9 ( ) 9 5 I II III A B (0 ) 5 I II III A B (0 ), 6 8 I II A B (0 ), 6, 7 I II A B (00 ) OAB A B OA = OA OB = OB A B : P OP AB Q OA = a OB = b () OP a b () OP OQ () a = 5 b = OP AB OAB PAB a f(x) = (log ax) x (x > 0) () f(x) x = b 5 a b e () () a f(x) dx () () a b y = f(x) (0 < x b) x = b x C : y = sin x () sin x + cos x sin x () n = 0,,, C nπ x (n + )π x n a n S n = a k lim n S n () n = 0,,, C 0 x nπ x V n lim n V n k=0
α β α =, β =, α β = β α () β α ( β α) 8 () α + β () n 8 α n + β n n 5 f(x) = ( ) log x x cos (x ) () f (x) = 0 x () f(x) = 0 () t t f(x) dx t 6 a A(0, a) B(a, 0) C(a, a) OB P(t, 0) BC Q y a A C APQ = 90 Q 0 < t < a O P a B x () PBQ S a t () S t a 7 P O () 7 P () 6 P 7 P
8 O C P C A(a, b) () C () C B(c, d) ac + bd 0 OB C O Q Q () B Q () A B AOQ = θ APQ OQ θ
() P A B : OA = a OB = b OP = OA + OB + = a + b = a + 6 b B( b) Q B ( b) P () Q AB () OP = 8 a + b = 8 OQ O A( a) A ( a) OP OQ = 8 () OP = ( a + b) AB = b a OP AB ( a + b) ( b a) = 0 a a b + b = 0 a = 5 b = 5 a b + = 0 a b = OAB OAB = a b ( a b) = 5 = () OAB : PAB = OQ : PQ = : 5 PAB = 5 OAB = 5 = 5 6
5 () f(x) = (log ax) x (a > 0) f (x) = (log ax) (log ax) x (log ax) x = (log ax) ( log ax) x f (x) = 0 x = a, e a f(x) x (0) a e a f (x) + 0 + 0 f(x) 0 x = b 5 e 7a e b = e a, 5 e = 7a e a =, b = e () a = (log x) f(x) dx = dx = x (log x) (log x) dx = (log x) + C (C ) () () () S S = e f(x) dx = [ ] e (log x) = 8
6 () ( ) sin x + cos x = {e x (sin x + cos x)} = e x (sin x + cos x) + e x (cos x sin x) = e x sin x = sin x ( ) sin x + cos x = sin x () C : y = sin x sin x a n = sin x + cos x dx = + C (C ) nπ x (n + )π y 0 = (n+)π nπ 0 < e π < lim S n = lim n n [ sin x sin x + cos x dx = e (n+)π + e nπ = n k=0 + eπ e + e π a k = lim n e π + eπ = (n+)π e π n (e π ) k k=0 ] (n+)π nπ = + eπ + eπ = eπ e π e π e π e π = (e π ) n e π (e π ) () V nπ n π = = 0 nπ 0 ( ) sin x dx e x sin x dx = nπ 0 e x ( cos x) dx (e x sin x) = e x ( sin x + cos x) (e x cos x) = e x ( sin x cos x) {e x (sin x cos x)} = e x cos x e x cos x dx = e x (sin x cos x) + C (C ) V [ ] nπ n π = 8 e x ( sin x + cos x) = 0 8 ( e nπ ) π lim V n = lim n n 8 ( e nπ ) = π 8
7 () O α β A B ABC OA = OB = AB = AOB = π β α = β α = β α β α = ( cos π + i sin π ) = + i ( ) 8 β = ( ) 8 (cos π + i sin π) = 6 α w = β α α = β = α β = β α =, α β α = w =, w = w = ( w)( w) = w w + w = w + w = Re(w) = w + w = w Im(w) = w = w = + i () () β = ( + i)α α + β = α + ( + i)α = α + i = + = 5
8 () () β = α( + i) n 8 α n + β n = α n + α n ( + i) n ( + i) n = = α n + ( + i) n = + ( + i) n { ( cos π + i sin π )} n cos π + i sin π ) { ( cos π + i sin π ) } n 8 8 = ( ) n ( = ( ) n ( + i α n + β n = ) = ( ) n ( + i) (cos π + i sin π) n 8 α n + β n = + ( ) n ( + i) = + ( ) n + ( ) n i = { + ( ) n } + {( ) n } = + ( ) n + n + n = + n+ + n + n+ + n
9 5 () f(x) = ( ) log x x cos (x ) x x = e θ (θ 0) f(x) = {cos(log x) + } x f(x) = e θ (cos θ + ) ( ) θ dx dθ = eθ f (x) dx dθ = e θ (sin θ + cos θ + ) ( ) f (x) = e θ (sin θ + cos θ + ) f (x) = 0 sin θ + cos θ + = 0 (θ 0) θ ( sin θ + π ) = θ = π x x = e π () g(θ) = f(x) ( ) g(θ) = e θ (cos θ + ) g(θ) = f(x) θ ( ) g (θ) = f (x) dx dθ = e θ (sin θ + cos θ + ) = ( {sin θ + π ) + } n 0 g(θ) g((n + )π) = 0 g((n + π)) = e (n+ )π
0 g(θ) g( π) = e π e π < e < < 0 g( π) < 0 θ π g(θ) = θ 0 g(θ) 0 θ π g(0) = g(π) = 0 g(θ) = θ 0 f(x) = 0 () f(x) = cos(log x) + t x x t f(x) dx = [ sin(log x) + log x ] t = sin(log t) + log t
6 () θ = PAO = QPB tan θ = OP AO = t a () f(t) = t (a t) (0 a f(t) BQ = PB tan θ = (a t) t a = t (a t) a S = PB BQ = t (a t) (a t) = t a < t < a) f(t) = a (t at + a t) (a t) a f (t) = a (t at + a ) = (t a)(t a) a t (0) a (a) f (t) + 0 f(t) 7 a t = a 7 a t a t a t t + (a t) + (a t) t(a t)(a t) t(a t) 8 7 a t a (a t) 7 a t = a t = a t t = a S 7 a
7 () 7 ( ) 7 7C = 5 8 () x y P (0, 0) (7, ) (i) (6, 0) 5 y ( ) (0, 0) (, ) (, 0) (, ) (, 0) (5, ) (6, 0) (0, 0) (, ) (, 0) (, ) (, 0) (5, ) (6, 0) (0, 0) (, ) (, 0) (, ) (, 0) (5, ) (6, 0) (ii) (6, ) (5, ) y 0 x (0, 0) (, ) (, 0) (, ) (, 0) (5, ) (6, ) (0, 0) (, ) (, ) (, ) (, 0) (5, ) (6, ) y O 5 6 7 x (i) (ii) 5 ( ) 7 = 5 8
8 () P OA ( ) a P, b C P ( x a ) ( + y b = ) B y O C C θ P Q A x () B(c, d) OB dx cy = 0 AQO C AQO = 90 AQ A(a, b) c(x a) + d(y b) = 0 cx + dy = ac + bd B C : x + y = c + d = Q x = ( c(ac + bd) Q, c(ac + bd), y = d(ac + bd) ) d(ac + bd) () PO = PQ POQ = PQO = θ APQ = POQ + PQO = θ + θ = θ () OQ = AO cos θ = cos θ