A (1) = 4 A( 1, 4) 1 A 4 () = tan A(0, 0) π A π

4 4.1 4.1.1 A = f() = f() = a f (a) = f() (a, f(a)) = f() (a, f(a)) f(a) = f 0 (a)( a) 4.1 (4, ) = f() = f () = 1 = f (4) = 1 4 4 (4, ) = 1 ( 4) 4 = 1 4 + 1 17

18 4 4.1 A (1) = 4 A( 1, 4) 1 A 4 () = tan A(0, 0) π A π

4.1. 19 4.1 = log (1) e () (a, log a) a = log = 1 (a, log a) log a = 1 ( a) a 1 (1) 1 e 1 a = e a = 1 e = log 1 log 1 ( e = e 1 ) e log a a 1 = e () 1 (0, 0) 0 log a = 1 (0 a) a log a = 1 log a a = e 1 1 a 1 = 1 ( e) e = 1 e = log

130 4 4. = e (1) 1 () e a A = e 1 a

4.1. 131 B 4.1 + = 4 ( 3, 1) m + = 4 + = 0 0 = = 3 = 1 ( 3, 1) m = 3 1 = 3 4. 8 + = 1 (, 1) 8 + = 1 8 + = 0 0 1 = 4 (, 1) 4 1 = 1 1 = 1 ( ) = 1 +

13 4 4.3 A (1) + 8 = 1 A( 1, ) A 1 () = 1 A(, 1) 1 A

C A A 4.1. 133 A A(a, f(a)) = f() A(a, f(a)) f (a) f (a) 0 A f 1 (a) =f() = f() (a, f(a)) f (a) 0 f(a) = f 1 0 ( a) (a) f (a) = 0 = a 4. = e (1, e) f() = e =e f () = e f (1) = e 1 a (1, e) e = 1 ( 1) e = 1 e + e + 1 e

134 4 4.4 A (1) = e A( 1, e) A( 1, e) =e 1 ( ) π () = sin A 6, 1 A ( π, ) 1 6 =sin

4.1. 135 4.1. A f() = f() A(a, f(a)) B(b, f(b)) AB f() (a, b) =f() A B AB B 1 f(b) f(a) AB b a c f A (c) f(b) f(a) b a = f (c), a < c < b f() [a, b] (a, b) a c b f(b) f(a) b a = f 0 (c), a < c < b c 4.3 f() = + 1 [0, ] (0, ) 5 = + 1 f() f(0) 0 f (c) = c = 5 1 = c = c = 1 1 1

136 4 4.5 c (1) f() = + 4 a = 0 b = 3 c 3 () f() = 3 a = 1 b = 1 c

4.1. 137 B 4. a > 0 1 a + 1 < log(a + 1) log a < 1 a log(a + 1) log a log(a + 1) log a = (a + 1) a f() = log f() = log > 0 f () = 1 [a, a + 1] log(a + 1) log a (a + 1) a a < c < a + 1 = 1 c 1 c a > 0 1 a + 1 < 1 c < 1 a 1 1 a + 1 < log(a + 1) log a < 1 a 4.6 a < b e a < eb e a b a < eb

138 4 4.1.3 A f() [a, b] (a, b) 1 1 (a, b) f () > 0 f() [a, b] (a, b) f () < 0 f() [a, b] 3 (a, b) f () = 0 f() [a, b] 1 [a, b] a 1 < b 1 f( ) f( 1 ) = f (c)( 1 ), 1 < c < c (a, b) f () > 0 1 f (c) > 0 1 > 0 f( ) f( 1 ) > 0 f( 1 ) < f( ) 1 1 I 1 1 < f( 1 ) < f( ) f() I 1 < f( 1 ) > f( ) f() I

4.1. 139 4.7 3 3 f() g() (a, b) g () = f () f() g() [a, b] g() = f() + C C 4.8

140 4 f () f() 4.3 f() = f() 0 f () = 1 1 1 = f () = 0 =f() = 1 1 4 f() 1 0 1 f () 0 + f() 0 1 f() 0 1 1 4.9 (1) f() = e = 1 () f() = log 1 1

4.1. 141 (3) f() = + sin (0 π) π π B f() = a f() = a f(a) f() = b f() = b f(b) b a f() f() = a = a f (a) = 0 f() f (a) = 0 = a f () = a a f () + 0 f() a f () 0 + f() f (a) = 0 = a f () f(a) f (a) = 0 f(a) 160 4.6()

14 4 4.4 (1) f() = e () f() = + 4 (1) f () = 1 e + ( e ) = (1 )e f () = 0 = 1 f() e > 0 1 f () + 0 f(1) = 1 e 1 = f() 1 e 1 e f(1) = 1 e () 0 f () = 1 4 = 4 = ( + )( ) f () = 0 =, f() > 0 0 f () + 0 0 + f() 4 4 f( ) = 4 f() = 4 4.4 (1) () 4 1 e 1 4 =

4.10 4.1. 143 (1) f() = 4 3 + 1 1 11 16 3 () f() = e 4 e

144 4 (3) f() = log 1 e 1 e 1 (4) f() = + =

4.1. 145 f() = a = a 4.3 f() = + 1 0 = < 0 = 1 0 f() = + 1 f () = + 1 + + 1 = ( + 1) + + 1 = 3 + + 1 > 0 f () > 0 1 < 0 f() = + 1 f () = 3 + + 1 f () = 0 = 3 f() 1 3 0 f () + 0 + f() 0 1 3 3 9 0 ( f 3 ) = 3 9 f(0) = 0 3 9

146 4 4.11 (1) f() = 1 1 1 () f() = + 4 6 9 4 3

4.4 f() = + + a 1 4.1. 147 = 1 a f() = 1 f ( 1) = 0 a f () = ( + 1)( 1) ( + + a) = 1 a ( 1) ( 1) f() = 1 f() = 1 f ( 1) = 0 a 4 = 0 f ( 1) = a 4 a = f() = + + 1 f () = 3 ( 1) = ( + 1)( 3) ( 1) f() ( 1) > 0 1 1 3 f () + 0 0 + f() 1 7 a = = 1 f( 1) = 1 f(3) = 7 ( ) a = 1 7 4.1 f() = + a = 1 a f()

148 4 C 4.5 = cos + sin (0 π) = cos ( sin ) + cos = cos (1 sin ) 0 < < π = 0 cos = 0 = π, 3 π sin = 1 0 π cos = 0 = π, 3 π sin = 1 = π 0 π 3 π π + 0 0 + 1 1 = π, = 3 π 1 =cos + sin 3 π π π

4.1. 149 4.13 (1) = (1 + cos ) sin (0 π) () = 4 3 + 1 (1 4)

150 4 4.1.4 A f() f () C = f() C f () f () f () f () f () f () > 0 f () C f () < 0 f () C C = f() f 00 () = f() f() f () 1 f () > 0 = f() f () < 0 = f()

4.1. 151 4.4 = 3 3 + 4 = 3 6 = 6 6 = 6( 1) 4 = 3 3 + 4 1 0 + 1 4.4 (1, ) = f() f (a) = 0 = a f () (a, f(a)) = f() f (a) = 0 (a, f(a)) = f() f() = 4 = 4 f () = 4 3, f () = 1 f (0) = 0 = 0 f () (0, 0) = 4 4.14 (1) = 4 + 3 + 1 1 1

15 4 () = e (, e ) (3) = cos (0 < < π) π + 1 π 1 π π (4) = 4

4.1. 153 B 4.6 = e = 4e = 4{e + ( 4e )} = 4(4 1)e = 0 = 0 = 0 = ± 1 1 0 1 + + + 0 + 0 0 + 1 e 1 1 e ( 1 ) ( ), 1 1 e, 1 e lim = 0 lim = 0 + 1 1 1 1 e

154 4 4.7 = 1 = + 1 + 1 1 = 1 = 1 ( 1) = ( 1) 1 = ( 1) ( 1) 3 ( ) ( 1) 0 1 + 0 0 + + + + 0 4 lim = lim = 1 0 1+0 = 1 lim { ( + 1)} = 0 4 { ( + 1)} = 0 = + 1 lim = + 1 1 1 1 =1 = + 1 + 1 1

4.1. 155 4.15 (1) = e

156 4 () = 4 + 1

4.1. 157 (3) = + 1 1

158 4 C f() f () f () 1 f (a) = 0 f (a) > 0 f(a) f (a) = 0 f (a) < 0 f(a) 1 f (a) > 0 a f () > 0 f () f (a) = 0 =f() < a f () < 0 > a f () > 0 a f ()<0 f () 0 + f () + f() f(a) a f ()>0 f (a)=0 f (a)>0 4.5 f() = 3 + 3 f () = 3 + 3 f () = 6 f () = 0 =f() f(1) = 1, 1 f ( 1) = 6 > 0 f (1) = 6 < 0 f( 1) f(1) 1 1 f( 1)

4.1. 159 4.8 f() = + cos (0 π) f () = 1 sin f () = cos 0 < < π f () = 0 = π 6, 5 6 π sin = 1 ( π ) f = ( ) 5 3 < 0 f 6 6 π = 3 > 0 ( π ) f = π 6 6 + ( ) 5 3 f 6 π = 5 6 π 3 4.16 (1) f() = 4 6 + 5

160 4 () f() = + sin (0 π) f() f (a) = 0 f (a) = 0 f(a) 4.6 (1) f() = 4 () f() = 3 f () = 4 3 f () = 1 f () = 3 f () = 6 f (0) = 0 f (0) = 0 f (0) = 0 f (0) = 0 f(0) f(0) = 4 = 3

4.1. 161 4.1.5 1 p 0 = 4p ( 1, 1 ) 1 = p( + 1 ) (1) = 4 ( 1 )

16 4 () = + 4 3 = 4 + a 3 + 3a + 1 a [ ] 1 p = 4p 0 (1) = = 1 3 () = = 3 a < 0 8 < a = 0 + a + a = 0 1

4. 4..1 4.. 163 A 4.5 > 0 e > 1 + > 0 f() = e (1 + ) f() = e (1 + ) f () = e 1 > 0 e > 1 f () > 0 f() 0 > 0 f() > f(0) = 0 f(0) = e 0 (1 + 0) = 0 e > 1 + 4.17 > 0 (1) e > 1 + + 1

164 4 () log( + 1) < > 0 4.17(1) e > e >, e < 0 e lim =, lim e = 0 n e lim =, n lim n e = 0

4.. 165 B 4.6 a e = a = 0 = e = a = 0 0 f() = e 0 1 f ( 1)e f () = () 0 + f() f() e lim f() =, lim f() =, lim +0 f() = 0 = t t ( e t lim f() = lim t t = lim 1 ) = 0 t te t = f() = a = e a =a e a > e a = e a < 0 1 0 a < e 0 1

166 4 4.18 a 3 = a( 1)

4.. 167 4.. A P P t =f(t) = f(t) t t + t f(t + t) f(t) t t 0 t P v v = d dt = f (t) P v > 0 v < 0 v v v t α α = dv dt = d dt = f (t) P t = f(t) t P v α v = d dt = f 0 (t), α = dv dt = d dt = f 00 (t) 4.9 P t = sin(πt a) t v α a v v = d = π cos(πt a) dt α α = dv dt = π sin(πt a) α = π P

168 4 4.19 v 0 m/ t m = v 0 t 1gt g t v m/ α m/ B P t P (, ) t P PQ PR Q R P(, ) R t d Q dt d dt Q P v ( ) R d v = dt, d P dt Q t P ( ) P(, ) v = d d, dt dt θ tan θ = d dt d dt = d d v P P d dt P θ d dt v

4.. 169 v v P d d dt dt ( ) α = d d dt, dt t P α α P P t (, ) t t P v v α α ( v = d, d ) (d ) ( ) d v = + dt dt dt dt ( ) (d ) ( ) α = d dt, d dt α = d dt + dt 4.10 P t = a cos ωt, = a sin ωt (a, ω ) P t P t v α v d = aω sin ωt d = aω cos ωt dt dt v v = ( aω sin ωt) + (aω cos ωt) = a ω (sin ωt + cos ωt) = aw α d dt = aω cos ωt d dt = aω sin ωt α = aω (cos ωt, sin ωt) α α = aω cos ωt + sin ωt = aω

170 4 4.10 P + = a aω a v P α ωt 4.10 α = ω (, ) a a P α P a 4.0 t P t = 3 P (1) = t + 1, = t 4t () = cos πt, = sin πt

4.. 171 4..3 A f(a + h) f() = a f (a) f(a + h) f(a) lim h 0 h = f (a) h 0 f(a + h) f(a) h f (a) f(a + h) f(a) + f (a)h h 0 f(a + h) h 1 1 h 0 f(a + h) f(a) + f 0 (a)h = f() A(a, f(a)) =f() f(a + h) P f(a) + hf (a) T = f(a) + f (a)( a) f(a) A h 0 h hf (a) Q P T a a + h 4.7 h 0 sin(a + h) 1 (sin ) = cos h 0 sin(a + h) sin a + h cos a

17 4 4.1 h 0 1 (1) cos(a + h) () tan(a + h) B 0 1 a = 0 h 0 1 0 f() f(0) + f 0 (0) 4.11 p 0 (1 + ) p 1 + p 3 1.006 f() = (1 + ) p f () = p(1 + ) p 1 f(0) = 1 f (0) = p f() f(0) + f (0) 0 (1 + ) p 1 + p 3 1 1.006 = (1 + 0.006) 3 3 1 1.006 1 + 0.006 = 1.00 3

4.. 173 4. 0 1 (1) e () log(1 + ) (3) 1 1 + 4.3 1 (1) 4 16.1 () log 1.01 (3) 1 0.998

174 4 4..4 4 P t = a(ωt sin ωt), = a(1 cos ωt) (a ω ) 5 8cm 3 t r cm S cm V cm 3 r = dv dr dt dt ds dt

4.. 175 6 1 5 4 π 3.1416 3 1.731 (1) sin 31 () tan 1 4 aω 5 dv dt = 8 (cm3 / ) dr dt = 1 π (cm/ ) ds dt = 8 (cm / ) 6 (1) 0.5151 () 0.0175 [ ( (1) sin 31 = sin π6 + 180 π ) sin π 6 + π 180 cos π 6 f () = cos 1 f (0) = 1 f(0) = 0 tan () f() = tan ]

176 4 4.3 4.3.1 A 1 (1) = + (, 0) ( π ) () = tan 4, 1

4.3. 177 (1) = log () = 3 3

178 4 3 (1) = ( 1) + 1

4.3. 179 () = sin + cos (0 π)

180 4 4 A r θ θ θ l θ r θ B C

4.3. 181 5 a = + a (1) a > 0 () a = 0 (3) a < 0

18 4 6 > 0 (1) cos > 1 () sin > 3 6

4.3. 183 4.3. B 7 = P Q P PR QR 8 = a + b = log A(e, 1) A a b

184 4 9 10 3 f() = 3 + 3a + 3b + c C C A C A P A P Q Q C

4.3. 185 11 P t = 4 cos t = sin t 0 t π P 4 4 1 5m A 0.3m B A 5m 4m B A

186 4 9 k 1 A B A = B = d dt = 0.3 1 (1) = 1 4 1 = 4 + 8 () = π + 1 = 1 + π 8 + 1 (1) = 1 1 e e 3 () = 3, 0, 3 0 = 1, 1 [ (1) = ( 1) ( + 1) = 4( 3) = 1 ( + 1) 3 ] () = ( sin 1)(sin + 1) = cos (4 sin + 1) (1) () 3 3 1 = 1 π 6 π 5 π 6 π 3 1 1 3 3 3 4 [ AB = r cos θ BC = r sin θ l = 4r cos θ + r sin θ dl dθ 5 (1) 3 () 0 (3) 1 ] = 4r( sin θ 1)(sin θ + 1) [ = ( 3a) ( + a) 3 ]

6 4.3. 187 [ ) (1) f() = cos (1 > 0 f () > 0 f(0) = 0 ) () g() = sin ( 3 6 ] (1) g () > 0 g(0) = 0 7 P t = t + (t + 1) t 8 a = 1 e b = 1 f() = a + b g() = log f(e) = 1 f (e) = g (e) [ 9 : 1 S() h k k = π h S() = π + πh S () = (π3 k) S() = 3 ] k π 10 A ( a, a 3 3ab+c) A C = g() g() = f( a) (a 3 3ab+c) = 3 +3(b a ) g( ) = g() 11 [ (d ) ( ) d 5 + = ( 4 sin t) + ( cos t) dt dt = ] 4{4 sin t + (1 sin t) } [ 1 0.4m/ A B A = B = + = 5 d t d dt + d dt = 0 ] d = 0.3 = 4 = 3 dt dt = 0.4