CHAPTER 53 METHODS OF DIFFERENTIATION
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1 CHAPTER 5 METHODS OF DIFFERENTIATION EXERCISE 5 Page 67. Differeniae wih respec o : (a 5 5 (b.4.5 (c 5 4 (a If 5 5 hen (5 4 ( (b If.4.5 hen (.4.5 (.5 (c If hen. Differeniae wih respec o : (a 4 (b 6 (c 4 (a If 4 hen ( (b If 6 hen 0 (c If hen ( 4 8 or 8. Differeniae wih respec o : (a (b 5 (c 4 (a If hen ( (b If 5 5 hen 5 ( 5 5 (c If hen (4 9 04, John Bir
2 4. Differeniae wih respec o : (a (b ( (c sin (a If hen 4 4 ( 4 4 (b If ( ( ( + + hen or ( (c If sin hen (( cos 6 cos 5. Differeniae wih respec o : (a 4 cos (b e 6 (c e5 (a If 4 cos hen ( 4( sin 8 sin e6 (b If e 6 hen ( 6 ( 6e (c If e 5 e5 hen (( 5e 5 5e 5 5 e5 6. Differeniae wih respec o : (a 4 ln 9 (b e e (c (a If 4 ln 9 hen (4 4 (b If e e e e hen e ( e e+ e e + e (c If hen , John Bir
3 7. Fin he graien of he curve a he poins (0, 4 an (, , hen graien, If 4 A (0, 4, 0, hence graien 8(0 + 9(0 A (, 8,, hence graien 8( + 9( 6 8. Fin he coorinaes of he poin on he graph 5 + where he graien is If 5 +, hen graien 0 When he graien is, 0 i.e. 0 5 an When, Hence, he coorinaes of he poin where he graien is is, 4 9. (a Differeniae + ln (cos 5 + sin e (b Evaluae when π, correc o 4 significan figures. (a + ln (cos5 + sin e + ln cos5 6sin e Hence, 4 + ( 5sin 5 6( cos ( e (b When π, 4 6 0sin 5 cos e 4 5π π sin cos + π π e π , John Bir
4 .0, correc o 4 significan figures 0. Evaluae s, correc o significan figures, when π given s sin +. 6 If s sin + sin + hen s cos 0 + cos+ s π cos + π When, 6 6 ( π /6.9, correc o significan figures. A mass m is hel b a spring wih a siffness consan k. The poenial energ p of he ssem is given b: p k mg where is he isplacemen an g is acceleraion ue o gravi. The ssem is in equilibrium if p 0. Deermine he epression for for ssem equilibrium. If p k mg hen p k mg If p mg 0 hen k mg 0 i.e. k mg an isplacemen, k. The curren i flowing in an inucor of inucance 00 mh is given b: i 5 sin 00 amperes, where is he ime in secons. The volage v across he inucor is given b: v L i vols. Deermine he volage when 0 ms. If i 5 sin 00 hen i (5(00 cos 500 cos00 Volage across inucor, v L i ( L(500cos00 When L 00 mh an 0 ms, volage, v ( 00 0 ( 500 cos( cos 7.0 vols 94 04, John Bir
5 EXERCISE 6 Page 68. Differeniae wih respec o : sin + If sin, hen ( ( cos ( sin ( cos + sin. Differeniae wih respec o : e If e +, hen ( ( e ( e ( e + e or e ( +. Differeniae wih respec o : ln If ln, hen ( + ( ln ( + ln or ( + ln 4. Differeniae wih respec o : cos If cos +, hen ( ( sin ( cos ( 6 6 sin + 6 cos 6 ( cos sin 5. Differeniae wih respec o : ln If ln ln, hen + ( ln + ln ln + + ln + ln 95 04, John Bir
6 6. Differeniae wih respec o : e sin 4 If e sin 4 +, hen ( e ( 4cos 4 ( sin 4( e e ( 4cos 4+ sin 4 7. Differeniae wih respec o : e 4 ln If e 4 ln, hen ( e 4 + ( ln ( 4e 4 e4 + 4ln 8. Differeniae wih respec o : e ln cos If e ln cos, hen ( eln ( sin + ( cos ( e + ( ln ( e cos e ln sin + + cosln e + ln cosln sin 9. Evaluae i, correc o 4 significan figures, when 0., an i 5 sin Since i 5 sin, hen i (5 (cos (sin (5 + When 0., 45 cos + 5 sin i 45(0. cos sin , correc o 4 significan figures (noe 0. is raians 0. Evaluae z, correc o 4 significan figures, when 0.5, given ha z e sin 96 04, John Bir
7 Since z e z sin, hen (e (cos (sin (6e + 4e cos+ 6e sin When 0.5, z 4e.5 cos+ 6e.5 sin (noe is raian , correc o 4 significan figures 97 04, John Bir
8 EXERCISE 7 Page 60. Differeniae wih respec o : sin If sin, hen ( ( cos ( sin ( ( cos sin. Differeniae wih respec o : cos If cos, hen ( ( ( ( ( ( + 6 6sin cos 6 sin cos 6( sin + cos or ( sin + cos. Differeniae wih respec o : + If +, hen ( + ( ( ( + 4 ( + ( + ( + ( ( + 4. Differeniae wih respec o : cos If, hen cos cos ( cos ( ( sin ( cos cos + cos sin 98 04, John Bir
9 5. Differeniae wih respec o : sin If, hen sin sin 9 sin ( sin ( 4cos ( 9 sin cos 4sin { } sin 4 cos 4sin 6. Differeniae wih respec o : ln If ln, ( ( ln ln ln ( ln ln 7. Differeniae wih respec o : e 4 sin If e 4 sin ( ( ( ( ( ( (, hen sin 4 e e e + cos ( sin 8 e4sin e4sin e4cos + sin e4 sin { + 4 sin cos } ( 8. Fin he graien of he curve 5 a he poin (, 4 If 5 hen graien, ( 5 ( ( ( ( ( ( A he poin (, 4,, hence graien (4 5 0 ( ( , John Bir
10 9. Evaluae a.5, correc o significan figures, given + ln + If hen ln When.5, ( ln (4 ( + ln ( ( ln 5 (0 [(.5 + ] , correc o ( ln significan figures 90 04, John Bir
11 EXERCISE 8 Page 6. Differeniae wih respec o : ( 6 If ( 6 hen 6( 5 ( i.e. ( 5. Differeniae wih respec o : ( 5 5 If ( hen 5 ( 5 ( 6 5. Differeniae wih respec o : sin( If sin( hen ( cos( ( i.e. 6cos( 4. Differeniae wih respec o α: cos 5 α α i.e. 0 cos4 α sinα α If cos 5 α hen (( 5cos 4 α( sinα 5. Differeniae wih respec o : ( + 5 If ( 5 ( hen 5( ( 5 ( 6 ( ( ( Differeniae wih respec o : 5e + If 5e + hen (5( e + ( i.e. 0e+ 9 04, John Bir
12 7. Differeniae wih respec o : co(5 + If co cos sin ( sin + cos hen (sin ( sin (cos (cos cosec sin sin sin Thus, if co ( 5 +, hen ( cos ec ( 5 + ( 0 0cosec ( Differeniae wih respec o : 6 an( + If 6 an( + hen (6 ( sec ( + ( i.e. 8sec ( + 9. Differeniae wih respec o : e an If e sec ean ean, hen ( an sec 0. Differeniae: sin( π wih respec o, an evaluae, correc o significan figures, when π π If sin When π, π π hen ( cos + sin ( π cos π π sin π π + π cos π + sin π , correc o significan figures 9 04, John Bir
13 . The eension, meres, of an unampe vibraing spring afer secons is given b: 0.54 cos( Calculae he spee of he spring, given b, when (a 0, (b s If 0.54 cos( hen (0.54 [ sin(0. 0.5]( sin( (a When 0, spee of spring, 0.6 sin( m/s 4. mm/s (b When s, spee of spring, 0.6 sin( m/s mm/s 9 04, John Bir
14 EXERCISE 9 Page 6. If fin (a (b (a If hen 6 + an 6 + (b + 7. (a Given f( eermine f ( (b Evaluae f ( when (a f( f ( f ( (b When, f ( ( 5 ( 5 4. The charge q on he plaes of a capacior is given b q CV e CR, where is he ime, C is he capaciance an R he resisance. Deermine (a he rae of change of charge, which is given b q, (b he rae of change of curren, which is given b q (a If q CV e CR, q ( CV e CR CR V e R CR (b If q V e CR, R q V e CR V e CR R CR CR 94 04, John Bir
15 4. Fin he secon ifferenial coefficien wih respec o he variable: (a sin + cos (b ln 4 (a If sin + cos hen ((cos sin 6cos sin an (6( sin cos sin cos or ( sin + cos (b If ln 4 hen ( an 5. Fin he secon ifferenial coefficien wih respec o he variable: (a cos (b ( 4 (a If cos, 4cos ( sin 4sin cos an (b If ( 4 an ( 4sin ( sin (cos ( 4cos 4sin 4cos +, 4( ( 8( 4( ( 48 ( 4( sin cos 6. Evaluae f ( when 0 given f( sec If f( sec, hen f ( 6 sec an an f ( ( 6sec ( sec + ( an ( 8sec an When 0, f (0 8sec + 8sec an 8 8 an 0 8 8(0 cos 0 cos , John Bir
16 7. Show ha he ifferenial equaion is saisfie when e If e hen ( (e (e ( e e + + an ( (e (e ( e 4e 4e ( 4 e + 4e 4( e+ e + 4(e 4 e + 4e 8 e 4e + 4e 0 8. Show ha, if P an Q are consans an P cos(ln + Q sin(ln, hen P cos(ln + Q sin(ln P sin(ln cos(ln sin(ln Q cos(ln P + + Q Pcos(ln Qsin(ln ( [ Psin(ln Qcos(ln ]( + Hence, Pcos(ln Qsin(ln + Psin(ln Qcos(ln + + ( Pcos(ln Qsin(ln + Psin(ln Qcos(ln + ( Psin(ln + Qcos(ln + P cos(ln + Q sin(ln Pcos(ln Qsin(ln + Psin(ln Qcos(ln Psin(ln + Qcos(ln 0 + P cos(ln + Q sin(ln 96 04, John Bir
17 9. The isplacemen s of a mass in a vibraing ssem is given b: ( s s naural frequenc of vibraion. Show ha: + ω + ω s 0 s + eω where ω is he If s ( + eω ( s ( + ωe + (e ( ωe ωe + e ω ω ω ω ω + ω e [( ω ωe + e ( ω] ωe an ( ( Hence, s s ω ω ω ω ( ( ω e + ω e ωe ωe ω ω ω ω ( ω e + ω e ωe ω ω ω + ω + ωs ω ω+ ( ω ω ω ω ω + ω ω ω ω + ω + ω + ω e e e [ e e e ] [( e ] ω e + ω e ωe ω e ω e + ωe + ω e + ω e 0 ω ω ω ω ω ω ω ω 97 04, John Bir
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