No1 1 (1) 2 f(x) =1+x + x 2 + + x n, g(x) = 1 (n +1)xn + nx n+1 (1 x) 2 x 6= 1 f 0 (x) =g(x) y = f(x)g(x) y 0 = f 0 (x)g(x)+f(x)g 0 (x) 3 (1) y = x2 x +1 x (2) y = 1 g(x) y0 = g0 (x) {g(x)} 2 (2) y = µ x 3 x 2 +1 (3) y = f(x) g(x) y0 = f 0 (x)g(x) f(x)g 0 (x) {g(x)} 2 (1), (2) (3) y = 1+x 1 x 2
No2 4 y =sinx (5) y = p sin(2x +3) (6) y = 1 tan(3x 2) (7) y =cos 2 (4x +5) (8) y = cos x 1+sinx 5 (1) y =sinx cos x 6 f(x) = sin(sin x) f 0 (π) (2) y =sin(x 2 2x) (3) y =cos 2x +1 7 f(x) = sin2 x x lim x 0 f 0 (x) (4) y =tan 3 x
No3 8 sin 2x (1) y =2 (9) y =log 3 x2 +3 (2) y = e 1/x (10) y =(logx) 3 (3) y = e 2x cos x (11) y = x log x (4) y = e e2x (12) y =log x a (5) y =log cos x (13) y = x x (x>0) (6) y = log(log x) (x>1) (14) y = x sin x (x>0) (7) y = log(x + 1+x 2 ) (15) y =(sinx) x (0 <x<π) (8) y =log s 1 sin x 1+sinx (16) y = e xx (x>0)
No4 9 x y (1) x 2 +2xy 5y 2 =1 11 y =1 x 2 e log y dy dx (2) e y = x 2 +1 (3) x =sin y 12 f(x) =1+e log x + e 2logx + + e n log x + (x >1) f 0 (x) (4) log(x + y) =x dy 10 dx (1) x = 1+t2 1 t 2, y = 2t 1 t 2 13 f(x) f 0 (x) sin(x 2 ) (x 6= 0) f(x) = x 0 (x =0) (2) ( x = a(cos t + t sin t) y = a(sin t t cos t)
No5 14 n (3) (2) x = t sin t, y =1 cos t (1) y = a x (2) y = xe x 16 y = 1 x 2 (3) y =logx (1 x 2 )y 00 xy 0 + y =0 d 15 2 y dx 2 (1) x 2 +2xy y 2 =1 17 y = e ax sin bx (b 6= 0) y 00 2y 0 +2y =0 a, b
No6 x (x 6= 0) 18 f(x) = 1+2 1/x 0 (x =0) (1) x =0 (2) x =0 20 x 1, x 2 g(x) g(x 1 + x 2 )=g(x 1 )+g(x 2 )+2x 1 x 2 (1) g(0) (2) x =0 g(x) g(x) x = a (3) x =0 g(x) g 0 (0) = m g 0 (a) ax +1 (x 1) 19 f(x) = x 2 x 2 + 1 3 x + b (x 1) a, b
No7 21 sin x sin a (1) lim x a x a 23 (1) lim h 0 a h a 2h h (a >0) (2) lim x 1 log x x 1 (3) lim h 0 (1 + h) 7 1 h (2) lim x a e x cos a e a cos x x a (4) lim h 0 e h 1 h 22 f(x) =logx lim x 1 f 0 (x) 1 x 1 (3) lim x 1 4 1 4 x 1/4 1 4x
No8 24 f 0 (0) = a lim x 0 f(3x) f(sin x) x (2) lim x 1 {f(x)} 2 4 x 1 25 f(0) = 2, g(0) = 1, f 0 (0) = 2, g 0 (0) = 1 2 2 f(x), g(x) lim x 0 f(2x) 2 g(3x) 1 27 f(x), g(x) x = a f(a), g(a), f 0 (a), g 0 (a) g(a) 6= 0 ½ 1 f(a + h) (1) lim h 0 h g(a + h) f(a) ¾ g(a) 26 1 f(1) = 2, f 0 (1) = 1 (2) lim x a f(x 2 ) 2 (1) lim x 1 x 1 x n f(x) a n f(a) x n a n (a 6= 0, n )
No9 e 28 lim (1 + h) 1/h = e h 0 (7) lim (1 1 x (1) lim (1 + 3h) 1/h h 0 2x )x+2 (2) lim h 0 (1 h) 1/h (8) lim x (1 + 2 3x )x+1 (3) lim x (1 + 1 x )x 29 a n = Z 1+1/n 0 x n dx, b n = Z 1 1/n (1) lim n na n (2) lim n nb n 0 x n dx (n =1, 2, 3, ) (4) lim x (1 + a x )x (5) lim x (1 1 x )x (6) lim x (1 + 1 x 2 )x
No10 30 f(x) =2logx 35 a<b f(e) f(1) e 1 = f 0 (c) (1<c<e) e a (b a) <e b e a <e b (b a) c x 36 x>0 < log(x +1)<x x +1 31 f(x) = 1 x (x>0) f(a + h) f(a) =hf 0 (a + θh) (0< θ < 1) θ h lim θ h 0 37 0 q<p, n 2 p n q n <np n 1 (p q) 32 f(x) f 0 (x) a<x<b f(x) 33 f(x) x lim f 0 (x) =3 x lim {f(x +1) f(x)} x 38 0 x 1 <x 2 <x 3 π sin x 2 sin x 1 x 2 x 1 > sin x 3 sin x 2 x 3 x 2 34 (1) lim x +0 e x e sin x x sin x 39 a, b sin a sin b a b 40 p, q (2) lim x{log(2x +1) log 2x} x log(p +1) log(q +1) p q
No11 41 y = 1+sinπx x =1 46 2 y = ax 2 (a 6= 0),xy=1 x a 42 x 2 + xy + y 2 =0 x =7 43 x =2cosθ, y =3sinθ θ = π 4 47 x + y = a x y A, B OA + OB O 44 y = ex x 45 y = e x2 (a, 0) 48 x =cos n θ,y=sin n θ P, Q PQ θ n (1) a n 6= 2 (2) a
No12 49 y = x2 +1 x +1 (1, 1) 53 y = x 2 + ax + b y = 8 x (2, 4) a, b 50 y 2 =4x (3, 0) 54 2 y = cx 2 (c ), y =logx 1 P (a, b) a, b, c 51 y = eax + e ax 2a (a ) P x M PM P y 2 55 y =2sinx y = a cos 2x 0 <x< π 2 a 52 x 2 +3y 2 =4 2 P (x 1,y 1 ) Q(1, 1) (1) P (2) Q 56 2 y = log(x +1),y= e x 1 1 (3) P, Q R P Q R
No13 2(x + a) 57 f(x) = bx 2 (a, b, c ) x =1 1 60 y = x + a + c x 2 1 a ( 3, 0) a, b, c x + a 58 y = x 2 x 2 a 61 y = ax + b x 2 +3 a, b x +1 59 f(x) = x 2 +2x + k k 62 f(x) = 4x a x 2 +1 1 a
No14 63 f(x) = x2 ax +1 x 2 + x +1 b 1 b a, b bx +1 65 y = x 2 (a>0, b>0) 2 1, 4 + ax a, b 66 k f(x) = 1 k2 x 1 3 x >1 x2 k 64 a f(x) = ax + a2 +1 x 2 4 x
No15 67 y = x + 1 x 2 sin x 69 a ±1 f(x) = (0 1+2acos x + a 2 π) x 68 f(x) = 3 x 3 x 2 (1) lim {f(x) (x + a)} =0 a x y = f(x) y = x + a (2) f(x) y = f(x)
No16 70 y =(1+cosx)sinx (0 x 2π) 74 y = sin x a +cosx a 0 <x<π 71 f(x) =cosx + x sin x ( π x π) x 75 f(x) =x + a cos x (a >1) 0 <x<2π 0 f(x) 72 y = 2sin 2 x + 3sin2x +2 π 2 x π 2 73 f(x) =a sin x + b cos x + x (0 x 2π) x = π 3 x = π (1) a, b (2) 76 f(x) =sinx a sin x (0 <x<π)
No17 µ 77 f(x) =logx 2(log x) 2 +(logx) 3 80 y = e 3x sin 3x + π 2(n 1) π <x< 2n π (n 6 3 3 ) 1 78 f(x) =(x 2 + ax + a)e x 0 a 81 0 <x<2π f(x) =ax + e x sin x (1) g(x) =f 0 (x) x (2) f(x) a 79 y = e 2x +2ae x +2x 18 a x 82 f(x) = 1 x e ax x >0 a
No18 83 f n (x) = 1 x n 1 x (1) a n (2) lim n na n (n n+1 ) 85 f(x) =e x cos x (x >0) x x 1, x 2,, x n, (1) x n X (2) y n = f(x n ) y n n=1 84 f(x) =e x sin x (x >0) x x 1, x 2, x 3,, x n, (1) x n (2) y n = f(x n ) y n X (3) y n n=1
No19 86 (4) f(x) = x x 2 +1 (1) f(x) =x cos x sin x (0 x 2π) (2) f(x) =x 2 e x ( 1 x 3) 87 f(x) = x +1 1+x 2 (3) f(x) =x + 1 x 2
No20 88 0 x π 91 x>0 P = x4 + x 2 +1 x 3 x + 1 + x x = t f(x) = 4cos2 x +2cosx +1 (1) P t 4cos 2 x 2cosx +1 (2) P 89 y =sin 3 x +cos 3 x 4sinxcos x (1) sin x +cosx = t y t (2) y 92 f(x) = ex e x (e x + e x 0 ) 3 x log 3 93 a, b a 3 + b 3 =2 90 0 <x< π 2 f(x) = 1 (log tan x +1) (1) a + b tan x (2) a 2 + b 2
No21 94 x f(x) =ax 2 +(2a 1)x log x (a >0) 1 x 2 2ax 96 a>0 f(x) = x 2 ax +1 a 95 0 x π f(x) =(a x)cosx +sinx a 0
No22 97 a x e f(x) =x log x x +1 99 0 x a (a ) y = 0 <a<e x +3 (1) a = 1 4 y 1 (2) y 2 1 3 a 98 f(x) = 3x x 2 +2 (1) f(x) (2) a x a +1 f(x) F (a)
No23 100 y =1 x 2 (x 1,y 1 ) x y 103 a θ S x 1 > 0 (1) S a, θ (2) S θ 101 O, r A P Q Q OA 104 1 h r R QR P (1) r h (2) V h (3) V 102 y = e x x =0,x=1 A, B AB P 4PAB 4PAB 105 am bm
No24 106 y = x 3 3x 2 +3 (1) 110 y = ax + b x 2 x =0 + c x =2 1 x = d e a, b, c, d, e (2) 111 x 4 y = f(x) 2 (2, 16), (0, 0) (2, 16) x f(x) 107 y = e sin x (a, e sin a ) sin a 0 <a< π 2 108 y = x 2 log ax (a >0) (1) (1) y =(x (2) a 2 +2x + a)e x 112 a 109 y = sin n x (0 < x < π 2, n = 2, 3, ) (a n,b n ) {a n }, {b n } (2) y = ax 2 + x +2sinx (0 <x<2π)
No25 113 114 (1) f(x) = x x 2 (1) f(x) =x + 1 x +1 2 (2) f(x) = x2 3x +2 x 2 (2) f(x) = sin x 1+sinx ( π 2 <x< 3π 2 ) (3) f(x) =x + 1 x 115 y 2 = x 2 (1 x 2 )
No26 116 (1) y = x3 x +1 (2) x 3 + ax + a =0 118 m log x = mx π 119 0 x 2 sin(cos x) = 1 2 x 1 117 a (a 1)e x x +2=0 120 a, q 0 q<1 x = q sin x + a
No27 121 a>1 e ax = x +1 x =0 1 0 124 log x = x 2 +3x + p p log 2 = 0.6931 122 a x 2 + ax =sinx 125 a, b (1) sin x = ax 0 <x< π 2 a (2) cos x =1 bx 2 0 <x< π 2 b 123 x a log(x + a)+ a 2 x2 x =0 a
No28 126 a, b x a sin x + b cos x =2a 129 f(x) =x 2 log x ax 2 + b (x >0), f(0) = b x 0 π 0 x 2 a, b f(x) f(x) =0 a b 127 log x = ax + b 130 y = e ax y = bx a>0, b>0 128 (1) e x = ax + b 131 y = e x (a, b) (2) (1) (a, b)
No29 132 x>0 x log x x +1 0 137 x>1 x 1 > x log x 133 0 x π 2 sin x 2x π 134 x>0 e x > 1 x 138 0 <a<1, b > 0 (a +1) b >ab 135 0 x<1 x + x2 2 + x3 3 + + xn n log(1 x) 136 0 <x< π (x 2 2ax +1)e x < 1 2 139 x>0, a > 0 log(cos x) < x2 2
No30 140 (1) x>0 e x > 1+x + x2 (2) lim x (3) lim x x e x log x x (4) lim x log x x +0 2 141 (1) log(x +1)< x +1 (2) lim x (x +1)1/x 142 (1) 0 <x<1 1+x<e x < 1 1 x (2) lim n n(e1/n 1)
No31 143 n (1) x>0 e x > 1+ x 1! + x2 + x3 + + xn 2! 3! n! (2) lim x xn e x =0 145 (1) x x>log(1 + x) >x x2 2 (2) a n = µ1+ 1n µ1+ 2n µ1+ nn (n =1, 2, 3, ) 2 2 2 lim n a n 144 (1) log(1 + x) <x(x >0) (2) log(n +1) log n< 1 n (n>0) X (3) n=1 1 n
No32 146 147 (1) x>0 2 x>log x (1) x>0 e x > 1+x + x2 2 (2) lim x log x x (2) lim x xe x (3) y = log x x (3) y = xe x
No33 148 (1) f(x) =x sin 2 x 150 (1) x>0 log(1 + x) > (2) x>a x a sin 2 x sin 2 a log(1 + x) (2) x>0 f(x) = x x 1+x (3) 0 <a<b (1 + a) b (1 + b) a 149 0 <y<x< π 2 151 0 < α < β tan x tan y x y π 2 α β < sin α sin β < π 2 α β
No34 152 π 4 x sin x + k cos x π 3 x k 154 x x log ax 0 a k 153 x (1 + x) 3/2 k(1 + x 3/2 ) k
No35 µ log x kx 155 x log 157 log(1 + x) x x2 x +1 x +1 2 + ax3 x 0 k a 156 x x>alog x a
No36 158 P t (x, y) x = πt sin πt, y = 1 cos πt (1) P (2) t = 2 3 x θ (2) P v (3) 160 xy P (x, y) t x =cost+sin t, y = cos t sin t (1) P 159 P t x = e t cos t, y = e t sin t (1) v (2) v OP O
No37 161 P (x, y) t 164 10m 1m 26m x = t 3 5t 2 4at, y = t 3 + t 2 +(8a 72)t +1(a ) (1) P 0 t a t (2) a (1) P P x 165 1m AB A 5cm/ O OA=50cm 162 y = x 2 P t =0 B t P x sin t y P P 163 t =0 xy (1, 0) P 1 t = π ( 1, 0) Q P (2, 0) PQ 16cm 3 4cm 1 x ( 2, 0) PQ R (1) R (2) t (0 <t<π) R V (t) (3) V (t) 166 20cm 10cm
No38 167 x 0 f(x) =(1+x) n 1 171 2 3cm, 4cm 30 2 1.02 4 1 µ 168 x 0 tan x + π 4 2 s l 172 T lcm T =2π 980 l =20cm l 1cm 169 f(x) = x 2 +3 x 1 f(x) a + b(x 1) + c(x 1) 2 a, b, c 170 a, b, c 173 1% f(x) =1+ c 2 log(a + bx), g(x) =(1+ax)b f(x) g(x) x a, b, c 0 2