[ ], IC 0. A, B, C (, 0, 0), (0,, 0), (,, ) () CA CB ACBD D () ACB θ cos θ (3) ABC (4) ABC ( 9) ( s090304) 0. 3, O(0, 0, 0), A(,, 3), B( 3,, ),. () AOB () AOB ( 8) ( s8066) 0.3 O xyz, P x Q, OP = P Q = /, P Q V.. () P x, y, z. () P xy (x > 0, y > 0), Q O. (), P OQ = θ, P Q x, y, θ. (b) P Q S, S. (3) V. ( ) ( s0703) 0.4 R 4 W = x x x 3 x 4 x + x x 4 = 0 x + x + x 3 = 0, W =,,., x, x, x 3 x, x, x 3 R 4 R 4. () W, W. () W + W, W W. ( 7) ( s700) 0.5 A(3, 0, 0), B(0,, 0), C(0, 0, ) α () α xy θ ( 0 < θ < π/ ) cos θ () ABC (3) α xy ( 5 ( s050)
0.6 A = ( 0 ),. () A. () x n+ = x n+ + x n (n ) {x n }, x, x., (x, x ) = (0, ) e, (x, x ) = (, 0) e, {x n }, T, e, e T A. (3) A n {x n }, (x, x ) = (, ) {x n }., A n A A A. 0.7 3 O, A, B, ( 5) ( s50) = OA, b = OB., (, b ), (= (, )), =, b = b. O, A, B P, p, q OP = p + q b ( ).,. () P AB A, B, ( ) p, q. () P AB, OP ( ). (3) AOB AB P, OP ( ). (4) A OB P, OP ( ). (5) OAB, (, b ). 0.8 f(x) f (x), f (x) = f(x + x) f(x) lim x 0 x ( 0) ( s0336)., f(x) = x 3 f (x) = 3x. ( ) ( s803) 0.9 r = (x, x, x 3 ) (x i ) t, A(r, t), B(r, t), E(r, t),. () A A i (, i =,, 3), () A (b) A (c) A A. () A = ( A) A, i. (3) B = c E t E = c, c. B t, E B. (4) B = A A = 0, () (3), A. ( 8) ( s8807)
0.0 D = {(x, y) (x, y) (0, 0)} f(x, y), f(x, y) = f xx (x, y) + f yy (x, y)., x = r cos θ, y = r sin θ (r > 0, 0 θ < π), z = f(r cos θ, r sin θ).,., D f(x, y),. () z r, z θ r, θ, f x, f y. () z rr + r z r + r z θθ = f. (3) f(x, y) = y x + y log(x + y ), f(x, y)., ; ( 8) ( s890) 0., 3 A(, 0, 0), B(0,, 0), C(0, 0, ) α, O(0, 0, 0) α H.,. () AC, BC. () α. (3) H. (4) H ABC. 0. ( 7) ( s7004) (), b, c = b = c =, + b + c = 0 b b c c () A(, ), B(b, b ), C(c, c ) A, B, C P (x, y) z = AP + BP + CP z x z = AP AP + BP CP + BP CP y (3) P z ()() AP B, BP C, CP A ( 6) ( s0605) 0.3 > b > 0 y A(0, ), B(0, b) P x AP B P 0.4 0 < t <, 4 ( 0) ( s00) A(t, t, 0), B(t, t, 0), C( t, 0, t ), D( t, 0, t ),. () AB E. () CDE S t. (3) ABCD V t. 3
(4) V t. ( 0) ( s003) 0.5 xy O r. A(r, 0), B AOB= 30.. () OAB x V (r). () AB x S(r). ( 6) ( s604) 0.6 xy, O C. x T (t, 0), 0 < t <. T l C A, B, l O. OAB S,. () l O h, h t. () h S. (3) S f(t) t. ( 8) ( s80) 0.7 T, l, g, T = π l/g. l, g l, g T T, T. T T = ( l l g ) g ( 7) ( s7405) 0.8 xy,, 3,., x y + = 0, x + y 4 = 0, 3 x + 3y + 3 = 0. (),, 3 y. (),, 3 xy. (3) () A, 3 B, 3 C, A, B, C. (4) A, B, C ABC. ( ) ( s65) 0.9 x, y, z O(0, 0, 0), A(, 0, 0), B(0,, 0), C(0, 0, ) OABC OA =, OB = b, OC = c () ABC G, OG = g g, b, c () ABC (3) OABC 0.0 ( 8 ( s08707) () () 3, 5 4, 7 7, 9 0, 3, 4
(b), 4, 6 3, () XOY = π/4, P P OY P OX P P OY P 3, P 4, P 5, () (b) Y P P 3 P 5 O X P 4 P (3) n= n + 4n + 3 ( ) ( s704) 0. ABC O r, b, c BAC, CBA, ACB A, B, C () B BA = r BA C = BAC, BAC = BOC () ABC S S = r (sin A + sin B + sin C) O ABC A A c O b B C ( ) ( s70) 0. P(x, x ) y = x A(, ), B(3, 9) APB 0.3 i, j, k OA = i + bj + ck OB = bi + cj + k A, B, C ( 4) ( s4304) OC = ci + j + bk 5
() AB () AB AC (3) ABC 0.4,. () α <<, ( + α) 3 + 3α. () cos(θ + θ) θ, θ 5. ( 8) ( s8305) (3) ( θ), cos(6 )., θ. ( ) ( s309) 0.5 O 3, A(0,, ) B(3, 3, 0).,. () AOB = θ, cos θ. () AB. (3) 3 O, A, B.,. (4) AOB. ( 5) ( s530) 0.6 xyz 3 O(0, 0, 0), A(,, ), B(3, 4, 3).,. () AOB = θ, cos θ. () OAB. (3) OAB. ( 6) ( s6303) 0.7 () y = f(x) x =, x = b x,. b f(x)dx = lim n k= n f(x k ) x, f(x) [, b], x = (b )/n, x k = + k x., () (), lim n 0 xdx.,. ( n + + n + + n + 3 + + n ). ( 7) ( s730) 0.8 3 xyz A(0,, ), B(, 0, ), C( cos t, sin t, 0).,, 0 t < π.,. () ACB = θ, t = π, cos θ =. () (), ABC. (3) ABC G. C t, cos t sin t. AG AC 6
( 8) ( s830) 0.9 U(x, y, z).,. () F = U. U(x, y, z) = exp [ (x + y + z ) ] () F = 0, x, y, z. (3) F.,. = ( x, y, ) z ( 6) ( s6304) 0.30 X A, B, C,. A B = (A \ B) (B \ A), B C = (B \ C) (C \ B), A C = (A \ C) (C \ A)., A \ B A B. ( 4) ( s43807) 0.3 OAB OA =, AB = b OAB = α ( 3) ( s3407) 0.3 V,, W V. x V V = W W (W W ) x = w + w (w W, w W ).. x w V V f, () x, x V, α, α R,. f(α x + α x ) = α f(x ) + α f(x ) () f f = f., f f f f. (3) x, y V, f(x), y = x, f(y). ( 0) ( s04504) 0.33 P, P (x, y ), (x, y ). P, P O P OP = θ, cos θ. 0.34 S ( 9) ( s9468) () A, B S P AB AP B () S n (n 3) ( 5) ( s5470) 0.35 x, y, z i, j, k, r = xi + yj + zk. C : r = cos θi + sin θj + k (0 θ π),. 7
() C. () ϕ = 4 x y C. (3) C S, A A = + z x i + z xy j + (x z y)k. ( ϕ) A + ϕ( A) S. 0.36 3 = (,, ), b = (,, 3) R 3,. () c = (,, ), c = p + qb p, q. () d = (,, ), b, b. (3), b, c, d, b, c, d. ( 8) ( s84704) ( ) ( s4908) 0.37 (fig.),. D C D F C F C E B E E θ A B A A fig. fig. fig.3 () fig. BAE θ AE fig.. ABE. () fig. ABE ABECD tn θ. (3) fig. BAD AF fig.3. AECF. ( 7) ( s7500) 0.38, [ ] xy O(0, 0) A(, ), B(x, y)., M =., OA, OB OA, OB, OA OB x y < OA, OB >. y () < OA, OB > OA, OB θ. () OB < OA, OB > x, y (3) OAB S, S = OA OB sin θ., 4S = M. M, M. (4) OAB, B. O θ A(x, y ) B(x b, y b ) ( 7) ( s75004) x 8
0.39, 3. 0, P. P 9, P. P, P 8, P 3.. () P, P, P 3. () P, P, P 3, P P P 3. 0.40 () rctn x x = 0 Tylor 0 ( 7) ( s75005) () BC = 0, AC =, C = π ABC B ( 6) ( s650) 0.4, A (, 0, ), B (, 0, ), C (3, 3, 5) ABC. 0.4 O(0, 0), A(, ),. ( 7) ( s7544) () A x B. () OA OB, OA OB. AOB = θ cos θ sin θ. (3) A 45 C. 0.43. (), b, c, + b + c = 0, = b = ( ) ( s5409) c ( + 3 ) 0. (), b θ., θ π θ π. OA = (,, 3), OB = (,, ), OC = (, 0, ), O ABC D. OD ABC S, OABC V. ( 4) ( s45405) 0.44 3 x, y, z. A(,, ), B(3, 3, 3 ),. () O, AOB. () O AB, P. AP P B., P. ( 4) ( s4544) 9
0.45 x y A(, ) B(, ),. () A y P. () O, OA OQ OA = OQ, Q. (3) OA OB., AOB θ, cos θ. ( 4) ( s4549) 0.46 O xyz, A(, 0, ) B(,, 0.. () OAB AOB., OAB S. () A, B l (3) O R x + y + z = R l Q, R Q. [ 0.47 OP = x y ] [, OQ = x OP OQ. () y ]. ( 7) ( s75404), OP OQ θ, cos θ. () OP Q S. [ 0.48 O P, Q OP = () OP OQ. (), OP Q S. 0.49 P (x, y, z), A. OP OQ. ( 7) ( s75408) ] 3, OQ = 3. ( 8) ( s85408) A A = (A x, A y, A z ) = (x, y, z), S,. () S n ( n = ). () S A n ds. S (3) S div A dv = A dv. ( 8) ( s85508) 0.50 AB = AD = DC =, ABC = DCB = θ ABCD S, θ S θ A D S B θ θ C 0
( 6) ( s66)