p.16 1 sin x, cos x, tan x a x a, a>0, a 1 log a x a III 2 II
2 III III [3, p.36] [6] 2 [3, p.16] sin x sin x lim =1 ( ) [3, p.42] x 0 x ( ) sin x e [3, p.42] III [3, p.42] 3 3.1 5 8 *1 [5, pp.48 49] sin x, cos x, tan x *1 5 8 306 305303 301 313 312 309308
sin x, cos x, tan x (1) (sin x) sin x =cosx lim =1 x 0 x (sin x) sin(x + h) sin x 1 cos h sin h = lim =sinx lim +cosx lim =cosx sin x 1 cos h lim =1lim =0 x 0 x { ( (2) (cos x) = sin x (cos x) = sin x + π )} ( 2 (cos x) = cos x + π ) ( x + π ) ( = cos x + π ) = sin x 2 2 2 ( π ) [5] cos x =sin 2 x (3) (tan x) = 1 ( ) sin x cos 2 x (tan x) = = (sin x) cos sin x(cos x) cos x cos 2 = 1 x cos 2 x sin 2 x +cos 2 x =1 3 sin x, cos x, tan x 1 2 sin x, cos x, tan x *2 3 3.2 3.2.1 cos x, sin x, tan x (1) (cos x) = sin x (cos x) = lim h 0 cos(x + h) cos x h = *2 3 P 3 6
cos h 1 sin h cos x lim sin x lim = sin x (2) (sin x) =cosx(1) { ( π )} ( π ) (sin x) = cos 2 x = sin 2 x ( 1) = cos x (3) (tan x) = 1 cos 2 3.1(1), (2) x (1) (3) 3.1 sin x, cos x *3 3.2.2 tan x 2 cos x, tan x, sin x sin x, tan x, cos x 2 (1) (cos x) = sin x 3.2.1 (2) (tan x) = 1 cos 2 x (cos x) = sin x 1+tan 2 x = 1 cos 2 x 2tanx (tan x) (tan x) (tan x) (x n ) = nx n 1 n (x n ) = nx n 1 n *4 [3, p.41] (1) 2 2 sin x cos 3 ( sin x) = x cos 2 tan x tan x = x cos x 2tanx (tan x) = 2 cos 2 x tan x (tan x) (tan x) = 1 cos 2 x (2) tan 2 x = 1 cos2 x cos 2 x tan x = cos( π 2 x) cos x (sin x) =cosx cos x, tan x, sin x (3) (sin x) =cosx sin x =tanxcos x (1), (2) 3.1 *3 1 x y f(x) =g(y) 2 f(x),g(x)[1, p.165] sin x cos x tan x cot x(= 1/ tan x) sec x(= 1/ cos x) csc x(= 1/ sin x) 1 *4
(2) (tan x) 1+ 1 tan 2 x = 1 sin 2 x 3.2.3 tan x 1 tan x, cos x, sin x tan x, sin x, cos x 2 (1) (tan x) = 1 cos 2 x (tan tan(x + h) tan x x) = lim = tan h tan 2 x tan h tan h lim = lim (1 tan x tan h) 1 tan 2 x 1 tan x tan h = 1 cos 2 x lim sin h =1 tan h lim =1 (2) (cos x) = sin x 3.2.2 1+tan 2 x = 1 cos 2 x 1 (1) 2tanx cos 2 x 2 cos 3 x (cos x) 3.2.2 (cos x) (cos x) 2tanx 1 cos 2 x = 2 cos 3 (cos x) x (cos x) (cos x) = sin x (3) (sin x) =cosx 3.2.2 (3) 3.2.2 sin x 1+ 1 tan 2 x = 1 sin 2 x 3.3 3 tan x 2 (cos x) tan x 4 a a>0 a 1e Napier
4.1 3.1 5 8 log a x,logx, a x, e x (1) (log a x) = 1 Napier e = lim x log a (1 + x) 1 x x 0 ( log lim a (x + h) log a x = lim log a 1+ h h 0 x ) 1 h = lim log a (1 + t) 1 1 tx = t 0 x log a (2) (log x) = 1 (1) a = e x (3) (a x ) = a x log a y = a x log y = x log a y y =loga(log x) = 1 x y y = a x log a (4) (e x ) = e x (3) a = e 1 4 1 2 301(2), (1), (4), (3) (2) (2) log a x = log x (1) [5] log a (2), (1), (4), (3) (2) (1) (2) (4) (1) (3) 309(1) (2), (2) (1) (3) (4) (3) (1)
4 2 III 4 *5 (1) (4) 4.2 4.1 4.14 (1) (log a x) = 1 x log a, (2) (log x) = 1 x, (3) (a x ) = a x log a, (4)(e x ) = e x 4.2.1 (2) log x (3) a x 4.1 (1) log a x (3) a x (1) log a x (4) e x (2) log x (4) e x (1) log a x (4) e x y = e x a log a y =log a e x = x log a e y y log a =log a e (log a x) =1/x log a y y = y log a log a e = y (e x ) = e x 4.2.2 (2) log x (1) log a x 4.1 (4) e x (3) a x X α = Y α log Y X a x = e x log a (a x ) =(e x log a ) = e x log a log a = a x log a *5 4 4 P 4 =24
X α = Y α log Y X 4.2.3 (1) log a x (3) a x 4.1 (2) log x (4) e x (3) a x (1) log a x (4) e x (2) log x 4.2.1(1) log a x (4) e x (2) log x (3) a x (1) (4) f(x) =e x, g(x) =logx = log a x f(x) g(x) log a e ( ) g 1 (x) = f (g(x)) loga x 1 = log a e exp(log x) = 1 x *6 (log a x) = 1 (2) log x (3) x log e a x f(x) g(x) (4) e x (1) log a x (3) a x (2) log x 4.3 4 2 2 4 2 1 2 4 *6 exp(x) =e x
2 2 4.1309 (1) (2) (1) (3) (1) (4) (1) 1 y = f(x) a y = a f(x) e y = e f(x) (4) e x (1) log a x xlog a 1 y =log a x e y e y =(x 1/ log a ) = log a α (x α ) = αx α 1 (2) (log x) =1/x 4. 4 Napier Napier e = lim (1 + x) 1 x x 0 Napier a h 1 e lim =1 a
1 5 *7 1 [6] [2] [4, p.39] *8 [6] 3 2016 2 IIA, B 45 [2] *7 Cantor *8 John Perry(1850-1920) 20
[1] 1993 [2] 49 pp.497 500 2016 [3] 2009 [4] 2010 [5] 2002 [6] PISA 2004