Chaptr Binomial Epansion Chaptr 0 Furthr Probability Chaptr Limits and Drivativs Chaptr Discrt Random Variabls Chaptr Diffrntiation Chaptr Discrt Probability Distributions Chaptr Applications of Diffrntiation Chaptr Eponntial and Logarithmic Functions Chaptr Drivativs of Eponntial and Logarithmic Functions and thir Applications Chaptr Continuous Random Variabls and Normal Distribution Chaptr Paramtr Estimation Chaptr 7 Indfinit Intgration and its Applications Chaptr 8 Dfinit Intgration Chaptr 9 Applications of Dfinit Intgration 00 Chung Tai Educational Prss. All rights rsrvd.
Find th following indfinit intgrals. ( 7)...... 7. 7 Find th following indfinit intgrals. (8 ) 8. ( + 9) 9. ( + ) 0. ( ). ( + 8) 7. 009 Chung Tai Educational Prss. All rights rsrvd.
. ( )( + ). + +. +. ln ln. 7. ( + ) 8. ( )( + ) 9. + 0.. + a + b. (a) Show that =, whr a and b ar constants, a 0 and a + b a Hnc find th following indfinit intgrals. (i) + b. a n+ n ( p ). (a) Show that ( p ) = p( n + ) Hnc find th following indfinit intgrals. (i) ( ), whr p and n ar constants, 0 p and n. 7. 009 Chung Tai Educational Prss. All rights rsrvd.
. (a) Show that = ln( + k), whr k is a positiv constant. + k Hnc find th following indfinit intgrals. (i) + + d. (a) Find ( 9) ( + ). Hnc find 9.. (a) Find d. Hnc find. d 7. (a) Find ln( + + ). Hnc find th following indfinit intgrals. + (i) + + + + Find th following indfinit intgrals. (8 ) 7 8. ( ) 9. ( ) 7. 009 Chung Tai Educational Prss. All rights rsrvd.
0. ( + ). ( ). +. 8. + Find th following indfinit intgrals. ( 8). ( + ). 9 + 8 + 7. ( )( + 7) 8. ( + ) 9. 0. + ( ). +. ( ) 8. +. +.. ( ) 7. + ln 8. 7. 009 Chung Tai Educational Prss. All rights rsrvd.
Find th following indfinit intgrals. (9 0) 9. 0.. Us th substitution u = to find. ( ). Us th substitution u = + to find. +. Givn that > 0, us th substitution u = to find. +. (a) If ( ) A Hnc find. ( ) B +, find th valus of constants A and B.. (a) If + ( + ) A B + + ( + ) C + ( + ), find th valus of constants A, B and C. + Hnc find ( + ).. (a) Find. Hnc find. + (c) Using th rsults of (a) and, find. 7. 009 Chung Tai Educational Prss. All rights rsrvd.
7. In ach of th following, S' ( t) is th rat of chang of S (t) with rspct to t. Find S (t). (a) S '( t) = t + ; S() = t S' ( t) = ; S(0) = (c) S '( t) = + ; (0) = t + t + S dy 8. It is givn that =. Whn =, y = 8. Find y in trms of. dy d y 9. It is givn that = k + +, whr k is a constant. Whn =, = 7 and y =. Find y in trms of. 0. Th slop at any point (, y) of a curv is +. If (, ) is a point on th curv, find th quation of th curv.. Th slop at any point (, y) of a curv is +. If th curv passs through (0, ), find th quation of th curv.. Th slop at any point (, y) of a curv is th curv.. If th y-intrcpt of th curv is, find th quation of d y 9. At any point on a crtain curv, = +. Find th quation of th curv if it passs through (, ) and th slop is at that point. 7. 009 Chung Tai Educational Prss. All rights rsrvd.
d y. It is givn that dy = +. Whn =, = 7 and y =. Find y in trms of. d y. At any point on a crtain curv, =. Find th quation of th curv if it passs through (, ) and (, ).. Th growth rat of th population of a city is givn by P'( t) = ( t 0), whr t is th tim masurd in yars from th bginning of 000, P (t) (in thousands) is th population at tim t. It is known that th population of th city was 900 thousand at th bginning of 00. (a) Find P (t). Find th population of th city at th bginning of 00. (Giv your answr corrct to significant figurs.) 0.0t 7. Th rat of chang of th numbr of flats sold in a privat housing stat can b modlld by dn 900 = ( t 0), 0.t 0.t ( + ) whr t is th numbr of days lapsd sinc th start of th slling of th flats, N is th numbr of flats sold at tim t. It is known that N = 00 whn t = 0. 0.t dn 900 (a) (i) Prov that =. 0.t ( + ) 0.t Using th substitution u = +, or othrwis, prss N in trms of t. Can th numbr of flats sold b 900? Eplain brifly. 8. Th rat of chang of th daily numbr of popl infctd with common cold in a town can b modlld by dn t( t ) = (0 < t < 7), whr t is th tim masurd in days with t = corrsponds to last Monday, N is th daily numbr of infctd popl. (a) Whn did th daily numbr of popl infctd with common cold bcom th gratst? If th daily numbr of infctd popl was 0 on last Monday, find th daily numbr of infctd popl on th day obtaind in (a). (Giv your answr corrct to th narst intgr.) 009 Chung Tai Educational Prss. All rights rsrvd. 7.7
dy 9. Th slop at any point (, y) of th curv C is givn by = 8. Th lin y = + is a tangnt to th curv at th point P. y y = + P O C (a) Find th coordinats of P. Find th quation of C. dy 70. Th slop at any point (, y) of th curv C is givn by = ( )( + ). Th y-intrcpt of C is 0. (a) Find th quation of C. 7 (i) Prov that th slop of C cannot cd. Find th point of C with th gratst slop. 7. Th rat of chang of th tmpratur of a city ystrday can b modlld by dθ kt = + h (0 t 0), whr t is th tim in hours masurd from 9:00 a.m., θ (in C) is th tmpratur at tim t. At 9:00 a.m., th tmpratur was 7. C, h and k ar positiv constants. dθ (a) (i) Eprss ln( + ) as a linar function of t. If th slop and th intrcpt on th vrtical ais of th graph of th linar function in (a)(i) ar 0. and rspctivly, find th valus of h and k. (Giv your answrs corrct to significant figurs if ncssary.) Tak h = 7. and k = 0.. Eprss θ in trms of t. (c) Find th gratst tmpratur. (Giv your answr corrct to dcimal plac.) 7.8 009 Chung Tai Educational Prss. All rights rsrvd.
7. Th rat of chang of th numbr of visitors in a library during a day can b modlld by dn 0(8 t) = (0 t ), t t + 00 whr t is th tim lapsd in hours sinc 8:00 a.m., N is th numbr of visitors in th library at tim t. Whn th library is just opn (i.. t = 0), thr ar 88 visitors. (a) (i) Lt u = t t + 00. Find du. Hnc prss N in trms of t. Thr is a priod of tim whr th numbr of visitors in th library cds 0. How long dos th priod last for? (Giv your answr corrct to th narst 0. hour.) (c) Can th numbr of visitors in th library rach 80? Eplain brifly. 009 Chung Tai Educational Prss. All rights rsrvd. 7.9
Ercis 7A (pag 7.). C. C. C. C. ln C. C 7 7. C ln 7 8. 9 C 9. C 0. C. 8 C. C. C. C ln. C ln. ln C 7. ln C ln 8. ln C 9. ln C 0. C. C. (i) C C ( ). (i) C 8 ( ) C. (i) ln( ) C ln(. (a) 9. (a) 7. (a) ) C ( 9) ( ) C (i) C ln( ) C ln( ) C Ercis 7B (pag 7.) 8 8. ( ) C 8 7 9. ( ) C 7 0. ( ) C. C 9( ). ( ) C 8. C. ln C. ( ) C. ( 8) C 7. ( 7) C 8 8. C ( ) 7.0 009 Chung Tai Educational Prss. All rights rsrvd.
+ 9. 0. ln. ln + 0 9. ( ) + ( ) 0 9. ( + ) ( + ) 7. ( ) + ( ) ( ). +. 7. 8. ( + ln ) 9. ( ) + ( ) 0. ( ) + ( ). ln. ( + ) ( + ) + ln( + ) +.. (a) A =, B = ln. (a) A = 0, B =, C = + + ( + ) 9. y = + + + 0. y = + +. y = + +. = y. y = + + +. y = + + +.. (a) 9 7 y = + P 0.0t 0.0 ( t) = 800 + 900 800 0 thousand 00 7. (a) N = + 80 0.t + No 8. (a) Last Friday 7 9. (a) (, ) y = + 8 70. (a) y = 9 + 0 + 0 87 (, ) dθ 7. (a) (i) ln( + ) = kt + ln h h = 7.89, k = 0. 0.t θ = t.8 +. (c). C 7. (a) (i) t N = 80 ln ( t t + 00) + 88 + 80 ln00. hours (c) No. (a) ln + ln (c) + ln Ercis 7C (pag 7.) 7. (a) S ( t) = t + t 9 ( ) t S t = + (c) S ( t) = ln t + + ln t + + 8. y = + 7. 009 Chung Tai Educational Prss. All rights rsrvd.