(1) 1 y = 2 = = b (2) 2 y = 2 = 2 = 2 + h B h h h< h 2 h

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1 O y A y y = f() y = f() b f(b) B y f(b) f() = b f(b) f() f() = = b A f() b AB O b y = 2 = 1 = 1 + h (1 + h) (1 + h) 1 2h + h2 = h h(2 + h) = h = 2 + h y (1 + h) O y = h 217

2 (1) 1 y = 2 = = b (2) 2 y = 2 = 2 = 2 + h B h h h< h 2 h h h h 2 lim 1 lim(2 + h) = 2 h (1) lim h 0 (4 h) = 4 (2) lim h 0 (3 + 3h + h 2 ) = 3 3h h lim limit h 0 h 0 0

3 6.2 (1) lim h 0 (6 + h) (2) lim h 0 (12 6h + h 2 ) C f() = = + h f( + h) f() h h 0 f() = f 0 () f() = f 0 () = lim h!0 f( + h) f() h 6.3 f() = 2 = 2 f (2) = lim h 0 f(2 + h) f(2) h 4h + h 2 = lim h 0 h = lim h 0 (4 + h) = 4 = lim h 0 h(4 + h) h = lim h 0 (2 + h) h 6.3 (1) f() = 2 = 1

4 220 6 (2) f() = 3 2 = 2 D f() = f () y = f() 2 y y = f() A(, f()) P( + h, f( + h)) AP f( + h) f() h f() = f( + h) f() = + h O A h P + h f( + h) f() h 0 P y y = f() A f() f () P AP A f () l l y = f() A A O A 1 f () l y = f() A(, f()) f() = f ()

5 y = 2 (2, 4) m f() = 2 m = f (2) 6.3 f (2) = 4 m = y = 2 (1) (1, 1) (2) ( 2, 4) f() = f () A f() = 2 = f () f () = lim h 0 f( + h) f() h = lim h 0 2h + h 2 h = lim h 0 ( + h) 2 2 h = lim h 0 (2 + h) = 2 f () = f (3) 1 = f() = 2 (1) f (3) (2) f (0) (3) f ( 2) 1 f () 1 f ()

6 222 6 f() f () f() f 0 () f() f () f 0 () 6.5 f 0 () = lim h!0 f( + h) f() h (1) f() = f () = lim h 0 ( + h) h h = lim h 0 h = 1 (2) f() = 3 f ( + h) 3 3 () = lim h 0 h 3 2 h + 3h 2 + h 3 = lim h 0 h = lim( h + h 2 ) = 3 2 h 0 (+h) 3 = h+3h 2 +h 3 3h h 2 0 (3) f() = 2 f () = lim h h = 0 6.5(3) 6.6 (1) f() = 3 (2) f() = 2

7 (3) f() = 4 y = f() y 0 dy d 3 3 ( 3 ) () = 1, ( 2 ) = 2 ( 3 ) = 3 2 n = 1, 2, 3 n n ( n ) 0 = n n`1 c (c) 0 = 0 n=1 0 =1 B f() f () f() f() = 3 g() = 2 f () g () y = 2f() y = f() + g() y = 2 3 y = 2f() { } y 2( + h) ( + h) 3 3 = lim = lim 2 h 0 h h 0 h = lim h 0 2( h + h 2 ) = lim h 0 ( h + 2h 2 ) = f () = 3 2 y = 2f() y = 2f ()

8 224 6 y = y = f() + g() y {( + h) 3 + ( + h) 2 } ( ) = lim h 0 { h ( + h) 3 3 = lim + ( + } h)2 2 h 0 h h = lim h 0 {( h + h 2 ) + (2 + h)} = f () = 3 2 g () = 2 y = f() + g() y = f () + g () y = f() g() y = f () g () k 1 y = kf() y 0 = kf 0 () 2 y = f() + g() y 0 = f 0 () + g 0 () 3 y = f() g() y 0 = f 0 () g 0 () 6.6 y = y = 3( 2 ) 4() + (2) y =(3 2 ) (4) +(2) = =3( 2 ) 4() +(2) = (1) y = (2) y = (3) y = (4) y =

9 y = ( + 1)( 2) ( + 1)( 2) = ( 2 2) = y = y = (1) y = ( + 2)( + 3) (2) y = 3( 2) 2 (3) y = ( + 2)( 2) (4) y = 2( + 1)( 3) f() f (0) = 3 f (1) = 1 f(2) = 2 f() = 2 + b + c b c f() = 2 + b + c f () = 2 + b f (0) = 3 b = 3 f (1) = 1 f(2) = b = b + c = 2 = 2 b = 3 c = 0 f() f() =

10 f() f (0) = 3 f (1) = 1 f(0) = 2 C y t s = f(t) s 0 f 0 (t) ds dt 6.7 t s = 1 2 t2 t ds dt = 1 2t = t 2 ds dt 6.10 r V S V = 4 3 πr3 S = 4πr 2 V S r

11 A 6.2 y = A(2, 1) (1) A l m (2) A l (1) f() = m = f (2) f() f () = m = f (2) = = 4 (2) l A(2, 1) y 4 l y 1 = 4( 2) 1 O 2 A y = y = A(2, 3) (1) A (2) A

12 228 6 y = f() A(, f()) y f() = f 0 ()( ) B 6.2 y = C(1, 0) 2 2 y (, 2 + 3) C y = y = 2 (, 2 + 3) 2 y ( 2 + 3) = 2( ) 1 C(1, 0) 0 ( 2 + 3) = 2(1 ) y = 0 ( + 1)( 3) = 0 = 1, 3 1 = 1 y 4 = 2( + 1) = 3 y 12 = 6( 3) ( ) y = y = O C 1 C(1, 0) m y = m( 1)

13 y = O (1) lim h 0 ( 1 + h) 2 ( 1) 2 h (2) lim h 0 2( + h) h

14 t (1) y = 3t 2 4t + 2 (2) f(t) = 1 (t 1)2 2 3 C(1, 0) y = 3 1 (1) 2 (2) 4 2 (1) y = 6t 4 (2) f (t) = t y = 0 y =

15 y O A f() = 2 4 f () = 2 4 = 2( 2) y = f() A(, f()) f () = 2( 2) y y = < 2 f () < 0 O 2 4 A 2 > 2 f () > 0 A f() = 2 4 < 2 > 2

16 232 6 f 0 () f() f() f () > 0 f () < 0 f () = 0 f() 6.8 f() = 3 3 y f () = = 3( + 1)( 1) f () = 0 = 1, 1 f () > 0 < 1, 1 < f () < 0 1 < < 1 f() 1 1 f () f() O f() = 3 f () = y 1 0 f () f() 0 O 1 f()

17 6.10 f() = 3 y f () = f () < 0 O 1 f() (1) f() = (2) f() = (3) f() = (4) f() = 3 B 6.8 f() = 3 3 f() f() = y y = f() f() = f() = b f() = b O b f(b)

18 f() = 3 3 = f () = 1 f() f() = f() = y = y = = 3( 2) 0 2 y y = 0 y = 0, y y 3 = 0 3 = 2 1 O 1 2

19 (1) y = (2) y =

20 236 6 (3) y = (4) y = f() f() = f 0 () = 0 f () = 0 f() = 6.9 f() = 3 f (0) = 0 f() = 0

21 f() = b = 2 6 b f() = 2 6 f (2) = 0 f(2) = 6 f() = b f () = f() = 2 6 f (2) = 0 f(2) = = b = 6 = 12 b = 10 f() = f () = = 3( + 2)( 2) ( ) = 12 b = f () f() f() = b = 1 8 b

22 A 6.4 y = ( 1 4) y = = 3( 2) y = 0 = 0, 2 4 y O 2 y y = 1, 2 4 = y 6.4

23 (1) y = ( 3 1) (2) y = ( 2 2)

24 cm 12cm cm (12 2)cm 1 cm cm cm y 3 y 1 cm y 3 > > 0 0 < < 6 1 y= (12 2) 2 = 4( ) y = 12( ) = 12( 2)( 6) 1 y y = 2 y + 0 y ( ) 2cm

25 cm 16cm 1 cm

26 242 6 B f() = 0 y = f() = 3 f() = y = f() y = y = y = = 3( + 2) y y = y y = y = 3 2 O 0 < < y y y = = = 0, 4 2 < 0, 4 < = 1

27 C f() 0 f() f() = ( 3 + 4) f() = ( 3 + 4) 3 2 f () = = 3( 2) 0 f() 0 2 f () 0 + f() f() = f() 0 ( 3 + 4) =

28 (1) y = (2) y = (1 ) 3 5 f() = b + c = 3 = 1 12 b c

29 6 18cm V cm 3 (1) cm V (2) V cm cm 7 y = (1) = = (2) 5 = 3 b = 9 c = 7 6 (1) V = 2 (18 2)π (2) 6cm 7 0 < < 32

30 A f() f() F () = f() F () f() 6.12 ( 2 ) = f() 1 F () f() F () + C f() F () + C f() f() d 2 f() f() f() f() F () = f() f() d = F () + C C C 2

31 ( () = ) ( 1 = 1 d = + C d = C 2 d = C 3 3 ) = 2 2 d = C 1 d 1 d 6.13 n = 0, 1, 2 n n n d = 1 n + 1 n+1 + C 0 d 1 d B F () = f() k (kf ()) = kf () = kf() kf () kf() 1 kf() d = kf () + C

32 248 6 F () = f(), G () = g() 1 kf() d = kf () + C k 2 {f() + g()} d = F () + G() + C 3 {f() g()} d = F () G() + C n 6.5 (1) 4 2 d (2) ( ) d (1) (2) 4 2 d = C = C ( ) d= C = C 6.21 (1) 6 2 d (2) ( 2 + 1) d (3) ( ) d (4) ( ) d (5) ( ) d (6) ( ) d

33 t 1 dt = t + C t dt = 1 2 t2 + C t 2 dt = 1 3 t3 + C F (t) 1 F (t) = 3(t 1) 2 2 F (1) = 0 1 3(t 1) 2 F (t) 2 C 1 F (t)= 3(t 1) 2 dt = (3t 2 6t + 3) dt = t 3 3t 2 + 3t + C F (1) = C = C C + 1 = 0 C = 1 F (t) = t 3 3t 2 + 3t F (t) 1 F (t) = 2(t 1) 2 F (0) = 0

34 f() = 2 F () 1 F (3) F (1) 1 A f() = 2 F () F () = 2 + C C F (3) F (1) F (3) F (1) = (3 2 + C) (1 2 + C) = 8 C f() 1 F () b f() F (b) F () F () b F (b) F () b f() d f() b 3 b b < b = b > b [ ] b F (b) F () F () F () = f() b f() d = [ ] b F () = F (b) F () d ( ) 1 3 = [ 2 3 d = 3 ] 2 1 = = =

35 6.23 (1) 3 d (2) 2 2 d (3) d 6.6 (1) 1 0 (1) ( 2 + 3) d (2) 1 0 [ 3 2 ( 2 + 3) d = = ( ) 2 12 = ] 1 ( + 4)( 2) d 0 ( ) (2) 2 1 ( + 4)( 2) d = [ 3 = ( 2 3 = = ] 2 ( ) d ) { ( 1) } + ( 1) 2 8( 1) 6.24 (1) 2 ( ) d (2) ( ) d

36 252 6 (3) 4 2 ( 2)( 4) d (4) 1 2 2( + 3)( 2) d B F () = f() G () = g() b {f() + g()} d = [ ] b F () + G() = {F (b) + G(b)} {F () + G()} = {F (b) F ()} + {G(b) G()} = b f() d + b g() d b b b k f() d = k b {f() + g()}d = {f() g()}d = f() d k b b f() d + f() d b b g() d g() d

37 k [ ( 2 + k)d = 2 3 d + k d = = k ] 2 1 = k [ 2 + k 2 ] ( 2 + 2) d = = = (2 2 1) d 3 {( 2 + 2) (2 2 1)}d ( )d [ ] 1 0 = = k (1) 4 (k 2 + 3)d (2) 1 ( + 2) 2 d ( 2) 2 d

38 254 6 b 1 f() d = 0 2 f() d = f() d b 3 b f() d = c f() d + b c f() d 3 b c 3 F () = f() c b [ ] c [ ] b f() d + f() d = F () + F () c c = {F (c) F ()} + {F (b) F (c)} = F (b) F () = b f() d

39 C f(t) t b F (t) = f(t) f(t) dt = [ ] b F (t) = F (b) F () b f(t) dt = b f() d f(t) F (t) = f(t) f(t) f(t) dt = F () F () F () (F ()) = f() F () (F ()) = 0 f(t) dt f() d f(t) dt f(t) dt d 6.8 f(t) dt = f() = f(t) dt = 0 f() = 2 3 = 0 0 = = 1, 2 ( ) f() = 2 3, = 1, 2

40 f(t) dt = 2 2 f() f() y = f() 1 f() = y y = y y = + 1 t + 1 Q 1 R O S(t) P t 1 O S b 1 y = + 1 y = t 4 t S(t) S (t) = t + 1 S(t) = 1 2 {1 + (t + 1)} t = 1 2 t2 + t S () = + 1 S () = f() 2 y = + 1 = = b S 1 S(t) S = S(b) S() S () = f() S S = [ ] b S() = b f() d 4 OPQR S S = 1 (OR + PQ) OP 2

41 A 2 y = 2 1 y = 2 = t t S(t) S (t) = t 2 S(t + h) S(t) h 1 h 0 h > 0 2 S(t + h) S(t) 2 APQB APRC h > 0 t 2 < ht 2 < S(t + h) S(t) < h(t + h) 2 S(t + h) S(t) h < (t + h) 2 h 0 (t + h) 2 t 2 1 t 2 h < 0 1 y y = 2 2 y y = 2 (t + h) 2 C h R S(t) t 2 B S(t + h) S(t) Q O t O A t P t + h

42 258 6 S(t + h) S(t) lim h 0 h = t 2 S (t) = t 2 y = 2 2 y = = b S y = 2 S S(t) S = S(b) S() S [ ] b b b O b S = S(t) = t 2 dt = 2 d y = f() (1) b f() 0 y = f() 2 y y = f() = = b S S = b f() d O S b 6.17 y = y = 1 = 2 S y = ] 2 2 [ S = ( )d = ( ) ( ) = = O S 1 2

43 (1) y = 2 2 = 1 = 3 (2) y = = 1 = 2 (2) y b f() 0 y = f() y = f() 2 = = b S S S = b { f()}d O S b y = f()

44 y = = 0 = 2, y 0 S 2 [ 3 ] 2 S = { ( 2 4)}d = ) } = ( { ( 2)3 + 4( 2) = y O 4 S 2 y = 2 4 y S = 2 { ( 2 4)}d (1) y = 2 1 (2) y = 2 2

45 B 2 1 y y = f() 2 f() g() b f() g() y = f() y = g() S y = g() 1 S O b S = = b b f() d b {f() g()}d g() d 2 y y = f() + k 2 2 y k S y = g() + k y = f() S = = b b {(f() + k) (g() + k)}d {f() g()}d (3) y b f() g() y = f() y = f() y = g() 2 = = b S S S = b {f() g()}d O O S y = g() b b y = g()

46 y = 2 2 y = = 1 = 2 S y S = = = {( 2 + 3) ( 2 2)}d ( )d [ ] = O y = S 1 2 y = y = 2 2 y = = 1 = y = 2 1 y = = = = 0 = 1, 2 y y = 2 1 S S = = = {( + 1) ( 2 1)}d ( )d [ ] 2 1 = S 1 O 1 2 y = + 1 1

47 (1) y = 2 y = + 2 (2) y = y = 2

48 y = ( α)( β) β α > 0 β α = k S 2 S = = k 0 k { ( k)}d 0 [ 3 ( 2 + k)d ] k = 3 + k ) = ( k3 3 + k3 = k3 2 6 k = β α > 0 S = β α { ( α)( β)}d = (β α)3 6 1 y = ( α)( β) 2 y = ( k) y y O α k S β k α O S 2 β α

49 (1) 1 1 (3 1) 2 d (2) 2 1 (t 2 5t + 4)dt 9 y = 2 (1) y = (2) y =

50 y = 2 1 y 2 = 2 = O (1) 8 (2) (1) 9 (2)

51 A 1 (1) y = (2 + 1)(1 2 ) (2) y = ( 2)( ) 2 y = A(3, 9) l (1) l (2) l B

52 y = 2 ( ) (1) > 0 (2) = 0 (3) < 0 4 f() = 3 + b 2 + c + d = 1 5 = 3 1 b c d

53 (1) 2 2 (2 3) 2 d (2) 2 2 (t 2 2)dt 6 f() = 1 (t 1)(t 2)dt

54 y = = 3 = y = (1) y = 0 y = 4 (2) y = 2 y =

55 B 9 f() = k k 10 k k 0 k

56 V V 1 V 2 O 10 { f() f() d} < {f()} 2 d

57 13 f() = f(t) dt f() 14 y = 2 4 3

58 y = O f () 0 12 f() = + b f(t) dt = t f(t) = t 2 + 2

59 (1) y = (2) y = (1) y = 3 18 (2) 1 3 (2) y = 3 3 (1) 0 (2) (3) [ y = 3 ( 23 )] 4 = 1 b = 6 c = 9 d = 1 f (1) = 0 f (3) = 0 f(1) = 5 f(3) = 1 5 (1) (2) = [ { }] 12 2 ( 2 + 5)d + ( 2 5)d (1) 49 (2) 39 [ 4 3 (1) {4 ( 2 3)}d { ( 2 3)}d ] (2) {2 ( )}d + {2 ( 2 3)}d 9 k k = 32 (y = k ) 0 [ 11 V 1 : V 2 = 8 : 27 V = 1 ] 3 π2 (20 ) (0 < < 20) [ 1 12 f() = + b f() d = 1 ] b, {f()} 2 d = 2 + b + b f() = 2 2 [ 1 1 ] f(t)dt = (t 2 + 2)dt = = 2, 2 [ > 0 ( 2 + )d = , < 0 ( 2 + )d = 4 ] [ 0 2 ] {( ) ( 6)}d + {( ) 2}d

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