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1 , nishioka/ : 5) i) [], : ) *1 e = ) fx) e x, x R : 7.1) fx) = a e b x, a = 13, b = , : ), *1,. e lim n 1 + 1/n) n.

2 : ,, i) 7.1) a, b? ii) 7.1) 7.1? 8.2. i),., 8.1), a) 8.1),. b) 8.1),.,, 8.1). ii),.., {a n }., D N. 8,,., 8.1. D f R x, 9 ): 9.1 a 0, a 1,, a n, 8.1) y, fx) = a 0 + a 1 x + a n x n, x R, y = fx)... x, y. D f f.,. R f {fx) : x D f } i) 1: a 0, b. fx) a x + b, x R. ii) 2: a 0, b, c. fx) = a x 2 + bx + c, x R. iii) 3: a 0, a 1, a 2, a 3 0. fx) = a 0 + a 1 x + a 2 x 2 + a 3 x 3, x R.

3 9.1. fx) = 2x 2 1 fx) = x 3 2x 2 x + 1 ) 2 : ii) n, f x) = fx), fx) = x n fx) = x n, n. x R. : i) n, f x) = fx),. v) : a 0, a 1,, a 3, a 4, fx) = a 0 + a 1 x + a 2 a 3 + x, a 3 + x 0, a 2 + a 3 x gx) = a 0 + a 1 x + a 4 + a 5 x + a 6 x 2, a 4 + a 5 + a 6 x hx) = 10x x. 9.2 fx) = x 3 [] fx) = 1 ) gx) = 10x + 1 ) : x

4 e = fx) e x, x R. fx) = exp{x}. ) vi) :. fx) = x, x 0, gx) = x + 1 ) 1/3, x 1, x R, :. e x e y = e x+y, e x ) y = e x y i), ii) HIV

5 9.3, OA X θ *2, XY A x, y), sin θ, cos θ, tan θ :! sin θ AB OA = x, OB cos θ OA = y, AB tan θ OB = y x. ' % " # $ & sin θ ) cos θ ) I. O 1 A. AOB, O ). *2.,. A ÂC r AOB = r. π, ĈD π. COD DOC II.. OAB OA = 1, sin 2 θ + cos 2 θ = AB ) 2 + OB ) 2 = OA ) 2 = 1 2 = 1. OAB = 90 θ sin π 2. θ) = sin OAB) = OB OA = cos θ π/4 π/3 π/2 π 3π/2 2π

6 " i) sin θ, cos θ. # & ii) sin 2 θ + cos 2 θ = 1. sin π 2 θ) = cos θ. iii) ) sinα + β) = sin α cos β + cos α sin β.! ' cosα + β) = cos α cos β sin α sin β. $ %. AO = 1, AOB = α, BOD = β. ACD = ABO = AEB = BDC = π/2., sinα + β) = AC. BECD, EC = BD. BAE = β, AB = sin α OB = cos α AE = AB cos β = sin α cos β. BD = OB sin β = cos α sin β 9.4,. [] f D f, R f.. AC = AE + EC = AE + BD, : sinα + β) = sin α cos β + cos α sin β. D f f R f E f "1 cosα + β) = OC,.

7 , E R f y E, fx) = y x D f : i) fx) = 2x, D f = R, ii) fx) = 1/x, D f = {x R : x 0}, iii) fx) = 1/2x), D f = {x R : x 0}, iv) fx) = x 2, D f = R, x = f 1 y), y E, f., x y : f 1 x), x E, ) y E f f 1 y) ) = y, fx) E x f 1 fx) ) = x. [ 9.7 ] 9.7 i) fx) = 2x, D f = R x = ff 1 x)) = 2f 1 x) f 1 x) = x 2.

8 9.7 ii) 9.7 iv) fx) = 1/x, D f = {x R : x 0},, x 0 x = ff 1 x)) = 1 f 1 x) f 1 x) = 1 x. fx) = x 2, D f = R. fx) = x 2, D f = R,, E = D f fx) = y x *3, f *3 fx) = 1 x x = ± iii) fx) = 1/2x), D f = {x R : x 0}., x 0 x = ff 1 x)) = 1 2 f 1 x) f 1 x) = 1 2x. E = {x : x 0} D f, : f 1 x) = x, E = D f 1 = {x : x 0},

9 9.5 f : D f R f g : D g R g R f D g, g fx) ) : D f R g g fx) ). [] i) fx) = x 2, gx) = x 3 gf)x) gf)x) = x 2 ) 3 = x 6, D gf) = R, R gf) = {x : x 0}. ii) fx) = x 2, gx) = x gf)x) gf)x) = x 2 = x, D gf) = R, R gf) = {x : x 0}. $ # #" % # $!!" %! #! 9.8. f, g gf)x). i) fx) = x 2, D f = R, R f = {x : x 0}, gx) = x 3, D g = R, R g = R. ii) fx) = x 2, D f = R, R f = {x : x 0}, gx) = x, D g = {x : x 0}, R g = {x : x 0}. 9.6, fx) = e x, D f = R, R f = {x : x > 0} f 1 x), log x : f 1 x) = log x, D f 1 = {x : x > 0}, R f 1 = R.

10 f 1 x) = log x, gx) = x, fx) = e x a, b > 0. i) log a a x = x, a log a x = x. ii) log a x y) = log a x + log a y, x, y > 0. iii) log a x y = y log a x, x > 0. iv) log a x = log b x log b a, x > 0. [] 9.4 e x e y = e x+y,. e x ) y = e x y. x R i) a > 0 log a x log x log a, x > 0, a, a = 10 log 10 x. ii), 9.9 log x e,. i), 9.1) 9.2) e log a = a! ' &,, log x ). & "#$%! 9.7 x log a e x e log a

11 . x e x log a = a x. 9.1) log e b = b, log a x = loge x log a ) = x log a , log a a x = 9.2), log ax log a = x log a log a = x. i) log e x = x, e log x = x. ii) logx y) = log x + log y, x, y > 0. a log a x = e log a) log a x = e log a) log x log a ) = e log x = x. iii) log x y = y log x, x > 0. ii) i) x = a log a x, y = a log a y, i) log a x y) = log a a log a x a log a y ) = log a a log a x+log a y = log a x + log a y. iii) i) x = a log a x, log a x y = log a a log a x ) y = loga a y log a x = y log a x. iv) [] : x ) , a, b, : a, b? fx) = a e b x, a = 13, b = 0.52 log b x log b a = log x/ log b log a/ log b = log x log a = log a x.

12 [ 9.13 ] fx) = a e b x x = 0 13 = f0) = a e b 0 = a a = 13. fx) :, x = 1 log fx) = log a e b x) = log 13 + b x log 22 = log f1) = log 13 + b. b = log 22 log 13 = log [] i) log e log 3 e = log e log log e/ log 3) = 1 log 3 + log 3 = 1. ii) log e 2 1 log 3e log 3 = 2 log e 1 log 3/ log 3e + 1 log 3 = 2 log 3 + log e log 3 iii) log 3 2 log 8 27 = log 2 log log 3 = log log 3 = 1. log 33 log 2 3 = log 2 log 3 3 log 3 2 log 2 = i) log e log 3 e, ii) log e2 1 log 3e log 3, iii) log 3 2 log 8 27.

4 4 4 a b c d a b A c d A a da ad bce O E O n A n O ad bc a d n A n O 5 {a n } S n a k n a n + k S n a a n+ S n n S n n log x x {xy } x, y x + y 7 fx

4 4 4 a b c d a b A c d A a da ad bce O E O n A n O ad bc a d n A n O 5 {a n } S n a k n a n + k S n a a n+ S n n S n n log x x {xy } x, y x + y 7 fx 4 4 5 4 I II III A B C, 5 7 I II A B,, 8, 9 I II A B O A,, Bb, b, Cc, c, c b c b b c c c OA BC P BC OP BC P AP BC n f n x xn e x! e n! n f n x f n x f n x f k x k 4 e > f n x dx k k! fx sin x cos x tan

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1 29 ( ) I II III A B (120 ) 2 5 I II III A B (120 ) 1, 6 8 I II A B (120 ) 1, 6, 7 I II A B (100 ) 1 OAB A B OA = 2 OA OB = 3 OB A B 2 : 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

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1 (1) ( i ) 60 (ii) 75 (iii) 315 (2) π ( i ) (ii) π (iii) 7 12 π ( (3) r, AOB = θ 0 < θ < π ) OAB A 2 OB P ( AB ) < ( AP ) (4) 0 < θ < π 2 sin θ 1 (1) ( i ) 60 (ii) 75 (iii) 15 () ( i ) (ii) 4 (iii) 7 1 ( () r, AOB = θ 0 < θ < ) OAB A OB P ( AB ) < ( AP ) (4) 0 < θ < sin θ < θ < tan θ 0 x, 0 y (1) sin x = sin y (x, y) () cos x cos y (x, y) 1 c

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