + n mn A = Y m + n mn m n m n X mny m + n, X Y mnx + m n Y m + n 4 l p = m n m + n q = mn l m + n x, y X, Y X = px + qy, Y = qx py X, Y G qx py = px + qy + apx + qy + bpx + qy + c y = x + ax + bx + c q = l p p py = p x + ap x + bpx + c y = p x apx bx c p x p = p G

.. 45 z = t r t r y + z r x + y r z = t, r t y r t, St St = = 4 r t r t r t r y r y = 8 r r t r t r t dy = 4 [ y dy = 4 r y r y x r y r y dy ] r t K z = t St = 4 r r t 4 r t = 8 r r t r t z 4 r t r r r y 4 r x r t r r t

46 V r V r = = r r r = 6 r St dt { } 8 r t r r t dt r r t dt 4 r t dt [ ] r r t t r r = 6 r πr 4 4 = 4πr 8 r V r = 4πr 8 r < r π π V r = 4π 8r = 8 r + r r π V r + V r V π π = 4π 8 π = 4 π

.. 47 4 azz + bz + bz + c = a c a zz + abz + bz + bb = bb ac az + baz + b = b ac az + b = b ac b ac > z + b a = b ac a z b a b ac a dz pz q = dz qz p id dzz + id p dqz + idq dpz + idpq d pq = i d d d a = id d, b = idq dp, c = idpq d pq a c azz + bz + bz + c = b ac = bb ac = idq dp idq d p id d idpq d pq = d p pq pq + q = dp q b ac dp q = a d d, b dp dp dq = idq = a id d d d dp dq z d d dp q d d ii d = d d d = d z pz q = z qz p z p z p z q = z p z q z q z p q

48 5 6 X X 5 5 X 4 = X X 5 5 5 X 4 = X 5 5 X 7 X 4 = X 4 X 5 5 5 X 4 =, 4, 5 X 4 5 5 X 7 X 4 =, 4, 5 5 5 X 6 X 7 X 4 = 5 X 4, 4, 5 5 n P X = = P X P X = 6 n 7 k 6 k k 6 P X = k = 6 6 EX = = = k= n n 6 6 { n n } 7 k 6 k kp X = k = k 6 6 k= 6 n 7 k 6 n 6 k k k 6 6 k= 5 n 6 k 5 n 6 k k + k 6 6 k= k= = 5 6 k n 6 n k= k= = 6 n n + n + n + 4 n + 5 n + 6 n 4 EX = 6 n n + n + n + 4 n + 5 n n n 5 5 < EX < 5 6 6 n log 5 6 < logex < log 5 + n log 5 6 log 5 6 < n logex < n log 5 + log 5 6 lim n n log 5 + log 5 = log 5 6 6 lim n n logex = log 5 6

.. 49 5 k 6 P X n = k = P X n k P X n k n n k k = 6 6 n P X n = = k = 6 n n k k k 6 P X n = k = 6 6 6 6 { n n } k k EX n = kp X = k = k 6 6 k= k= 6 n k 6 n k = k k 6 6 k= k= 5 n k 5 n k = 6 + k k + 6 6 k= k= 5 n k = 6 6 k= = 6 6 n n + n + n + 4 n + 5 n EX + X n = EX + EX n = 6 n n + n + n + 4 n + 5 n + 6 n = 7 + 6 6 n n + n + n + 4 n + 5 n

5 6 i j k n 5 n 5 4n 4 8n 4n i m j n k i = No i Yes Yes Ans k + j k k + j No i [i/] j j i i [i/] = i k k + j i i j j i j + k = i j + j + k = i j + k ii i [i/] = i i i j j i j + k = i j + k = i j + k i ii i j + k i j + k m n + = mn i i = k + j Ans mn 4 m l = l+ + l d l +

.. 5 7 θ = θt f t = tan θ t f t = θ cos θ f t > θ > θt P G n = sin θ, cos θ π < θ < π cos θ = + tan θ, sin θ = tan θ + tan θ f t n = + {f t}, + {f t} OQ = OP + n OQ = t + f t + {f t}, ft + {f t} αt = t + f t + {f t}, βt = ft + {f t} b b b L = + {f t} dt = + tan dt θ dt = a a a cos θ αt = t + sin θ βt = ft cos θ α t = + cos θ dθ dt β t = f t + sin θ dθ dt {α t} + {β t} = + tan θ L = b L L = a dθ = tan θ + sin θ dt = tan θ + cos θ dθ = dt {α t} + {β t} dt = b θb θa [ dθ = θ ] θb θa a dt cos θ + + cos θ dθ dt cos θ + dθ dt θb θa = θb θa π 9 dθ

5 8 < x < < < < x x x e t dt < x e t dt < e t tdt < 6 dt < e x < 6x x x 6t dt < e x x < x t dt < e x x x < x < x < < fx < x n > < n < < f < n < n f n lim n n = [ ] gx dx = e x dx = e x = e K n = n f < n n lim n n f n k= { } gx dx K n = e ne n } n = e {ne e n n = e n n n n ne n = nf { } n gx dx K n = e { n e n n f = n e k n = n e e n + n n + } n k k = lim n n gx dx K n = e + = e

4 9 x p 9 = 6 y q 4 9 = 6 9 6 6 4 x = 6 9 p y 6 4 q.. 5 x = p q y = p + q m q = m y = p + m = m p m n p m = n p = m + n 4 p 5 q 5 m, n 4 m, n =,,,,,,,,,,, p, q p, q =,,,,,, 4,,, 4, 5, 4 d 6 x, y 4 d d p q p = d p q = d q x, y x = d p q, y = d p + q p = q = x = d p = q = y = d d d 6

54. x = et e t y = et + e t y + x = e t y x = e t y + xy x = e t e t y x = y > y = x + Pa, b [, a] C OP : y = b x a b = a + y St x + b a x = a x + x M a + At a O a x = aa x + + x a + a x At = + b a x dx = [ x x + + log x + x + b ] a a x = { a a + + loga + } a + ab = {ab + loga + b ab} = loga + b = e t log e t + et + e t = t b a P C x At + St = a St = a At = et e t t 4 ft = At St ft = t et e t 4 f t = et + e t = e t 4 4 et 4e t + f t = t t log + t = log + f t + ft t = log +

.. 55 b p k i k b = p i p i p n i n k =,, n fb = + p + p + + p i + p + p + + p i = p i + p + p n + p n + + p n i n p i + p p n i n+ p n a = m b a = m p i p i p n i n fa = m+ = m+ fb p i + p p i + p p n in+ p n p a pq q fa pq + q = p + q a pq q pq q = p q = a = m r b = n s fa = m+ fr, fb = n+ fs fa = b fb = a m+ fr = n s, n+ fs = m r m+ fr = n+ s, n+ fs = m+ r m+ n+ s = m+ s, r = n+ r s, r fr = n+ s, fs = m+ r 4 fs { m+ + }s = m+ s fr { n+ + }r = n+ r 5 6 6 s r 4 5 r s r = s 5 6 m+ n+ r = s = r s r = n+ s = m+

56 y hx = x log x log y + x + y h x = log x + log y + = log x log y h x = x = y hx hx x log x x log y x + y x = y x y h x + hx [a, b] fx > gx > b a fx log fx fx log gx fx + gx {fx log fx fx log gx fx + gx} dx b M = b a a b a b a { fx + gx} dx = fx log fx dx fx dx b a b a fx log gx dx fx dx = b a M dx gx = M b a fx log fx dx b b a > b a b a a fx log M dx = log M fx log fx dx log M b a b a b a fx dx fx dx = M log M

.. 57 4 BAC = θ < θ < π sin θ > ABC = AB AC sin θ = AB AC cos θ = AB AC AB AC cos θ = AB AC AB AC BCD = π ABCD z FBC = π FBCD J ABFE BC E A B D E A,, B,, D,, E cos θ,, sin θ AC =,, AP = x + cos θ, y, sin θ O H I θ A P D F G B y C x AC =, AP = x + cos θ + y + 4 sin θ AC AP = x + cos θ + y ACP = AC AP AC AP = y x cos θ + 8 sin θ x y y x i < cos θ < θ < π y x = ACP 9 4 cos θ 4 cos θ ii cos θ π θ π y x = cos θ ACP sin θ iii cos θ < π < θ < π y x = ACP 9 + 4 cos θ 4 cos θ

58 5 Az Bz Cz A BC z z z z z z z z z + = z z z z w = z + z + z z = z = z = w z z z + w z z z = z + z z z + Az w Bz w Cz z + z z z = z + z + z + z z z z z = z + z z z + w z + z z z = w B CA C AB w ABC C z = w = z z z w = z z z = w C w z z z + w z z z = z z z z z z + z z z z z z = z z z z z z + w C w z z z z = z z

.. 59 B C z z z z z z z z z = z z z z w w w w = z + z + z z z z w z z z w z z z = z z + z z z z z z = z z + z z z z z z = z z + z z z z z z { } z z + z z z z z z z z + z z z z z z z + z z z z z = z z w w w BC Az Bz Cz ABC O z + z + z z + z + z w z z z z BC w w + z z + z w + z + z + z = z + z + z w + z z + z w + z z + z z z + z + z = 9

6 6 O n, k 45 n + k 45 n k n k = n + k n + k n + k n + k n + k = m m 45 m 45 m k n m O n, k n C n+k O n, k y = k Aa, k a n A n, k + A n, k y = k N O n, k, k, k n, k N + N = N O n, k, k, k n, k O n, k + A B A : T 9 = B : ii =,,, 7 T i k P A B = P A B P A = P A P A B P A = P A B P A O 9, 9 C 6 P A = 9 C 6 9 O 9,,, 7, O 8, 4 8 C 6 P A B = 8C 6 9 P A B = 8C 6 = 8 7 9C 6 9 8 7 =

,.. 6 7 f = P, fπ = P P C P C α a α < α π f = 5 8 P, dx dθ = f θ cos θ fθ sin θ, θ = π α + β a b = α + β a b < + β α < + β b a b 5 f θ = θ π 8 dy α < 4 dx dθ = 5 8 dy dθ = π dy dx = dy dθ l y = 4 5π x + 5 8 π f π dθ = f θ sin θ + fθ cos θ dx = 4 dθ 5π = π D x k, D x k P D k + y b = k D P k + 5 64b = kx k P D + 5y k 8b = 4 5π = 8b k l 5 k 8b k 5 k b = πk k k k + 5πk = 5π + 64k = k k = 5π + 64 < 64 + 64 = 4 P D k, k <

6 8 α = a fx = x p α x p gx α = x x p α x p px p p α x p 4 α = x α xx αx + αx + α px p α = x α px p α xx + αx + α px p α = x α p x + p αx + p α px p α p p gx a = x a p x + p a x + p a px p a p = gx a = x a x a x + a = x a x + a x + a a gx = a x x + a x + a x = a = 9 < 9 <. 9 g = 9 + + 9 < 9 + + 9 <. +. + 9 < < 9 g < fx = x 9 x f x = x + 9 x g < 9 < g +. f = 9 =, f = + 9 = 5 4 g = f f = 5 4 =.8.8 < 9 <.8 9 4.8

.. 6 9 det A = ad bc A A = A A A = E A a + da + ad bce = O A = A ad bc = a + d A = O A O a + d = a + d = a + d = ad bc = A = A ad bc = A a + d =, a b A = tr A = a + d c d det A tr A = A = E det A = tr A = tr A + B = tr A + tr B A B A + B tr A =, tr B =, tra + B = A + B = E A B A =, B = a b A = c a a a bc = B = a b c a

64. x = e t cos t y = e t sin t t x = e t sin t + cos t = e t sin t + π 4 y = e t sin t cos t = e t sin t π 4 x y t π 4 π x + x e π 4 e π t π 4 π y + y e π 4 y e π e π O π e 4 e π 4 e, π 4 e, π 4 x x t = π 4 π 4 e y t = π 4 π e 4 C π π {rt} r t sin t cos t dt = {rt } sin t cos t dt = [ ] π {rt} sin t cos t = = = π π π π {rt} sin t cos t dt {rt} sin t cos t sin t dt {rt} { sin t sin t sin t} dt {rt} sin t sin t dt

y = e t sin t α = t π 4 π e 4 β = e π y = e t sin t.. 65 π 4 t π V = π = π = π = π = π α π β y dx π y dx 4 π dt dt π π π α π y dx dt dt = y dx y dx 4 π dt dt π rt sin t {rt cos t} dt rt sin t {r t cos t rt sin t} dt rt r t sin t cos t dt + π π π V = π rt sin t sin t dt + π = π π rt sin t dt = π π rt sin t dt π e t sin t dt rt sin t dt r = rθ θ π x V V = π 9 π {rθ} sin θ dθ xy x r = + cos θ θ π C C x x V = π π + cos θ sin θ dθ = π [ 4 + cos θ4 ] π = 4 π

66 x + y 4,,,, x 4 + y 5 A x + y x 4 y 5 A y 8 O 5 4 x fx, y = x 4 + y 5 gx, y = x 4 + y 5 fx, y A x, y x > y > gx, y = f x, y g±x, ±y = fx, y gx, y B x, y A = ±x, ±y B B A A x y B 5 y 5 4 O 5 8 8 5 5 4 x y B x, y x, y =, 4,, 5,, 6, 4,, 4,, 4, 4, 4, 5, 4, 6, 4, 7, 4, 8, 5, 4, 5, 5, 5, 6 x log x y = m n m n x m n = y x m = y n x, y = 4,, 4, 4, 4, 8, 5, 5 x, y 8 x, y = 4, ±, 4, ±4, 4, ±8, 5, ±5

k x k + y = x p y = x p x = k x = k + i y i+ y = x p k p i < k + p y n y = x p n.. 67 y = x p k +, k + p. y = i + y = i. k, k p y = x p y y = i y = i+ S i m i i n m i < S i < m i + n n n m i < S i < m i + i= i= i= M n < S n < M n + n S n = n x dy = n y p dy = y n i + i p p + n p+ p O m i y =x p x y = i i =,, n m i +, i,, i,, i,, m i, i y = n [n p ] +, n,, n,, n,, [n p ], n D n L n n L n = m i + + [n p ] + = Mn + n + [n p ] + i= n p < [n p ] n p M n + n + n p < Ln M n + n + n p +

68 S n + n p < Ln < S n + n + n p + p p + n p+ p + n p < Ln < p p + n p+ p + n + n p + p p + + n < n p+ p Ln < p lim n lim n p p + + n p p + + n = p p +, p + + n p + n + n p+ p p + n + n p+ p = p p + lim n p+ p Ln = p n p + D n y = i i n [i p ] + n L n = [i p ] + i p < [i p ] i p n n i= n i= i p i= < Ln n i p + i= i p < n p+ p Ln n n lim n n lim n n i= n i= i p = x p dx = n n p + n p+ p = i p + n p + n p+ p n p p + lim n n p+ p Ln = p p +

.. 69 4 OG AB OG BC OG AB =, OG BC = z C c a, b, c a, b, = a = b O B b y a b c, b, c = b = c a >, b >, c > x A a a = b = c D BC : AD = OD OA = OA + OB + OC P AD A t OP = OA + t AD APQ G OA + OP + OQ = OG OQ = OB + OC OP OQ = OA + OB + OC t AD = a, b, c + t a, b, c OR = a, b, c d = a, b, c OQ = OR + t d

7 OR d θ θ π OQ OQ = OR sin θ OR d OR d sin θ = OR d OQ = OR d OR d d d R OQ d OQ d OR d + t d = t = OR d t = a + b + c 9a + 4b + c d OR d OR d = b c + 4c a + a b OQ = b c + 4c a + a b 9a + 4b + c = b c + 4c a + a b 9a + 4b + c θ Q O

.. 7 5 z = t + ai y z = t + ai = t a + ati x = t a y = at t x = y 4a a 4a x + a = y a a O a x m l θ zcos θ + i sin θ = t + aicos θ + i sin θ = t cos θ a sin θ + it sin θ + a cos θ x = t cos θ a sin θ y = t sin θ + a cos θ t x sin θ y cos θ + a = i sin θ = y = ±a ii sin θ x sin θy 4a cos θy 4a a sin θ = y D/4 = 4a 4 cos θ + 4a sin θa sin θ = 4a a sin θ < a < sin θ < a sin θ = a a < sin θ i ii sin θ <, < sin θ < a sin θ =, a a < sin θ

7 6 y = xa x S P a P a = S = S i < a P a = a xa x dx = a 6 y a 4 ii < a P a = xa x dx = a O a a x P = 48 5 P + P + 48 P 5 = 88 P a 9 7 P a 9 7 P a E E = k C k P a k P a k = P a k= E P a P a P a a ii a 4 a 5

.. 7 7 A = A = AX = B X = A + a a + b + B = = + b a + b Qa + b +, a + b R, RQ OP = a + b + a + b 4a b = 9 9 a + b a + b RQ OP a = b RQ OP P P O a b a = b < a P b O a A λ 4λ + = λ =, u = B = a + b a b + u + v A B = a + b A a b + u + A v = a + b v = a b + u + v OQ = 9a + b u + 4 a b + v = 9 4 a + b + a b + a b a + b a b + OQ a + b = a b + = a = b = P,

74 8 PQ PR x θ θ m = tan θ, m = tan θ m < < m < θ < π < θ < π tan θ = tanθ θ = tan θ tan θ + tan θ tan θ = m m + m m y = 4 x y = x Q α, 4 α Q m m = α α = m Q m, m Q y m = m x m y = m x m R y = m x m P m m a = m + m, b = m m m m = m + m 4m m = a 4b > m m < m m = a 4b G tan θ = a 4b θ = π 4 b + b = a 4b O + b a + b + b = a 4b b + a = 8, b < G

.. 75 9 k n k= k+ k k+ k dx x < k dx n x < k= logn + < S n k n+ dx x < S n k + k < k n k= k dx x n k < + k= k k S n < + log n dx x S n < + n dx x logn + < S n < + log n a k b k + b k+ a k = a k b k+ b k + b k+ a k+ a k = a k+ b k+ a k b k n n a k b k + b k+ a k = a k+ b k+ a k b k k= k= = a n b n a b n n a k b k = a n b n a b b k+ a k k= k= a k = S k b k = k a k = k + b k = kk n k= ks k = S n nn S n = nn S n n k= k k= = nn S n nn 4 = S n nn kk + k + P n = nn

76 n ks k = S n nn k= k= n ks k log n = S n log n nn logn + log n < S n log n < < log + n < S n log n < 4 4 < n ks k log n < nn k= n ks k log n nn < k= + + + + + + = + + + + 4 5 + 6 + 7 + + 8 > + + + + + + + + n= n {}}{ = + + + = + n n n= = γ = lim n n k= k log n Euler s constant γ.5775665 6 γ

.4 4 I =! x e x dx = [ ] e x dx = e x = e.4. 4 77 I n = n n! = n n! x n e x dx = n x n e x dx n! [ ] x n e x n x n e x dx n! = n n e + n n! n! x n e x dx = n n e n! + I n I n I n = n n e n! n n e I = I + n! n= = e + e! + 4e! + 8e! = e x x n e x x n e n =,, n! x n e x dx n! x n e dx = [ ] e x n+ = e n+ n +! n +! n n+ n +! = 4 n + n = n = n n! x n e x dx e n n =,, n n I k I k = e k k k= n k k k= k! k= k! = + I n I e = I n e + e n k k k= k! = I n I e I n e n n =,, 4 lim n k= lim I n = n n k k = k! e 4

78 a + b 4 = a 4 + 4a b + 6a b + 4ab + b 4 a b 4 = a 4 4a b + 6a b 4ab + b 4 A λ aλ + a b = A α = a + b, β = a b i α β b F = A βe α β { = a b b G = A αe β α { = a b b b a b a b a + b a } = } = F = F G = G F G = GF = O A = αf +βg A 4 = α 4 F + β 4 G A 4 = a + b 4 ii α = β b = A = ae A 4 = a 4 E i ii b = A 4 = a + b 4 + a b 4 + a b 4 b = A = ae A n = a + b n + a b n 4 x n = a + bn + a bn b = a a + b = a b = a < a < < a < < a b < lim n x n =

.4. 4 79 P xt = yt = { cos t + π 4 = cost = t < π t = π 4, 5 4 π x t = sin t + π 4 y t = sint { x { π 4 = y x 5 π π = 4 4 = y 5 π = 4 t = π, t = 5 π, 4 4 t = t t < π C P x, y x y Qx, y R x, y x = cos { t + π 4 = cos π t } + π 4 y = cos t = cos π t x = cos { } t + π 4 = cos t + π + π 4 y = cos t = cos t + π Q t π, t = π t C π < t < π t = 7 π t C R t < π, t = t + π C π t < π t = t π C x = cost + π y = cost 4 y = cos t = sin t + π 4 = sin t + π 4 y = x x x y = x x cos t + π 4 = ±x x y x y + y O x C 4 S S = 4 x [ x dx = 4 x ] = 8 π P Q R 4

8 4 OA = a, b, OC =,, z n = b, a, a b n = a ab + b e n e = ± n = ± b, a, a b a ab + b x a B c O d A b C y e bc + a bd OB = ± a ab + b bc a b d e OB = bc + a bd a ab + b OAC S S = OA OC OA OC = a ab + b OABC OAC e OB V V = S e OB = bc + a bd b a c d b + a b i b > a b > V = a 6 6 6 V a =, < b <, c =, d = 6 ii b = V = ad 6 6 V a =, c, d = 6 6 iii a b = V = bc 6 6 V 6 b = c =, d V A B 6 A, b, B,, < b < A,, Bc,, c A,, B,, d d

.4. 4 8 5 n n,,, n n, n + n = n n n n n n n m n C m P m m n P m = n C m n = n C m n 4 E E = n m= m P m = n n m= m n C m = n n n = n n k nc k = k= n n k nc k = n n C k = n n k= k=

8.5 5 y = sin x y = cos x y = π x π x = ± π x = π C y sin π = x π y = x + π x = π C y sin π = x + π y = x + π a > a = π V y V = π π + a π = π π = [ π π π = π π π cos x dx x sin x ] π sin x dx a l C a O π x

.5. 5 8 p = A p + b E A p = b E A = E A = p = E A b = = p n+ = A p n + b p = A p + b p n+ p = A p n p qn = p n p qn+ = A q n A = A A + 4 E = O A E = O B = A E = B = O A = E + B A n = n n k nc k E k= B k = E + n B + n B n = n n n k nc k E k= + n n B k = n n 4 q n = A n q q = p p = a p = qn = n n p n = q n + p = n = = n n n + n n +

84 z + tz + t = D < t 4 t < < t < 4 z = t ± 4t t i z = zt z + z = t zz = t t zz + z + z = z + z + = z + = z + = zt zt C z = + cos θ + i sin θ < θ < 8 w = iz i{ + cos θ + i sin θ} = z + + cos θ + i sin θ + i + icos θ + i sin θ = cos θ + i sin θ = i{cos θ + i sin θ} + i = cos 7 + i sin 7 {cos θ + i sin θ} + i = i + cos7 θ + i sin7 θ 9 < 7 θ < 7 w i O i i O

[ ] [ ] 5 5 4 log + cos θ log + cos θ log [ ] 5 + cos θ 5 + cos θ < cos θ < π < θ < π < θ < π [ ] + log sin θ + log sin θ log sin θ log sin θ.5. 5 85 < θ < π π 4 θ 4 π 5 + cos θ 7 [ 5 + cos θ ] =,, log [ 5 + cos θ ] =,, log + log sin θ [ ] + log sin θ [ ] [ ] 5 log + cos θ + log sin θ log [ 5 + cos θ ] =, [ ] 5 + cos θ = [ 5 + cos θ 5 + cos θ < cos θ < < θ < π π < θ < π ] =,

86 [ ] + log sin θ + log sin θ log sin θ log log sin θ log sin θ < θ < π α θ π α 4 sin π > sinπ α π < π α 4 π < θ π α [ ] 5 + cos θ = 5 + cos θ < cos θ < π < θ < π < θ π 5 [ ] + log sin θ = + log sin θ log sin θ log log sin θ log sin θ < θ < π 5 6 π < θ π π 4 θ 4 π 6 θ π < θ π α

5 C : { x = t y = e t Sa = a a t y y dx = a = e t t dt = [ ] a = t + e t t dx e dt dt a te t dt.5. 5 87 y x=a Sa O a C x = a + e a + S a = ae a S a = ae a Sa Sa a S a + + + S a + Sa 4 e O a lim a a + e a = lim Sa = a Sa 5 < e < S = e = e S = 4e = 4e > S <.5 < S < { 4 = 4 <.5 } = 86 5 5 >.5 Sa =.5 a < a <

88.6 6 fx = log x x f x = log x x f x = x = e x e f x + fx e y O e α n e β n y = n x lim fx = x y = log x x n < log x e x = n x > n e n e ne fe n = ne n ne > n > < n < fe n f < fαn < fe n fne = log ne ne > log e n = n fne > n fne > fβ n fx < x < e e < x lim n e n < α n < e n, ne < βn = lim α n = n

.6. 6 89 P AB α : α OP = α a + α b Q OP OQ = s OP a A α R P Q O α OQ = s{ α OA + α OM} = s α OA + sα OM M b B Q AM s α + sα = s = + α OQ = + α { α a + α b} = α α a + b + α + α OR = t a α RQ = + α t a + AM = OM OA = b a RQ AM RQ t α + α AM = { α + α t a + α b + α α } a b + + α α + α b = a = b = a b = a b cos θ = 6 = 4 t α { α + + α + α t α } 9α + + α + α = t = α + α R OA OR = α a + α α + α = α = 5

9 a n b n A n = a = =, b = + = a = =, b = + = 7 n = A n = k k A a k = M b k = N ± M, N a k+ = a k b k = M N ± = MN ± b k+ = a k + b k = M + N ± = 8M + 9N ± 6N + = 6M + N ± N +, n A b = b n+ = a n + b n b n a = b = n m a n b n a m+ b m+ p p a m+ = a m b m a m p b m+ a m = b m b m p b m p b m+ b m = a m a m p, a m b m p n a n b n

4 fx = sin x =.6. 6 9 sin x = ± sin x =, π x π x = π,, π, π i sin x π 6 x 5 6 π fx = sin x = sin x = sin x ii sin x π x π 6 5 6 π x π fx = sin x = sin x i ii y π π O π 6 π π 5 π 6 x x π π x π f x π = sin x π = cos x = cos x { fxf x π sin x cos x x π = 6 sin x cos x π x π 6 a π 6 F a = a π π a π 6 F a = 6 sin x cos x dx = cos a 4 sin x cos x dx + a π 6 = 4 cos a + sin a sin x cos dx

9 5 fx = 4x + 4x = 4 x + [, ] fx [, ] fx [a, b] fx a fa, fb b a 4a a, 4b b b < a < b < < a 4 b < i < a [a, b], f = / [a, b] ii < a 4 [a, b] fx fa b, fb a 4a 4a + b, 4b a + 4b 4b + ab a 5b a < a < b < b a > a + b 5 4 < a 4 b i ii [a, b] fx Logistic map [a, b] x n+ = fx n, a x b a x n b n =,,,... {x n } x n+ = λx n x n : λ 4, x

.7. 7 9.7 7 fx = xe x f x = x + e x Pp, pe p y pe p = p + e p x p y = p + e p x p e p gx = p + e p x p e p hx = fx gx x hx = xe x p + e p x + p e p h x = x + e x p + e p h x = x + e x x h x > h x hx x hx x fx gx x p h x + hx Sp = = = L L L {fx gx} dx L L xe x dx p + e p x dx + p e p xe x dx L p + ep + Lp e p dx S p = L p + ep + Lp + pe p = L ep p + p L L Sp p Sp p = L S p + Sp

94 A = A B BA + C A = + p + p + p = + p = + p + p + p A = A B BA + C + p A = + p + p + + p = + p + p + p = A n + p + + p p + p + p A n+ = A n B BA n + C A n+ = A n+ B BA n+ + C p + + p p A n+ A n+ = A n+ A n B BA n+ A n A A = p, A A = p n A n+ A n = p n

.7. 7 95 n = n = k p k A k+ A k = A k+ A k+ = A k+ A k B BA k+ A k p k = + p + p p k + p k+ p k = p k+ = n = k +, n n p n n p k A k+ A k = k= k= n p k A n = + k= = pn p p k n = n n = + pn p < p < lim n = + n p = p p

96 AB D a + a, a + a, b + b,, b DC = OC OD = a, a, b,, b = a, a, b AB = OB OA = a, a, b a, a, b = a, a, DO = OD =,, b =,, b DC AB = a a + a a + b = DO AB = a + a + b = DC AB DO AB DC AB a > ABC = AB DC = a + a + a + a + b = a a + b b > cos θ = DC DO DC DO = b a + 4b b = sin sin θ = cos θ = DO AB OA = OB b a + b a a + b OAH OBH HAD HBD DC AB H CD a OH = DO sin θ = b a + b = ab a + b O C α θ H A D B

.7. 7 97 OH S α O P ABCP PH = PO + OH = OA + OH = a + b + ab a + b ABC PH = a a + b = a a + b + a + b a + b + ab P ab a + b O C A H B α

98 4 b 4 6 4 b 4ac > ac < b 4 = 9 ac < b 4 9 ac 8 ac,,, 4, 5, 6, 8 ac a, c ac = ac = ac = ac = 4 ac = 5 ac = 6 ac = 8 a, c =, a, c =,,, a, c =,,, a, c =, 4,,, 4, a, c =, 5, 5, a, c =, 6,,,,, 6, 4 a, c =, 4, 4, D = b 4ac b ac = D = b 4 b ac = D = b 8 b = ac = D = b b = 4 ac = 4 D = b 6 b = 5 ac = 5 D = b b = 6 ac = 6 D = b 4 b = 5 ac = 8 D = b b = 6 + + + + 4 + 6 = 5 6 = 5 7

.7. 7 99 5 sin x sin x = sin x sin x < sin x = sin x y = sin x y π O π π π π x x sinx+p = sin x x = sin p = p p = π x sinx + π = sin x π x fx + p = fx x = p f p p = f f x = sin mx sin nx = sin mx sin nx = fx f p p = f p f = f p = p f = sin mp sin np = sin mp = sin np = mp = kπ np = lπ k l

i mp = kπ k fx + p = sin mx + p sin nx + p = sinmx + mp sinnx + np = sinmx + kπ sinnx + np = sin mx sinnx + np fx + p = fx sin mx sinnx + np = sin mx sin nx x np π ii np = lπ l sinnx + np = sin nx fx + p = sin mx + p sin nx + p = sinmx + mp sinnx + np = sinmx + mp sinnx + lπ = sinmx + mp sin nx fx + p = fx sinmx + mp sin nx = sin mx sin nx x sinmx + mp = sin mx x = sin mp = mp π mp π i ii mp π np π

.7. 7 mp = kπ np = lπ k l p = kπ m = lπ n kn m = l m n k m k = mk k p = k π fx + p = fx + k π = sin mx + k π sin nx + k π = sinmx + mk π sinnx + nk π = sin mx sin nx cos nk π + cos nx sin nk π = nk sin mx sin nx = nk fx 4 i n k = 4 fx + π = fx π ii n k = 4 fx + π = fx π n π n π

.8 8 fx = ex e x + = e x + f x = f x = e x + e x + e x e x + f x = ex e x + + ex e x + = ex e x e x + x f x + + + f x + fx y O x lim fx = lim fx = y = y = x x y = ex < y < x e x + e x = y y x = log y y x f x = log < x < x < n + < n + < f x = log x f = logn + = logn + n + f = logn + = log n n + { } n f f = n{logn+ log n} = log + n n + n + n { } { lim n f f = lim log + n } n n + n + n n = log e =

.8. 8 k k 9 = k 9 k + k 9 = k + 9 9 n i n k n k n k 9 = k 9 ii k < n n k n + k 9 = k + 9 9 E E = k n= = k 9 = k 9 n k 9 + k n= n= n=k+ n + k + 9 9 n + 9 n k + 9 9 n=k+ n=k+ n n = k 9 k n= n + k + 9 = k 9 + 9 k{k + + } = 8 k + k + 99 n=k+ n

4 AQ : QB = OAP : OBP = a : b OQ = b OA + a OB a + b B P OQ : OP = OAB : OAP + OBP = a + b c : a + b OP = a + b OQ = b OA + a OB a + b c a + b c O Q A OA OB AB r OAP : OBP : ABP r P = OA r : OB r : AB r =OA : OB : AB B r r a : b : c = : 5 : 6 OP = 5 OA + OB + 5 6 = 5 OA + OB O Q A OP OAB O s OA OP = s OA + OB = s OA + s OB OB 5 AP OAB A t OP = OA OA + t OA + AB = OA OA + t AB + OB OA 6 = + t t OA + OB 6 6 s = + t 6 s 5 = t 6 OP = 5 OA + OB s = 5 t = 9

4 f x = x g x = x P x p fp = gp f p = g p a + log p = p p = p p = p p >.8. 8 5 p = 4p p = p > p = a > a = log P P, P l y = x y = x hx = fx gx x hx = a + log x x h x = x x = x x x x x = x x x + x x h x hx a hx = x = P S S = e log + log x dx [ = log x + x log x x [ = x log x x log y ] ] x dx e = e e [ ] x y = fx O e y = gx x

6 5 sin θ = 5 R < θ < π < θ < π cos θ = = 4 5 5 5 θ Q sin θ = sin θ cos θ = 5 4 5 = 4 5 < 5 < sin π 6 < sin θ < sin π 4 π 6 < θ < π 4 π < θ < π α = π BC = A = α B = C = α 5 B CA D BC = BD = AD = CD = x AB : BC = BC : CD + x : = : x x + x = A D x x > x = + 5 B C BCD x = x cos α = + x = x + = 5 + 4 5 + 4 4 5 = 5 5 = 5 > cos α > cos θ α < θ 5 π < θ 5 π < θ < π 4 < π θ < 5 4

.9. 9 7.9 9 C AB α OC = α OA + α OB = α, 6 + α, 4 = α, 4 + α OC AB AB =, 4, 6 =, α + 4 + α = α = C4, CP = OP OC = s OA + t OB OC = s, 6 + t, 4 4, = s + t 4, 6s + 4t CP = s + t 4 + 6s + 4t t = 4s + 5t + 6st 4s 4t + CP = 5t 4 6st + 4s 4s + = t 4 6s + 4s + 8s + 4 t 5 i 4 6s 5 ii 4 6s 5 s t = 4 6s 4s + 8s + 4 5 < s > t = 4s 4s + i ii CP s 4s + 8s + 4, s > 4s 4s +

8 M M + N, p = N M + N M M + N M N p = + = M + N M + N M + n M + N n + n + n n + n n + p n+ = p n M M + N + p n = M N M + N p n + N M + N N M + N M + N k M + N = + + + + k = kk + i k M N k M = + 4 + 6 + + k = k + k = k 4 + k k N = + + 5 + + k = = k 4 M N M + N = k + k k 4 4 kk + = k +

.9. 9 9 ii k M N k k + M = + 4 + 6 + + k = k { + k } = k 4 4 k + N = + + 5 + + k = = k 4 + k + 4 k k + k + 4 4 4 4 i ii M N M + N = M N M + N = k + k kk + = k k k 4 p n+ = M N p n M + N { p n } M N M + N p n = p M N M + N M p n = = n M + N M N M + N n M N + M + N n + n + k + p n = n + k k k

y = x y = x a A a,, a A x a + a y a = x + ay = a + a b B b, x + by = b + b x b ay = b a + b a b a y = + a + ab + b x + a + a + ab + b = a + a aba + b R b a R A aba + b x =, + a + ab + b a, + a x = a y = + a a = x y = + x C C C y x = + x x x = x = t x = t t 6 t = t + t = t = ± x = ± C y = 4 C C ±, 4

C : y = + x y = x y = x x y < C C C y C.9. 9 4 C O x C P, PA = a, a = a a, a C P, y cusp y = x, C C y S S = [ = = 88 5 { + x x + 9 x 5 x 6 x ] } dx

4 Y = X AX = Y AX = X AY = Z A X = Z AX = Z Z = X AZ = X AX = X 4 4 X = Y A X = X A Y = Y A X Y = X Y 5 X Y Y X Y X detx Y 6 5 {deta} detx Y = detx Y 6 {deta} = deta = 7 AY Z = Z X Y X Y//\X deta dety Z = detz X 7 dety Z = detz X detx Z + dety Z = detx + Y Z = Z = kx + Y k 8 X + Y Z 8 k 8 AZ = X AkX + ky = X kax + kay = X ky + kz = X ky + kkx + ky = X k X + k + ky = Y//\X k = k + k = k = Z = X + Y X + Y + Z =

.9. 9 4 X Y X Y X + Y = Z X + Y X + Y = Z Z X X + X Y + Y Y = Z Z X X = Y Y = Z Z = X Y = X Y π Z + X = Y Z X = Z X π cos X = sin cos i Y = π cos 4 sin π Z = π sin 4 π A π cos ii Y = A = cos 4 π sin 4 π π sin π sin π cos π Z = = cos π sin π A 4 π cos 4 A = π sin 4 π sin 4 π cos 4 π = i ii A = ±

4 5 dy dt = dx dx dy dx = ex dt dx v = dt, dy dt dt = dx dx, ex dt dt = dx dt, ex v = dx dt > = dx + e x dx dt dt = + e x v = + e x, ex =, + e x e x + e x s, e s v =, e s + e s + e s t α = dx e x e x, dt + e x + e x e x e x =, + e x + e x + e x = ex + e x, e x 4 + e x s, e s α α = es + e s, e s + e s

.9. 9 5 4 α = e x + e x ex, α = e x + e x ex + = e x + e x fx = α fx f x = ex e x + e x 5 x f x + fx f x = e x = x = log fx x = log ex = + = 9

6. ABC C cos A = b + c a bc b a APC CP = b + tc b tc cos A A tc c P B CP = b + tc b tc b + c a bc = b + t c tb + c a = ta + tb + t tc CP = a a = ta + tb + t tc t a b + tc = t b a t =, b a b < a t = AB t t b a B A b a c < b < c + a B < 9 c A B < 9

4 5 6 6 6 6 6 E 4 6 5 6 E = 6 + 6 + 6 + 4 6 + 5 4 6 + 6 5 6 = 5 8 6 6 4 5 6 5 6 6 6 6 E 4 6 6.. 7 4 5 6 4 5 6 4 5 6 4 5 6 4 5 6 5 6 6 4 5 6 4 5 6 4 5 6 4 5 6 4 5 6 5 6 6 6 6 6 6 6 6 E = 6 + 6 + 6 + 4 6 + 5 4 6 + 6 6 = 5 8

8 n n 6 En E = 6 6 k k= En = n {k + + k + + + 6} + 6 6 k= En + n 5 En + = 6 6 k n 6 k=n n {k + + k + + + 6} + 6 k= n 5 6 k=n+ En + En = 6 {n + + n + + + 6} 6 n = 6 6 nn + 7 6 n = 7 n n + 4 = {4 nn + } 7 E E >, E E >, E4 E <, E5 E4 <, E6 E5 < E < E < E > E4 > E5 > E6 k

.. 9 y = x y = x Q b, b Q y P y b = x b b y = x b + b P a, a C Q P C a a = a b + b a b + a = b b a b = a Q a, 4a Q x b P x b P x 4a P 4a, 6a Q P = a, a a, = a, 4a 4a Q P = 4a, a, = 6a, 6a 4a 6a P Q P S a > S = a 6a 6a 4a = 8 a P n x a n n =,,, P n Q n P n+ S n S n = 8 a n {a n } a 4 a n = 4 n a 8 S n = 4 n a = 8 n a 4 4 S n a 4 n= 8 S n = a n= 4 = 7 8a O a 4a x

4 Px, y t C x T OT = TP = at at OC = a CP CP = CT t cos t sin t CT sin t cos t cos t sin t a sin t = = sin t cos t a a cos t OP = OC + CP at = + a a sin t a cos t = at sin t a cos t P y a at sin t, a cos t P a t C O at T πa πa x

.. y S = πa y dx x = at sin t dx = a cos tdt x t S = = πa π y dx a cos t a cos t dt π = a cos t + cos t dt π = a cos t + + cos t = a [ t sin t + 4 sin t ] π = πa O dt x t S πa x dx = a cos t dt πa π C L dx = a cos t dy = a sin t dt dt π dx dy L = + dt dt dt π π = {a cos t} + a sin t dt = a cos t dt π = a sin t π dt = a sin t [ dt = a cos t ] π = 8a

5 9X = Y X a = Y c b d x y = ax + by cx + dy c x + 4cdxy + d y 9ax 9by = y = x d x 4 + 4cdx + c 9bx 9ax = x d =, 4cd =, c 9b =, 9a = a =, b = 9 c, d = A Px, x Q 9 c x, cx 9X = Y c A = 9 c c c X + Y = X + Y Y = X a b x ax + by = = Y c d y cx + dy a + c x + ab + cdxy + b + d y cx dy = y = x b + d x 4 + ab + cdx + a + c dx cx = x b + d =, ab + cd =, a + c d =, c = a =, b =, c =, d = A =

.. y = x,, L y = x P, P, P p P p Q q Q q A p p = q q deta det p p = det q q L det q q = det p p = deta = p p A L A λ a + dλ + deta = a + d a + d L L x, y a b x x = a + d c d y y d c b a P t, t A a b t = c d t L b = d =, L x y cx ay = = at + bt ct + dt a c detab = deta detb detab = detba deta = A a L c

4. PH Pt, t y = x y t = x t y = x + t + t y H S y = x Q y = x P H t + t, t + t O S t x Qt, t PQH = PH = PQ = t t = 4 t t t + t S S = = t x x dx PQH [ x x x ] t = 4 t 6 t t 4 t + 6 4 t t t + t S = x x dx = [ x x ] x = 6 S = S 4 t 6 t t 4 t + 6 = 6 t{ t t } = + t > t = + t = = + 9

.. 5 fx = x + x + a e x f x = x + ax ae x a > fx x a a f x + fx a e a a + e a a + e a a e a gx = x e x g x = x xe x x > g x < x gx x gx g x x e x 7e 7e lim x x < x e x 7e x = lim x x e x = y = x + x + y = ke x + a x + x + = ke + a x + x + a e x = k y = fx y = k x a lim fx = lim fx = lim + + x e x = x x x x k i < a a e a < k < a + e a ii < a < k < a + e a i < a ii < a y y a+e a a+e a a O a e a a x a O a a e a x

6 a = a a = = a = a a = = a 4 = a a = = a, a =, a n = n n tan π = tan π π 4 = tan π tan π 4 + tan π tan π 4 = + = a n = tan θ n a n+ = tan θ n tan θ n = tan θ n a = tan π a = tan π a = tan π 5 a 4 = tan 5 π a 5 = tan 4 5 π a 6 = tan 8 5 π = tan 5 π a 7 = tan 6 5 π = tan π 5 a tan π 5 tan 5 π tan 4 5 π tan 5 π a n+k = a n n =, 4, 5 k 4

.. 7 4 O,, x + y + z + lx + my + nz = A,, m + n + = B,, l + n + = C,, l + m + 5 = l =, m =, n = x + y + z x y z = x + y + z = 7 D,, A B C α AB = OB AD = OD OA = OA =,,,,, AB AC α n = 6,, AD n θ DF = AD cos θ, cos θ = ABC = DF = AD n n AC = OC OA =,,, AD n AD n α A n θ D = 6 + + 6 + + = 7 AB AC AB AC = 5 = 7 F ABC DF = 7 7 =

8 5 4C = 6 n = 4 6 6 = 6 n = 4 {, 4} {, } {, } {, 4} 6 = 8 n = i ii i {, k} k =,, 4 6 = ii C 6 = 4 i ii 4 X P X i C 6 = 6 + 4 = ii 6 = 9 i ii P = 6 + 9 = 5 8 4 P = P = P 4 = P 4 k= = 54 kp k = 5 8 + + + 4 54 = 9

.. 9. x x + y = 4 y y = ± 4 x O y V x { V = π + 4 x dx π } 4 x dx { = π 5 x + 4 x dx 5 x } 4 x dx { } = π 5 x dx + 4 x dx + 4 x dx I = 4 x dx I = I = 4 π = π I = π 6 = π y y=+ 4 x y= 4 x 4 x dx y O I x y O π 6 I x V = π { [ ] 5x x π + π + } = 6 π + 6 π

x + y = 4 x y x + y = 4 x r = 4 cos θ π θ 6 π r θ O 4 x θ π 6 V V = π π r sin θ dθ = 6 π π sin θ + cos θ sin θ π sin θ + cos θ 4 π dθ = 6 sin 4 θ + π π dθ = 6 π [ = 6t 6 sin t cos t 4 sin t cos t = 4π + 9 4 ] π sin 4 t dt π π sin θ + cos θ sin θ + cos θ dθ = [ 4 sin θ + ] cos θ 4 π = 9 4 π 4 sin θ + cos θ sin θ = 4π + 9 π sin θ + cos θ sin θ = π + 9 8 V = 6 π π + 9 = 6 8 π + 6 π

.. 7 8 9A + 5A = 9 + 5 5 9 8 4 5 A = A = 64 8 7 8 7A + A = 7 + 5 9 4 A = A = 8 4 5 5 A = 8 5 8 5 7 B + B = + 6 8 48 6 B = B = 56 7 5 5 B = B = 6 6 6 5 B = 7 6 A = B = 5 8 5 6 5 7 6 5 8 5 6 5 7 6 = = A 7 5 9 8 = B 8 A = 5 7 B = 6 7 5 9 8 8 5 7 = 6 5 = 8 5 = 6 5 7 6

AB = 5 8 5 6 5 7 6 = 9 7 X = AB X + X + E = O X EX + X + E = O X = E AB = 6 5 5 7 BA = = 7 6 8 5 9 AB + BA = E BA = AB E ABA = ABA = A AB E = A B A = A B, BAB = BAB = AB EB = AB B = A B, ABAB = ABAB = A BB = AB B = AB E, BABA = BABA = A BA = A BA = E AB E = AB, ABABA = ABABA = AB EA = ABA A = A B A = B, BABAB = BABAB = ABB = AB = A, ABABAB = ABABAB = BB = E, BABABA = BABABA = BABABA = AA = E. 6 A, B, AB, AB E, A B, E 5 6 5 9 7,, 8 5 7 6 7 8,, 9 5, 6

.. AB = E A B A = E B = E ABABAB = E, BABAB = A, ABAB = BA, BAB = ABA, AB = BABA, B = ABABA, E = BABABA A, AB, ABA, ABAB,, B, BA, BAB, BABA, 6 E, A, BA, ABA, AB, B 5 7,,, 8 5 9 8 9 7 6 5,, 5 7 6 6

4 ax + x + + n = nx + a = x = a y = x + x + n n y = x + a x y = x + x + n = x < y = x x + n = n k = y = kx + a k i x y x + + n 4 x + n 9 4 x + x + n = kx + x + kx + n k = k, k 4 n k = k = n ii x < y x x + n = kx + x k + x + n k = k + k = n <, { k + } 4 n k =

k = n, n.. 5 y k k n, n k n O, n 4 x n a n, n n a n n + a <, < a n n a = a n n + a n n {b n } {c n } n b n = n +, c n = b n = + n, c n = n n n {b n } {c n } a b a lim n c n 5 a

6 4 x + px + q = α + β = p, αβ = q a n = α n β n = αβ n α n + β n + = q n + α n + β n q α n + β n a n α + β = p, α + β = α + β αβ = p q = p q, α + β = α + β αβα + β = p q p = p + pq p q a a a α β > i α > β > α > β > lim a n+ n a n = lim α n+ β n+ n α n β n α β α = lim n β n n = αβ = q α n β n ii α < β < α < β < lim a n+ n a n = lim α n+ β n+ n α n β n = i ii lim a n+ n a n

.. 7 α β α β < lim a n+ n a n = + 5 α β α >, β < α >, β < lim a n+ n a n = lim α n+ β n+ n α n β n α α β n+ = lim n n α n βn = α α > α = + 5 i α = + 5 α + 5 + p + 5 + q = p + q + + p + 5 = p + q + p + p + q + =, p + = p =, q = x = ± 5 ii α = + 5 α + 5 + p + 5 + q = p + q + + p + 5 = p + q + p + p + q + =, p + = p =, q = x = ± 5 i ii p, q = ±,

8 5 A B B A B p p = C 4C = 6 A B p p = C C 4C = A B p p = C 4C = 6 T n A x n y n z n x n y n z n A B A B A B A B A B 6 6 A B A B A B x n+ y n+ z n+ x n+ = + x n + 6 + y n + 6 6 z n y n+ = + x n + + y n + z n z n+ = 6 x n + 6 y n + 6 z n

.. 9 x = p y = p z = p 7 5 x n+ 8 8 6 5 y n+ = 9 8 z n+ 8 9 6 x n y n z n, x y z = 6 6 x y z = = 7 8 5 9 8 7 8 5 9 8 5 8 8 9 5 8 8 9 6 6 6 6 x y z 6 6 = 5 8 8 9 q = x q = z q = z q = 5 8, q = 8, q = 9 x y z = 7 8 5 9 8 5 8 8 9 6 6 x y z, x y z = 5 8 8 9 z z = 8 5 8 + 9 8 + 6 9 = 8

4. C C P x y x = a x x = a P a, a x = x l y a = a x a y = x a + a C : x = y y a [ ] a x dy PQR = y dy a y a a = a 4 = a a a O Q R P a l C C x a C P l = a x x l a a > < a < < a < l l tan α = a, tan β = a π 4 < β < < α < π 4 < α β < π θa = α β tan α tan β tan θa = tanα β = + tan α tan β = a a + a = a a lim tan θa = lim a a = lim θa = a a lim a sin θa = lim a tan θa cos θa a a = lim a cos θa = cos = a a a

.. 4 a = OA c = OC e a e c e = OP a c e OP = x a + y c + z e x, y, z x = 4 y = OP = 4 a + c + z e M N OM = OA + OP = a + 4 a + c + z e = a + c + z e ON = OC + OP = c + 4 a + c + z e = a + c + 8 4 z e α β γ α a + β c + γ e OM ON α + β + zγ =, 8 α + 4 β + zγ = α : β : γ = z : z : PQ v = z a + z c e PQ t OP + t v = 4 a + c + z e + tz a + z c e = 4 + zt a + + zt c + z t e Q BC + zt =, z t = z = ±, t = ± OB = a + c OQ = 5 a + c 4 BQ = 4 a 5 CQ = a 4 BQ : QC = 4 : 5 4 = : 5 OP = 4 a + c + z e 4 a + c + z e z = ± OP = 6 + 4 + = 4

4 i j i j 6 i j 6 i j = i j = 5 i, j =, 6, 6, 6 = 8 L R i j LRi, j i i + j 6 i + j LRi, j = 6 i + j ii 7 i + j i + j 6 LRi, j = i + j 6 i,ii LRi, j = i + j 6 LRi, j = i + j 6 = j 7 i + 6 6 6 i,j 6 i,j 6 i,j 6 = j i + 6 i,j 6 = j i + + j i + + j i + 6 = 6 = 6 = 6 = 6 = 6 i<j 6 i<j 6 i<j 6 i<j 6 j<i 6 i=j 6 j i + + i j + 6 j<i 6 j i + + j i + 6 i<j 6 j i + 6 = j 6 j i + 6 6 j= i= 6 {jj jj } + 6 6 j j + 6 = + 6 6 6 7 6 7 6 = 9 9 j= j=

.. 4 LRi, j = i + j 6 + 8 + 6 + 4 4 + 5 + 6 6 = 76 6 = 9 9 i j 4 5 6 4 4 4 5 4 5 6 4 5 6 L R a f a b c d e f L R i j a e i, j 6, 5 6, 4 6, 6, 6, f i, j i + j = 6 5 a f L 6 5 + 5 6 = 5 8 4 OB x π T r D r, r OD = r OD + DB = r + r = r = D, C r y A O B, D S T r π x

44 S T y S T S : x + y + = T : x + y = 9 y O T S x x S T y = x, y = x + x V V = π x x + x dx = π x dx π x dx I = x dx I I = π 6 + = π + 8 V = π [ x x ] = π 7 8 π π I π + 8 = π 8 π y π 6 O, x

.. 45 5 AB = x t x y x 5 4 6 5 = 5x 6y x 5t 4x 5y x 4t trab = 5x 6y + x 4t = x y t, detab = 5x 6yx 4t 4x 5yx 5t = x ty xy AB x y tab + x ty xye = O AB x y t AB AB,, 4x 5y =, x 5t = x = 5t, y = 4t x y 5t 4t A = = = tb t x x t 5t 5t detab = deta detb detab = deta detb = x + ty + xy = x ty xy x y t A = tb B = E A = tb = t B = t E, A 4 = A = t E = t 4 E A 4 = E t 4 = t = ± A = E x y t =

46 tra = A A + detae = O A = detae, A 4 = {deta} E deta {deta} = deta = { x + ty + xy = x y t = y = x t tx = t + t x = t + t, y = 4t + t 4 x t = t y t = 4t x t t y 4t t t t = ± t = ± x = ±7 y = ±5 7 5 7 5 A =, 7 7

.4. 4 47.4 4 fx = x sin x f x = cos x a, b y y = fx l cos a = a = π b O a π x b = fa = π π, π l π y = x π y = x + π 6 a = π V V π = = π π x sin x dx x x sin x + sin x dx [ = x + x cos x sin x + x sin x 4 = π 8 + π 9 8 π V = π 8 + π 9 8 ] π

48 a k r r =,, r = a = k = k r = a = k + = k + k + r = a = k + = k + 4k + + a a b c a + b = c a + b a b l m a = l, b = m l + m = c c = l + m c c a b c a + b = c a b c a = n A, b = n B, c = n C n A B C n A + n B = n C A + B = C A B C A B C a + b = c a b c

.4. 4 49 x + 6 + y 4 = y = x + a x + + 6 x + a 4 = 5x + a x + 4a a = D D/4 = {a } 5 4a a D/6 = a 5a a = a + 6a + D a + 6a + 5 a + 5 x + y = y = x + x + x i x y = x + y = x + y = x y ii x y = x + y = x + y = x O x i ii x + y = x, y

5 x + y = k 4 k,, k k,, k x + y = k x, y y = x + + 5 + 5 x a = + 5 y + 5 x a x = 5 = { + 5 } 5 = + 8 5 5 y = + 8 5 5 + + 5 = 5 + 5 5 x y +,,,,, +,,, x + y + 8 5, 5 + 5 + 5 5 5,

.4. 4 5 4 A B X Y X 5 5 8 8 8 8 Y 5 5 4 4 4 4 A B p q p = 8 4 + 8 4 + + 4 8 4 + 4 + = 4 8 q = 8 4 + 8 4 + 8 4 + 8 4 = 4 X Y A Z Z P Z Z X Y 5 5 5 5 5 5 5 5 5 5 5 5 P Z Y X 5 5 8 4 5 8 4 8 4 5 8 4 P Z = = + + = P Z = 5 = + = 6 P Z = = P Z = 5 = q = 8 P Z = = E P Z = 5 = + = 4 P Z = = + + = 7 8 4 8 4 8 4 8 4 8 4 8 4 8 4 8 4 8 4 8 4 8 4 8 4 E = 5 6 + + 5 8 + + 5 4 + 7 = 55 6

5 5 n f n x = x x nx k =,,, n f n = f n = k + k f nc k =, k + < c k < k c k c k n f nx = n n,,,, n, n f nx = x f n x f nx = n n!x c x c x c n x = c k f nx f n c k k =,,, n f n x k + < x < k

g n x = f nx n! = x x x n.4. 4 5 k =,,, n g n x k + < x < k n log g n x = log x j j= g nx = g n x n j= x j h n x = n j= x j g nx = g n xh n x n h nx = j= x j k =,,, n k + < x < k h nx lim x k+ + hx =, lim x k hx = h n c k =, k + < c k < k c k k + < x < c k hx > c k < x < hx < k g n x k + < x < k g nx x = c k g n c k k + < x < k

4 4 n 4 5 4 5 6 III 6 9 6. π 7. 6. 6 5 4. 9 5. 9 5 6. 7. 5 III 55

56. II fx = ax + a + bx + b + x fx fx f x = D D a = fx = bx + b + x fx x > fx b > f fx a fx x > a > f x = D i a = ii a > D iii a > D > i iii III x, y X, Y p, q Taylor expansion 58 l : mx + ny = n d P p s t p = s n + t d P l Q q q = s n + t d l A A p = q As n + t d = s n + t d sa n + ta d = s n + t d A n = n, A d = d A

.. 57 4 G l G P l P l P G G G P l P l m G P G l m i G P l A P A G C C l C C C loop G m P C C A G l ii G P m B P B G C C P C C l C C C G C m B P B C C G l i ii G l m P G P Q Q P Q Q Q l Q Q 4 P Self-Intersection G m P Q Q Q fx = ax + bx + cx + d x = p fx = fp + f px p + f px p + ax p f p = p = b y = fx a y = fp + f px p + ax p p, fp 4 Q l

58 ft C k x x t k x { } x t f k k t dt = f k t dt a k! a k! [ ] x t k x x = f k x t k t + f k+ t dt k! k! = x ak f k a + k! a x x x t k x f k x t k t dt f k+ t dt = a k! a k! k =,,,... n x a f t dt fx = x a n k= x t n n f n t dt = n! k= x a k f k a + k! x a a a x t k f k+ t dt k! x a k f k a k! x ak f k a k! x t n f n t dt. n! f n t M m x x t n f n t dt n! a x a x t n M dt = M n! n! x an, c x a x t n n!.. fx = n k= x a k k! x a x t n m dt = m x an n! n! f n t dt = f n c x a n. n! f k a + f n c x a n. n!. fx x = a Taylor expansion a = fx = n k= x k k! f k + f n c n! Maclaurin s expansion x n

.. 59 III z = t r t r y + z r x + y r z y y x z = t, r t y r t, r y x r y x = r y x = y St r St = r t r t x x dy = V r = r r r t r t Stdt r y dy < r V r 4 dz pz q = dz qz p d dzz + d p dqz + dq dpz + dpq d pq = i id dzz + id p dqz + idq dpz + idpq d pq = i d d d a = id d, b = idq dp, c = idpq d pq a c azz + bz + bz + c = ii d = d d d = d z pz q = z qz p z p z p z q = z p z q z q

6 5 III C P X = = P X P X EX 4 5 EX + X n = EX + EX n 6 B i 4 m 7 III Pt, ft f t tan θ P G n = sin θ, cos θ tan θ = f t π < θ < π cos θ = + tan θ, sin θ = tan θ + tan θ OQ = OP + n L L L L III

.. 6 8 III < x < < < < x x x e t dt < x e t dt < e t tdt < e dt < e x < ex x x et dt < e x x < ex et dt < ex x x < ex 6 < x < < fx < ex 6 < x < x < x dx < x e t dt < x e dt < x < x 6 < fx < ex 6 gx dx = e x dx = e, K n = n n e k n = n e e n k= 9 C 4 d 6 d

6. III y + x y x k x + k dx = x x x + k x x + k dx = x x x + k + k dx + k dx x + k x + k dx = x x + k + k dx x + k dx x x + k = log + x + k + C x + dx = x x + + logx + x + + C At St I p a pq q III y x x y fx fy gx = M

.. 6 x log x x log y x + y x y y log y y log x y + x y log y log x + x y a k >, p k > k =,,, n p + p + + p n = M = p a + p a + + p n a n fx = log M log x + x M x > fx x = M n n p k fa k = p k log M log a k + a k M k= n = p k log M p k log a k p k + p ka k M k= n p = log M log a k k + M M k= k= = log M log a p a p a n p n p k fa k k =,, n a k = M log M log a p a p a n p n p a + p a + + p n a n a p a p a n p n a = a = = a n p = p = = p n = n a + a + + a n n a a a n n

64 4 B 79 a b = a b a b AC =,,, AP = x + cos θ, y, sin θ AC AP = sin θ, sin θ, y x cos θ AC AP = y x cos θ + 8 sin θ ACP = AC AP = y x cos θ + 8 sin θ AC =,,, AP = x + cos θ, y, sin θ AC =, AP = x + cos θ + y + 4 sin θ AC AP = x + cos θ + y ACP = AC AP AC AP = y x cos θ + 8 sin θ y x cos θ

.. 65 5 Az Bz Cz A BC z z z z z z z z z + = z z z z w = z + z + z w B CA C AB w ABC C z = w = z z z B C z z z z z z z z z = z z z z w w w 6 A O n, k 45 n + k 45 n k O n, k nc n+k n C n k O n, k y = k Aa, k a n A n, k + A n, k y = k

66 Random walk[ ] 9, 8 84 = n 4 5 6 7 8 9 9 8 7 9 6 8 5 7 6 4 6 8 5 84 4 5 56 5 6 k 6 7 5 6 4 5 56 5 84 4 6 8 5 7 6 6 8 7 9 8 9 n 4 5 6 7 8 9 8 9 8 9 8 9 k 6 9 6 4 7 4 5 55 5 8 4 6 8 5 7 6 6 8 7 9 8 9 ii =,,, 7 T i O n, k n C n+k O n, k, k, k n, k O n, k + n k nc n+k n C n+k = n C n+k n k n nc n+k = k n nc n+k O n, k k n k n