(2000 )

(000)

< > = = = (BC 67» BC 1) 3.14 10 (= ) 18 ( 00 ) ( ¼"½ '"½ &) ¼ 18 ¼ 0 ¼ =3:141596535897933846 ¼ 1 5cm ` ¼ = ` 5 = ` 10 () ` =10¼ (cm) (1) 3cm () r () () (1) r () r 1 4 (3) r, 60 ± 1

< > µ AB ` µ ± = ` (1) µ =360 ± ¼ 360 ± =¼ ¼ ; 3:14 () µ =180 ± ¼ 180 ± = ¼ (3) µ =90 ± 1 4 ¼ 90 ± = ¼ µ = 180± ¼ ; 57:3± 360 ± ; 180 ± ; 90 ±

< > 1 1 ( ) 0 ± 360 ± (1) 40 ± =360 ± +60 ± =¼ + ¼ 3 () =7 3 ¼ () () 510 ± = 360 ± 150 ± = ¼ 5 6 ¼ () = 17 6 ¼ () (1) 540 ± () 40 ± (3) 600 ± (4) 390 ± (5) 840 ± (6) 405 ± 3 1 () (1) r ` () r S ` = S = 3

< 3> µ r OAB ` OAB S (1) µ =¼= 360 ± ` ` =¼r S S = ¼r () µ = ¼= 180 ± (1) ` = ¼r S = 1 ¼r (3) µ = ¼ = 90± (1) 1 4 ` = 1 ¼r S = 1 4 ¼r µ ` OAB S µ r ` = S = 4

< > y =sinx y =cosx y =tanx (1) y =sinx () y =cosx (3) y =tanx 5

< > A; B; C y = A sin(bx + C) ³ sin x + ¼ =sinxcos ¼ +cosxsin ¼ cos ¼ =cos90± =0; sin ¼ =sin90± =1 ³ sin x + ¼ =cosx y =cosx y =sinx y =cosx y =cosx y =sinx x ¼ cos x sin x ¼ sin x cos x ¼ y = A sin(bx + C) Bx + C ³ y =sin x ¼ 6

< > y =sinx y =sinx y =sinx y =sinx y =sinx y x y = 3sinx 7

< > y =sinx ¼ x y =sinx ¼ y =sin(x) ¼ y =sin(x) ¼ y =sin(3x) 8

< > p y = p x p x 0 x y p x ( x = 0) p 0 y y = 0 y y = p x +1 p 0 x +1 = 0 ) x = 1 : x = 1 p = 0 : y = 0 (1) y = p x + () y = p x 1 9

< > y = p 3 x p 0 3 x = 0 ) 3 = x : x 5 3 p = 0 : y = 0 y = p x 1 p 0 x 1 = 0 ) x = 1 : x = 1 p = 0 p 5 0 : y 5 0 (1) y = p x +1 () y = p x + 10

< 3> y = p x +1 x +1= 0 : x = 1 p p 5 = 0 : y 5 y = 1+ p x x = 0 := x p = 0 1+ p = 1 : y = 1 (1) y = 1+ p x +1 () y =1 p x 11

< > y = 1 x x =0 0 0 : x 6= 0 y = 1 x xy =1 y 0 y = 1 x 6=0 y 0 : y 6= 0 y = 1 x 0 x 6= 0 : x 6= : y 6= 0 y = 1 x x + y = 1 x +1 1

< > y =3+ 1 x 6= 0 : x 6= 1 6=0 y 3= 1 x 6=0 y 3 6= 0 : y 6= 3 y = 1 x x +y +3 x x =x y =3 x =;y =3 y =3+ 1 x y =+ 1 x +1 13

< 1> y x y = f(x) x y x y y = f(x) x y f(x) =+ p x 1 p 0 x 1 = 0 f(x) x = 1 y =+ p x 1 p = 0 y = f(x) =+ 1 x 1 0 x 1 6= 0 x 6= 1 f(x) 1 y =+ 1 x 1 1 6=0 y 6= x 1 (1) f(x) = p x 1 () f(x) =+ p 1 x (3) f(x) =1+ 1 x + 14

< > f(x) =log 4 (x ) =log 4 =4 > 0 = x > 0 f(x) x> y =log 4 (x ) x =4 y y f(x) = x +1 x f(x) y = x +1 x > 0 x +1> 1 y>1 (1) f(x) =log (1 x) () f(x) = x 1 (3) f(x) = 3 x 15

< 3> 1 f(x) =3+sinx x f(x) 1 5 sin x 5 1 5 sinx 5 3 5 3+sinx 5 3+ 1 5 3+sinx 5 5 1 5 y 5 5 f(x) =1+3cos(x) cos x x f(x) 1 5 cos(x) 5 1 3 5 3cos(x) 5 3 1 3 5 1+3cos(x) 5 1+3 5 1+3cos(x) 5 4 5 y 5 4 (1) f(x) =3+sinx () f(x) =1+cosx (3) f(x) =+3sinx (4) f(x) =+4cos(3x) 16

< 4> 1 µ tan µ = Y X µ 90 ± 90 ± x 0(X =0) 0 tan µ 90 ± ; 70 ± ; 450 ± ; tan µ µ 6= ¼ n¼ (n =0; 1; ; 3; ) y =tanx 5 5 f(x) =tan(x) tan( ) 6= ¼ n¼ (n =0; 1; ; ) x 6= ¼ n¼ x 6= ¼ 4 n ¼ f(x) ¼ 4 n ¼(n ) (1) f(x) =tan(3x) () f(x) =tan(¼x) 17

< > y = f(x) x y ( ) x 1 6= x f(x 1 ) 6= f(x ) y = f(x) 1 1 f(x) =x 1 y = f(x) 1 1 f(x) =x y = f(x) 1 1 x 1 = 1 ;x =1 f(x 1 )=f(x )=1 ( ) x 1 ;x 1 1 1 1 1 (1) y =3x () y = x 3 x (3) y = 1 x (x 6= 0) 18

< > f(x) 1 1 y b b = f(a) x a 1 a = f 1 (b) f(x) =x 1 y = f(x) 1 1 b = f(a) f(a) =a 1 b =a 1 a a = 1 b + 1 a = f 1 (b) f 1 (b) = 1 b + 1 f(x) y = f(x) 1 1 f 1 (b) b (1) f(x) =3x () f(x) = 1 x + (3)f(x) =p x 19

< > y = f(x) 1 1 y b x f 1 (b) y = f 1 (x) y = f(x) f(x) =x +1 b = f(a) () a = f 1 (b) b =a +1 () a = 1 b 1 = f 1 (b) f 1 (x) = 1 x 1 y = f(x) y = f 1 (x) y = x f(x) f 1 (x) (1) f(x) =3x + ()f(x) = 1 x (3) f(x) = p x 0

< > y = f(x) 1 1 f f 1 f f 1 f(x) =x +1f x = 0 y = f(x) 1 1 b = f(a) () a = f 1 (b) b = a +1 () a = p b 1=f 1 (b) (a = 0 ;b= 1) (a = 0 ;b= 1) b x f 1 (x) = p x 1 x = 1 y = f(x) y = f 1 (x) y = x f(x) =(x +1) f x = 1 y = f(x) 1 1 1

< > y = f(x) 1 1 y = f(x) (a ; b) b = f(a) () a = f 1 (b) (b ; a) y = f 1 (x) y = f 1 (x) y = f(x) y = x f(x) =3 x b = f(a) () a = f 1 (b) b =3 a () a =log 3 b = f 1 (b) f 1 (x) =log 3 x y =log 3 x y =3 x y =log 3 x y =3 x y = x µ 1 x y = y =log1 x

< > y =sinx 1 5 y 5 1 ¼ 5 x 5 ¼ 1 1 y =sinx ¼ 5 x 5 ¼ ; 1 5 y 5 1 y =sin 1 x y =arcsinx 1 5 x 5 1 ; ¼ 5 y 5 ¼ y =sin 1 x y =sinx y = x y =sin 1 x a =sin 1 b () b =sina µ 1 sin 1 µ µ 1 µ =sin 1 () 1 =sinµ ¼ 5 µ 5 ¼ sin 1 µ µ = ¼ 6 sin 1 µ 1 = ¼ 6 Ãp! (1) sin 1 = () sin 1 Ãp 3! µ = (3) sin 1 1 = 3

< > y =cosx 1 5 y 5 1 0 5 x 5 ¼ 1 1 y =cosx 0 5 x 5 ¼; 1 5 y 5 1 y =cos 1 x y =arccosx 1 5 x 5 1 ; 0 5 y 5 ¼ y =cos 1 x y =cosx y = x y =cos 1 x a =cos 1 b () b =cosa µ 1 cos 1 µ µ 1 µ =cos 1 () 1 =cosµ 0 5 µ 5 ¼ cos µ 1 µ µ = ¼ 3 cos 1 µ 1 = ¼ 3 Ãp! Ã p! µ 3 (1) cos 1 = () cos 1 = (3) cos 1 1 = 4

< > y =tanx ¼ + n¼n ¼ <x<¼ 1 1 y =tanx : ¼ <x<¼ ; : y =tan 1 x y = arctan x : ; : ¼ <y<¼ y =tan 1 x y =tanx y = x tan 1 x a =tan 1 b () b =tana tan 1 ( p 3) µ µ =tan 1 ( p 3) () p 3=tanµ ¼ <µ<¼ tan µ p 3 µ µ = ¼ 3 tan 1 ( p 3) = ¼ 3 (1) tan 1 (1) = () tan 1 Ãp 3 3! = (3) tan 1 p 3 = 5

< > a 1 ;a ;a 3 ; ;a n a n n fa n g fa n g n a n n n!1 1 1 ; 3 ; 4 3 ; ; n +1 ; n n!1a n = n +1 n 1 1 ; 1 ; 1 4 ; ; µ 1 n 1 ; µ n!1a n = 1 n 1 0 1 ; 3 4 ; 5 6 ; ; n 1 n ; 6

< > fa n g n!1a n fa n g n!1 a n! lim a n = n!1 fa n g lim n!1 n +1 n µ =1; lim 1 n 1 =0 n!1 n!1 1 n! 0 µ n +1 lim = lim 1+ 1 =1; n!1 n n!1 n lim n!1 3n n +1 = lim n!1 3 + 1 n = 3 +0 = 3 n!1 n! 0; 1 n! 0 n +n 1+ lim n!1 3n +1 = lim n n!1 3+ 1 = 1+0 3+0 = 1 3 n. (1) n +1 lim n!1 n 1 = () lim n!1 n +3n +1 n +1 = 7

< > fa n g fa n g 1; 4; 9; 16; ;n ; n!1a n = n fa n g n!1 a n!1 ( a n! +1) lim a n = 1 ( lim a n =+1) n!1 n!1 ; 0; ; ; 4 n; n!1a n =4 n fa n g n!1 a n! 1 lim a n = 1 n!1 lim n!1 4 n = 1 (1) lim n!1 (n 3 + n) = () lim n!1 (5 n ) = 8

< 4> 1 1; 1; 1; 1; ; ( 1) n ; 1; ; 4; 8; ; ( ) n 1 ; n fa n g (1) lim a n =+1 n!1 () lim a n = 1 n!1 (3) (1) lim n!1 µ 4 n µ =+1 () lim 3 n = 0 (3) lim 3 n!1 4 n!1 (4) 4 3 ; 16 µ 9 ; ; 4 n ; 3 µ 4 n = 1 3 (1) 3 ; 9 4 ; 7 8 ; 81 µ n 3 16 ; ; ; () 3 ; 4 9 ; 8 7 ; 16 µ n 81 ; ; ; 3 (3) 3 ; 4 9 ; 8 7 ; 16 µ 81 ; ; 3 n ; (4) 3 ; 9 4 ; 7 8 ; 81 µ 16 ; ; 3 n ; (5) 3 µ n 3 ; 9 4 ; 7 8 ; 81 16 ; ; ; 9

< > fa n g (1) a 1 + a + a 3 + + a n + fa n g S n = a 1 + a + a 3 + + a n n S 1 ;S ;S 3 ; ;S n ; S ( lim S n =) n!1 (1) S a 1 + a + a 3 + + a n + = S S 1 + 1 4 + 1 8 + + 1 n + S n S n = 1 + 1 4 + 1 8 + + 1 + 1 n 1 n 1 S n = 1 S n = 1 1 4 + 1 8 + 1 16 + + 1 + 1 n n+1 S n =1 1 n n! 1 1 n! 0 1 n+1 S = lim S n = lim µ1 1 =1 n!1 n!1 n () S 1 3 + 1 9 + 1 7 + + 1 3 n + 30

< > 1 1 3 1 =0:333 3 3 4 11 4 =0:363636 3 6 11 () 0:333 =0:_3 ; 0:363636 =0:_3_6 0:513131313 = 0:5_1_3 0:_1_ =S S = 0:_1_ =0:111 = 0:1 + 0:001 + 0:00001 + µ µ 1 1 µ 1 3 = 1 +1 +1 + 100 100 100 n S n µ µ µ n 1 µ 1 1 1 1 S n =1 +1 + +1 +1 100 100 100 100 µ 1 1 µ 1 3 µ 1 n µ 1 100 S n =1 +1 + +1 +1 100 100 100 100 µ µ 99 1 1 n+1 100 S n =1 1 100 100 S n = 1 99 1 99 µ 1 n µ 1 n n!1! 0 100 100 ½ 1 S = lim S n = lim n!1 n!1 99 1 µ 1 n ¾ 99 = 1 100 99 = 4 33 n n+1 S =0:_7 =0:777 31

< > f(x) x a a f(x) x! a f(x)! lim f(x) = x!a x! a f(x) a x 1 lim x 3x 3x =10; lim x!5 x!1 x +1 = 3 ; lim h!0 h +3 h +1 =3 3 x + x (x 1)(x +) lim = lim x!1 x 1 x!1 (x 1)(x +1) = lim x + x!1 x +1 = 3 (3 + h) 3 lim h!0 h =lim h!0 (9 + 6h + h ) 9 h = lim h!0 (6 + h) =6 7x +3 (1) lim x! x +1 = x x () lim = x! x 4 x x 3 (3) lim = x!3 x 3 (4) lim h!0 (4 + h) 4 h = 3

< > (a + b) 3 =(a + b)(a + b) =(a + b)(a +ab + b ) = a(a +ab + b )+b(a +ab + b ) = a 3 +a b + ab + ba +ab + b 3 = a 3 +3a b +3ab + b 3 (1) (a + b) 4 =(a + b)(a + b) 3 =(a + b)(a 3 +3a b +3ab + b 3 ) = a 4 + a 3 b + a b + ab 3 + b 4 ³ () (a + b) 5 =(a + b) a 4 + a 3 b + a b + ab 3 + b 4 = a 5 + a 4 b + a 3 b + a b 3 + ab 4 + b 5 (a + b) n (a + b) 0 =1 1 (a + b) 1 =1 a +1 b 1 1 (a + b) =1 a + ab +1 b 1 1 (a + b) 3 =1 a 3 +3 a b +3 ab +1 b 3 1 3 3 1 (a + b) 4 = a 4 + a 3 b + a b + ab 3 + b 4 (a + b) 5 = a 5 + a 4 b + a 3 b + a b 3 + ab 4 + b 5 (a + b) 6 (a + b) 6 = a 6 + a 5 b + a 4 b + a 3 b 3 + a b 4 + ab 5 + b 6 33

< > f(x) x f(x) f(x) f 0 (x) f 0 (x) =lim h!0 f(x + h) f(x) h f(x) 1 f(x) =1 (1) 0 = f 0 f(x + h) f(x) 1 1 (x) = lim =lim =0 h!0 h h!0 h f(x) =x (x) 0 = f 0 f(x + h) f(x) x + h x h (x) =lim = lim = lim h!0 h h!0 h h!0 h =1 3 f(x) =x x 0 = f 0 f(x + h) f(x) (x + h) x (x) =lim = lim h!0 h h!0 h =lim h!0 x +xh + h x h = lim h!0 (x + h) =x 4 f(x) =x 3 x 3 0 = f 0 f(x + h) f(x) (x + h) 3 x 3 (x) =lim = lim h!0 h h!0 h =lim h!0 x 3 +3x h +3xh + h 3 x 3 h =lim h!0 (3x +3xh + h )=3x f(x) =x 4 f(x) (x 4 ) 0 = f 0 (x) = 34

< > 1 3334 f(x) =x 5 f 0 (x) () (x 5 ) 0 = f 0 (x) = f(x) =x 6 f 0 (x) () (x 6 ) 0 = f 0 (x) = 3 x 0 =1 f(x) x 0 x 1 x x 3 x 4 x 5 x 6 f 0 (x) 4 n x n (x n ) 0 (x n ) 0 = 35

< 3 > f(x);g(x) k kf(x) 0 = k f 0 (x) f(x)+g(x) 0 = f 0 (x)+g 0 (x) f(x) g(x) 0 = f 0 (x) g 0 (x) () () () (1) (x 5 + x 6 ) 0 =(x 5 ) 0 +(x 6 ) 0 =5x 4 +6x 5 () (7x 4 ) 0 =7 (x 4 ) 0 =7 4x 3 =8x 3 (3) (6x 5 +5x 4 ) 0 =(6x 5 ) 0 +(5x 4 ) 0 =30x 4 +0x 3 (4) (x 7 4x 5 +5x 8) 0 =(x 7 ) 0 (4x 5 ) 0 +(5x ) 0 (8) 0 =7x 6 0x 4 +10x (5) (x +3)(x 4) 0 =(x 4 x 1) 0 =4x 3 x (1) (x 3 x 5 ) 0 () (1x 5 ) 0 (3) (4x 3 +8x 5 ) 0 (4) (7x 6 3x 4 +5) 0 (5) (10x 4 5x 3 + 16) 0 (6) (4x 6 7x 4 +8x 3 9x) 0 (7) (x )(x +3) 0 (8) (x 4)(x 5) 0 36

< 1 > f(x) a x f(a) >f(x) f(x) x = a f(x) b x f(b) <f(x) f(x) x = b f(b) 3 y =x 3 9x +1x y 0 =6x 18x +1 =6(x 1)(x ) x = 1 x = y 0 =0 x 1 y 0 + 0 0 + y % 3 & % x =1 y =3 x = y = 3 y = x 3 3x x = y = x = y = x y 0 y 37

< > 4 y =3x 4 16x 3 +18x +8 3 y 0 =1x 3 48x +36x =1x(x 4x +3) =1x(x 1)(x 3) x =0; x =1; x =3 y 0 =0 x 0 1 3 y 0 0 + 0 0 + y & 8 % 13 & 19 % x =1 y =13 x =0 y =8 x =3 y = 19 (1) y = x 4 x +3 x y 0 y () y = 3x 4 +4x 3 +1x x y 0 y 38

< > f(x) = 1 x +3 f 0 (x) f(x) = 1 x +3 f(x + h) = 1 x + h +3 µ 1 0 = f 0 f(x + h) f(x) (x) = lim = lim x +3 h!0 h h!0 1 x + h +3 1 x +3 h =lim h!0 (x +3) (x + h +3) (x + h +3)(x +3) h =lim h!0 h (x + h +3)(x +3) h =lim h!0 1 (x + h +3)(x +3) 1 =lim h!0 1 (x + h +3)(x +3) = 1 (x +0+3)(x +3) = 1 (x +3) µ 1 0 1 = x +3 (x +3) ³ (x +3) 1 0 = (x +3) f(x) = 1 x (1) f 0 (x) ( ) f 0 (x) = () (1) 39

< > f(x) = p x +3 f 0 (x) f(x) = p x +3 f(x + h) = p x + h +3 ³ p p px +3 0 = f 0 f(x + h) f(x) x + h +3 x +3 (x) =lim = lim h!0 h h!0 h p p p p x + h +3 x +3 x + h +3+ x +3 = lim h!0 h p x + h +3+ p x +3 p p x + h +3 x +3 = lim h!0 h p x + h +3+ p x +3 (x + h +3) (x +3) = lim h!0 h p x + h +3+ p x +3 h = lim h!0 h p x + h +3+ p x +3 = lim 1 p p h!0 x + h +3+ x +3 = 1 p x +0+3+ p x +3 = 1 p x +3 ³ px +3 0 = 1 p x +3 ³ 0 (x +3) 1 = 1 (x +3) 1 f(x) = p x (1) f 0 (x) () f 0 (x) = () (1) 40