( )5 ( ( ) ) 4 6 7 9 M M 5 + 4 + M + M M + ( + ) () + + M () M () 4 + + M a b y = a + b a > () a b () y V a () V a b V
n f() = n k= k k () < f() = log( ) t dt log () n+ (i) dt t (n + ) (ii) < t dt n+ n + (iii) + f() f( ) log 5 n+ (n + ) () log 4 z = cos + i sin z α = z + z () α () () α
5 n m S (S ) (a) (b) (c) (a) S j = n m + (b) j t (c) t S t j t S (d) j (e) j n (b) j > n () n = m = 6 (b) t 4 6 5 S () n = m = 6 S 8 7 5 9 (b) t () S (b) 6 a b c A(a,, ) B(, b, ) C(,, c) () P AP ( BP + CP) = P Q () () Q A B C () () P ABCP
4 7 m (cm) σ =.5(cm) () 7.cm.cm m = (cm) () m n 95% m [9.8,.79] n () P ( U u ) u.....4.5.6.7.8.9.6.445.446.4474.4484.4495.455.455.455.455.4545.7.4554.4564.457.458.459.4599.468.466.465.46.8.464.4649.4656.4664.467.4678.4686.469.4699.476.9.47.479.476.47.478.4744.475.4756.476.4767..477.4778.478.4788.479.4798.48.488.48.487
5 8 O r( ) θ (r, θ) ( () C : r = cos(π+θ) C : r = (cos θ+) π < θ < π C C () P P E (r, θ ) (r, θ ) (, ) OEP OP Q Q P P P P Q EOP = P OQ () O P P OP RP R P P P P (r, θ ) (r, θ ) r = r = r P P (4) P (r, θ) k kp k (kr, θ) k < ( k r, θ+π) () C C V k(v (V V )) C k ) 9 E P () QP = E Q () R = S = P R a a a a 4 b b b b 4 c c c c 4 () A B P AX = P B X AX = B (4) y z y + z = a + 4y 5z = b 6 5y + 4z = c A AX = B P (5) (4)
6 () M + M =, ( + ) = +, ( + ) = + + + M, +, +, + + ( ) + + M () ( ) M =, ( + ) = +, ( + ) = +, ( + + ) = + +, ( + ) = +, ( + )( + ) = + + +, ( + )( + ) +, ( + )( + + ) + M + + + + M (), +, + +, + +, + + () 4 + + M (a b ) 4 + + ( + )( + a + b + ) = 4 + (a + ) + (a + b) + (b + ) + a +, a + b, b + a b 4 + + M M M M () 4 + + ( + + ) 4 + +
7 () y = a + b + y = ay y a + b = ( ) y ( ) ( ) 4a( a + b) = b = a + 4a ( ) y = a = a a < a < a > = ± 4a a b a >, b = a + ( 4a, ± ) 4a, a () V b V = π dy = π b (b y) d = π [ by ] b a a y = b π a = π ( a + ) = π a 4a a (4a + ) () () dv da = π V a > { a 4 (4a + ) + } a (4a + ) 8a = π a 4 (4a + ){ (4a + ) + 6a } = π a 4 (4a + )(4a ) a = a ( ) dv + da V 4 π 9 b = 4 V 9 π
8 () t + n k= < ( ) n + t k dt = k= [ n k k= n k= ] k k = t k = tn t t dt [ log( t) = log( ) ] t dt t dt f() = n k= k k < () (i) f() = log( ) t dt t dt [ dt = + (n + ) ] = n+ (n + ) (ii) dt ( < ) t = u t dt = du t u t dt = ( u) n u n ( du) = ( )n+ + u + u du t dt = u n + u du u n du = n+ n +
9 (iii) () f() + log( ) = t dt, f( ) + log( + ) = t dt + f() f( ) log = t dt t dt t dt + t dt = ( t) n + t ( dt) + t dt = = + t dt + dt + t dt tn dt = 5n+ (n + ) + f() f( ) log 5 n+ (n + ) ( ) ( ) + ( ) f( ) f( ( )) log 5 n+ ( ) (n + ) + f() f( ) log 5 n+ (n + ) ( )
() n = = 5 n+ (n + ) = 5 ( 8 f() f( ) = = k= ) 4 = 5 648 < 5 5 = k k { + log + = log + ( ) n = 4 = 56 8 log 56 8 5 n+ (n + ) = ( 4 f() f( ) = = k= k= ( ( ) k = k ) } = 56 8 = log < ) 5 = 486 < 4 4 ( ) k k k { + k= ( k ) } = = 56 8 log + = log + = log ( ) 56 8 log < 4 ( + ) ) ( +
4 () z = cos + i sin α = z + z α = (z + z) = z + z + (z + z) = (cos 6 + i sin 6 ) + (cos 6 i sin 6 ) + α = + α α α = α = α = cos cos θ = 4 cos θ cos θ θ = cos θ = cos θ = α ( α ) = 4 α () f() = f () = = ( + )( ) f() f () + + f() y = f() α α = y O f( ) = > f() = < f() = p (p q p q ) q = ( ) ( ) p p q q = p = q (p + q) p q q = p p = p(p ) = p ( ) ( ) < < < < < < p p = p
() α + b + c = f() + b + c + s t + u (s t u ) f() = ( + b + c)( + s) + t + u = α tα + u = t α = u t t = u = f() = ( + b + c)( + s) s f() = () α f() = = cos θ f( cos θ) = ( cos θ) cos θ = (4 cos θ cos θ) = cos θ θ 8 f( cos θ) = θ θ = 6,, 4 θ =,, 4 = = cos, cos, cos 4 cos.997 =.8794 cos (.76) =.47 cos 4 (.766) =.5
5 () n = m = 6 j = n m + = 6 + = 5 j t S 5 4 4 j = 5 t = 4 S t 4 6 4 j = j + t = S S 7 6 64 j = j + t = 6 S S 8 684 j = j + t = S j 9 684 j = j + t = S S 5 5684 j = j + t = 5 S S S 5684 () n = m = 6 j = n m + = 6 + = 5 ( ) j t S 5 j = 5 t = S t 6 5 5 j = j + t = 5 S S 7 7 75 j = j + t = 7 S S 8 75 j = j + t = S S 9 5 759 j = j + t = 5 S 5 j 8 8759 j = j + t = 8 S S t, 5, 7,, 5, 8 () t T (a) T j = n + (b) j (c) j S S j T j S S T (d) j > n m + (b) j n m +
4 6 () A(a,, ) B(, b, ) C(,, c) (a, b, c ) OA OB = OB OC = OC OA = AP ( BP + CP) = ( OP OA) ( OP OB OC) = OP ( OA + OB + OC) OP = OP OA + OB + OC OA + OB + OC = 6 6 OQ = OA + OB + OC 6 OQ = 6 (9 OA + OB + 4 OC ) = 6 (9a + b + 4c ) OP OQ = OQ QP = 9a + b P Q r = + 4c AP ( BP + CP) = 6 AP ( OP OB OC) = BC : D 9a + b + 4c 6 OD = OB + OC AP ( OP OD) = AP DP = P AD Q r OQ = ( OA + OD) = 6 ( OA + OB + OC) r = OQ OA = OA + OB + OC 6 = 9a + b + 4c 6 () OQ = OA + OB + OC = OA + AB + AC 6 6 Q A B C () AB = ( a, b, ) AC = ( a,, c) AB = a + b AC = a + c AB AC = a ABC = AB AC ( AB AC) = a b + b c + c a QP ABC ABCP ABC r = (a b + b c + c a )(9a + b + 4c ) 6
5 7 () X X N(,.5 ) Z = X.5 Z N(, ) ( 7. P (7. X.) = P Z.5 = P (.8 Z )..5 = P (.8 Z ) + P ( Z ) = P ( Z.8) + P ( Z ) =.464 +.477 =.94.94 =.587 () 95% m [9.8,.79].96.5 n = 9.8, +.96.5 n =.79 =. [cm] n = 6 () y y = (8 + ) =.8 = (8 + ) =.8 y = (6 + 4 ) =.6 y = (6 + 4 ) =.6 y = (6 + + ) =.6 σ = =.8.8 =.6 =.4 σ y = y y =.6.6 =.4 σ y = y y =.6.8.6 =. ) σ y σ σ y =..4.4 = 6 4 (.6)
6 8 () C (, y) y = r cos θ, y = r sin θ C y θ π O = cos(π + θ) cos θ = cos θ = cos θ = cos(θ π) y = cos(π + θ) sin θ = sin θ cos θ = sin θ = sin(θ π) π < θ < π < θ π < π C (, ) ( ) C C (, y) y = r cos θ, y = r sin θ = (cos θ + ) cos θ = cos θ + cos θ = + cos θ + cos θ (, y) O θ 4 y = (cos θ + ) sin θ = sin θ cos θ + sin θ C = sin θ + sin θ π < θ < π C C : r = cos(π + θ) C : r = (cos θ + ) cos(π + θ) = (cos θ + ) cos θ = π θ π θ = π, 4π ) (, π ), (, 4π r = a( + cos θ) (cardioid)
7 () OEP OP Q OE : OP = OP : OQ : r = r : OQ OQ = r r y Q : P P P (r, θ ) P (r, θ ) Q θ + θ P P (r r, θ + θ ) () OP = OP = r P y P M OM = r cos θ θ OR = OM = r cos θ θ R (4) i) V θ + θ ( P P r cos θ θ ( ) π, V V = V (V V ) = (, k(v (V V )) = (k, π) O O r θ, θ ) + θ π + π ) = ( 4π cos π, θ r E(,) P (r, θ ) θ θ (, π + 4π C () k = ( ) 4π ii) V, V V = V (V V ) = (, k(v (V V )) = (k, π) 4π + 4π ) = ( 8π cos 4π, (, 4π + 8π M P (r, θ ) ) 4π ) = (, π) R : P P ) 8π ) = (, π) = (, π) i) C () k = i),ii) k =
9 () P = Q = () S = P R = = () () QP = E (4) a a a a 4 b b b b 4 c c c c 4 a a a a 4 b b b b 4 a + c a + c a + c a 4 + c 4 P AX = P B Q 4 5 6 5 4 QP AX = QP B AX = B y z = a b c 4 5 6 5 4 4 5 y z y z = = a b a + c a b c 8
9 (5) a 4 5 b 6 5 4 c a 4 5 b 6 5 4 c 6 a 4 a + b a + c a a + b a + c a + b a + b ( ) a + b + c (4) a + b + c = c = a b z = t (t ) ( ) t = a + b, y t = a + b = t + a + b, y = t + a + b, z = t (t )