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1 005/05/05 by. I : : : : : : : : : : : : : : : : : : : : : : : : :. II : : : : : : : : : : : : : : : : : : : : : : : : : 3 3. III : : : : : : : : : : : : : : : : : : : : : : : : 4 4. : : : : : : : : : : : : : : : : : : : : : : : : 5 5. : : : : : : : : : : 6 6. : : : : : : : : : : : : : : : : : : : : : : : : 7 7. : : : : : : : : : : : : : : : : : : : : : : : : 8 8. : : : : : : : : : : : : : : : : : : : : : : : : 9 9. : : : : : : : : : : : : : : : : : : : : : : : : 0 0. : : : : : : : : : : : : : : : : : : : : : :. : : : : : : : : : : : : : : : : : : : : : :. : : : : : : : : : : : : : : : : : : : : : : 3. (A): (B): (C):

2 . I a; b; c; : : :. x; y; z; : : :. (function) y x x y y = f (x) x y. a (m/sec) t x (m) x = at a t x x t f (t) =at x = f (t) x = x(t) x x(t) x. PV = nrt T P V P V P V (domain) x D, I interval. f (x) = p x [0; ).. (a; b) =fx j a<x<bg: [a; b) =fx j a» x<bg: (a; b] =fx j a<x» bg: [a; b] =fx j a» x» bg: (range) y = f (x) x y R, f (D), f (I) f (D) =ff (x) j x Dg :. f (x) =sinx I = R, f (I) =[ ; ]. R =( ; ). (A)

3 . II x y. y = x. x y. x = y. x = y = ±. x y. x = sin y. x =0 y = mß (m =0; ±; ±;:::). f x, x f :, x <x f (x ) <f(x ) f :, x <x f (x ) >f(x ) f :, f f :, x <x f (x )» f (x ) f :, x <x f (x ) f (x ). f (x) =x [0; ) ( ; 0]. [x] x Gauss [ß] =3,[0]= 0, [0:4]= 0, [ 0:]=. f (x) =[x] ( ; ) x>0 [x] x. (A) 3 3

4 3. III I f y f (I) f (x) = y x I y 7! x x f f (I). f x = f (y). y x.!! f (x) = f (x) f (x) y = f (x): f f (I), I. y = f (x) y = f (x) y = f (x) y = x.. y = f (x) =ax + b (a 6= 0) f (x) x = ay + b y y = f (x) = x b a.. f (f (x)) = f (f (x)) = x f (f (x)) = af (x) +b = a x b + b = x a f (f (x)) = f (x) b = ax + b b = x a a 3. (A) f (x) f (x) f (f (x)) = f (f (x)) = x 4

5 4. y = ax + b (a; b: a 6= 0) straight line y y = b y x x = b=a x x a b a =0 linear y = b 0 y x =0, y x = y = ax + bx + c (a; b; c: a 6= 0) 3 y = ax 3 + bx + cx + d (a; b; c, d: a 6= 0). n y = a n x n + a n x n + + a 0 (a 0 ;:::;a n : a n 6=0) (A) () x y 3 6 () 3 y = x, y =0:x, y =0:0x (3) 3 y = x, y =0x, y =00x (4) 3 y = x, y =(x ), y =(x +) (5) 3 y = x, y =(x ) +3, y =(x +) (6) 3 3 y = x 3, y = x(x ), y =(x )(x 3)(x +) 5. (A) () () (3) 3 5

6 5. f (x) g(x) y = g(x) f (x) f; g y = x x y 6. (A) a = ; ; y = a x a = ; ; y = a x 7. (B) a>0 y = a x y = x p a 8. (A) y = x y = y = a x a (a; p a 6= 0) x p 9. (A) y = x + y = a + q (a; q a 6= 0) x y = a x 0. (A) y = x 3 y = y = a x a + q (a; p; q a 6= 0) x p. y = x + x =+ 3 x x + x 0.. (A) y = x x ax + b y = ad bc 6= 0 (a; b; c; d c 6= 0) cx + d y = x 6

7 6. y = a x (a>0) a a x a n (n =; ;:::) a : =!!! y = a x a >, a =,0<a< y = a x y = x = a> 0 <a< 8 a x a y = a x+y >< a x =a y = a x y. a>0, b>0 (a x ) y = a xy (ab) x = a x b x >: (a=b) x = a x =b x b = a x y = a x = = b b x = b x. (A) y = e x y = e x 3. (A) () y = x, y =3 x, y =4 x () y = x, y =3 x, y =4 x 4. (A) a m n =( mp a ) n = mp a n p () a 3p a 4p a 3 () 3p a b 4 4p p q p 6 a b= ab 3 3 (3) a a 5. (A) 3, 3:, 3:4, 3:4, 3:45 ß 6. (A) () 4 x 5 x +4=0 () 9 x 3 x 3=0 (3) 4 x+ + x+ =0 7

8 7. y = a x (a 6= ;a > 0) x = a y (a 6= ;a > 0) y y = a x log a x a x x = a y, y =log a x (a 6= ;a > 0) y = a x x = a y x y y =log a x y = a x y = x y = log a x a> 0 <a< a = e log e x log e log, ln a =0 log 0 x 8 >< log a (xy) = log a x +log a y. a>0, a 6= log a (x=y) = log a x log a y >: log a x p = p log a x (p ) 7. (A) log a x 8. (A) () log a a p = p () log a a = (3) log a = 0 9. (A) 0. (A) : log b c = log a c (a; b; c > 0, a 6=,b 6= ) log a b. (A) a log a b = b (a; b > 0, a 6= ). (A) x [x] x Gauss. [ß] = 3, [99:99] = 99, m K(m) = [log 0 m]+ K(m) m m =00 K(00) = 3 00 () K(3), K(35), K(005), K(8899) [0] = 0. () log 0 =0: (3) K(m) m 3. (A) () log x =3 () log (x x) =3 (3) log x = (4) log x = 8

9 8. y = x a (a: ) a a a p x :3 ;x ß ;x ; np x 0: ;::: a x a x a = e a log x x>0 a =0 x 0 =(x>0) 0 0 = f (x) =x 0 x 0. y p = x = x. y = x (x 0) x 0, y 0.. y = 3p x = x 3. y = x 3 ( < x < ) <x<, <y<. 4. (A) () y = x (x 0) () y = x (x<0) (3) y = p x, y = p x, y = p x + p 9

10 9. x tan sin cos cos cos(ff + fi) =cosff cos fi sin ff sin fi: sin(ff + fi) =cos ( ß ff) +( fi) sin cos(ff fi) sin(ff + fi) =sinff cos fi +cosff sin fi: tan 3 sin cos. " cosec x = sin x ; sec x = cos x ; cot x = tan x 5. (A) sin cos tan 6. (A) tan sin, cos 7. (A) sin(nß), cos(nß), sin(n + )ß, cos(n + )ß 8. (A) t =tan x () sin x = t () cos x = t (3) tan x = t +t +t t 0

11 0. ( ) sinh x = ex e x ; cosh x = ex + e x ; tanh x = sinh x cosh x : (hyperbolic sine) (hyperbolic cosine) (hyperbolic tangent) e ix =cosx + i sin x sin x = eix e ix ; cos x = eix + e ix i i i " 9. (A) () x = cosh t, y =sinht () cosh x sinh x = x y = () tanh x =sech x x = cos t, y = sin t x + y = (3) sinh(x + y) =sinhxcosh y + cosh x sinh y (4) cosh(x + y) =coshxcosh y + sinh x sinh y III sech x == cosh x 30. (A) tanh(x + y) tanh x, tanh y 3. (B) sinh x ( <x<), cosh x (x 0), tanh x ( <x< ) arcsinh x, arccoshx, arctanh x arcsinh sinh arccosh; arctanh

12 . f (x) = sin x f (x +ß) =f (x) ß ß f (x + T )=f(x) f (x) T f (x) =sinx f ( x) = f (x). x; x 3 ;x 5 ;x 7 ;:::;sin x; tan x; sinh x; tanh x; : : : f (x) = cos x f ( x) =f (x). x ;x 4 ;x 6 ;x 8 ;:::;cos x; cosh x; : : : sin x cos x 3 3. (A) () () (3) (4) (5) (6) 33. (A) f (x) () f (x +)=f (x) () f (x +)=3f (x) (3) f (x) =f ( x) (4) f (x) = f ( x) (5) f (x + y) =f (x)f (y) (6) f (xy) =f (x) +f (y)

13 .. y = sin x [ ß ; ß ] [ ; ]. y = Arcsin x y = Arcsin x y = Sin x [ ; ], [ ß ; ß ] y [ ; ] y = sin x x x = arcsin y y = arcsin x [ ; ], ( ; ) y = arcsin x [ ß ; ß ] Arcsin x Arcsin x arcsin x x = sin y. y =cosx [0;ß] [ ; ]. y = Arccos x y = Arccos x [ ; ], [0;ß] y [ ; ] y =cosx x x = arccos y y = arccos x [ ; ], ( ; ) y = arccos x [0;ß] Arccos x Arccos x arccos x y = Cos x x = cos y. y =tanx ( ß ; ß ) ( ; ). y = Arctan x y = Arctan x y = Tan x ( ; ), ( ß ; ß ) y ( ; ) y = tan x x x = arctan y y = arctan x ( ; ), ( ß +nß; ß + nß) (n =0; ±; ±;:::) y = arctan x ( ß ; ß ) Arctan x Arctan x arctan x x = tan y. 38. arcsin; arccos; arctan 34. (A) p () arcsin 3 () arcsin( ) (3) arccos( ) (4) arccos 0 (5) arcsin( p ) 35. (B) () arccos(sin 6 5 ß) () arcsin(cos 4 5 ß) (3) arcsin(tan ) (4) tan(arcsin( p )) (B) arcsin x + arccos x = ß 37. (C) () f (x) = arcsin(sin x) () g(x) = arccos(cos x) (3) h(x) = arctan(tan x) 38. (C) m =0; ±; ±;::: () arcsin x =( ) m Arcsin x + mß () arccos x =( ) m Arccos x + mß +( ( ) m ) ß (3) arctan x = Arctan x + mß 3

. sinh x sinh x) = e x e x = ex e x = sinh x 3) y = cosh x, y = sinh x y = e x, y = e x 6 sinhx) coshx) 4 y-axis x-axis : y = cosh x, y = s

. sinh x sinh x) = e x e x = ex e x = sinh x 3) y = cosh x, y = sinh x y = e x, y = e x 6 sinhx) coshx) 4 y-axis x-axis : y = cosh x, y = s . 00 3 9 [] sinh x = ex e x, cosh x = ex + e x ) sinh cosh 4 hyperbolic) hyperbola) = 3 cosh x cosh x) = e x + e x = cosh x ) . sinh x sinh x) = e x e x = ex e x = sinh x 3) y = cosh x, y = sinh x y =

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