Chpter : Review Eercises Chpter : Review Eercises - Evlute the following integrls:..... 6. 8. ( + ) 9. +.. ( + ). ( ). 8. 9....... 6. 7. (csc + + ) sin tn 6. ( )( + ) 7. ) 8.. + ( + )( ). ( ) sin sin sec csc. + ( + tn ) (Hint: tn = sec tn.. sec (tn sec ) 9. (sec ) (sec tn + ). ( + ) ( ) ( + ) 7. sin ( + ) sec (sec + tn ) csc (csc + cot ). sin
Chpter : Review Eercises - 6 Evlute the following integrls: ( + ). p +. p ( + ) + +. p ( ) +. 7. 8. 9...... ( + )( + ). sin 6. sin 9. sin 6. 6. + sin ( + ) 6. csc 6.. 6. 7. 8. sec ( + ) sin sin cos + sin + 8. 7. sin + 9. sin sin. 6. 9. 6. sin ( + ) sin cos ( + ) sin ( + ) sec tn
Chpter : Review Eercises 6-7 Choose the correct nswer: sin is equl to + + + c + + c sin (tn ) is equl to cos (tn ) + c sin (tn ) + c 6. The vlue of the integrl () + + c (b) + + c 66. The vlue of the integrl () cos (tn ) + c (b) sin (tn ) + c p 67. The integrl + is equl to () + + c (b) ( + ) + c 68. The integrl is equl to tn + c + c () tn + c (b) tn + c 69. The vlue of the integrl () (b) ( + ) + c ( + ) + c sec is equl to cot + 7. The vlue of the integrl () sin + + c (b) sin + + c + sin cot tn sin + sin + + c Chpter : Review Eercises - Epress the following sums in terms of n:. nk= (k ). nk= (k + ) -8 Find the following sums:. nk= (k k + ). nk= (k + k + )
Chpter : Review Eercises. k= (k + ) 7. k= (k + k) 6. j= 8. i= (i ) j+ 9 - For the prtition P, find the norm k P k. 9. P = {,.,.,.,.6,, 6}. P = {,.,,,., 6} - 6. P = {,.,,.,., }. P = {,.,.9,.,.7,, } Find Riemnn sum RP for the given function f by choosing the mrk ω s follows: () the left-hnd endpoint, (b) the right-hnd endpoint, the midpoint,. f () = +, {,.,,.,,, 6}. f () = +, {,.,,.,,., }. f () =, {,,,.,,,.} 6. f () =, {,,,,,, 6} Find the re under the grph of f from to b by tking the limit of Riemnn sum. 7-7. f () = +, =, b =. f () = +, =, b = 8. f () =, =, b =. f () =, =, b = 9. f () =, =, b =. f () = + +, =, b = - 8 Let A be the re under the grph of the given function f from to b. Approimte A by dividing [, b] into subintervls of equl length nd using inscribed polygons (AIP ) nd circumscribed polygons (ACP ) for = / nd by tking the limit of Riemnn sum.. f () = +, =, b = 6. f () = +, =, b =. f () =, =, b = 7. f () =, =, b =. f () =, =, b = 8. f () = + +, =, b =
Chpter : Review Eercises 9 - Evlute the following integrls: 9.. 6. ( ) 7. ( )( + ) π ( + ) 8. (6 + ) 9... π sin. π. ( ) π p +.... b integrls: c f () =, If b f () = nd b b b f () 7. b f () g() 6. c. g() = where c (, b), evlute the following f ().. π csc π/ - 8 sec (tn sec ) b f () + g() f () + g() 8. f () + 7g() 9 - Use the properties of the definite integrls to prove the following inequlities without evluting the integrls: 9.... + ( + ) ( + ). ( + ). < 7. f () =, [, ] 8. f () = +, [, ] + p + - 9 Find the verge vlue of the function f on the given intervl.. f () =, [, ] 6. f () = 9, [, ]
6 Chpter : Review Eercises 9. f () = 6 +, [, ] Find the number z tht stisfies the Men Vlue Theorem for the function f on the given 6-6 intervl. 6. f () = +, [, ] 6. f () =, [, ] 6. f () =, [, 9] 6. f () =, [, ] 6-7 6. Find the derivtive of the following functions: sin t dt 68. 6. cos t dt t 69. dt sin (t + ) 7. dt 7. tn t dt dt t + By using the trpezoidl rule, pproimte the definite integrl for the given n, then estimte + 7-7 the error. 7. t + dt 6 67. t + dt 66. p, n= 7. 7., e, n = n= 7. p +, n=6 76-79 By using the Simpson s rule, pproimte the definite integrl for the given n, then estimte the error. π 76., n= sin 77. ln( + e ), 78. e, n=6 n=6
7 Chpter : Review Eercises π 79., n= 8-8 Find the minimum number of subintervls to pproimte the integrl the trpezoidl rule such tht the error is less thn 8. 8. 8-8 Find the minimum number of subintervls to pproimte the integrl the Simpson s rule such tht the error is less thn 8. Choose the correct nswer: 8-6 8. The sum nk= (k ) is equl to () n (n ) (b) n(n ) n (n +) 8. n (n ) 8. The sum lim nk= ( nk ) is equl to n () (b) 86. If nk= (k + α) = n (n ), then the vlue of α is equl to () n (b) 87. If k= (k + ) =, then the vlue of is equl to () (b) 88. If k= (αk + k ) =, then the vlue of α is equl to () (b) 89. If 6k= (k + k + α) =, then the vlue of α is equl to () (b) 9. The verge vlue of the function f () = + on [, ] is equl to () (b) 9. The verge vlue of the function f () = sin on [, π ] is equl to () π (b) π 9. The verge vlue of f () = on [, ] is equl to () (b) 9. The verge vlue of f () = sin on [ π, π] is equl to () π (b) π 9. If F() = () 8 + R t + dt, the F () is equl to (b) 8 + 8 + 9. The vlue of the integrl is equl to + by using + + by using
8 Chpter : Review Eercises () (b) 96. If f () =, f () = 7, f () = nd f () =, the vlue of the integrl is equl to (b) 6 () 6 t dt, then F ( π) is equl to π π 97. If F() = ( + ) f ( + ) cos π () (b) 98. If F() = sin t dt, then F () is equl to () sin 6 sin 8 (b) sin 6 sin 8 sin 6 sin 6 sin 6 + sin 8 99. Theq number z tht stisfies the Men q Vlue Theorem for f () = on [, ] is () 8 (b) 8. The number z tht stisfies the Men Vlue Theorem for f () = + on [, ] is () (b) +. If F() = tn(t ) dt, then F () is equl to () tn ( + + ) + tn ( + ) (b) tn ( + + ) tn ( + ). If F() = tn ( + ) tn ( ) p + t dt, then F () is equl to 6 6 (b) 6 (). If f ( t) dt =, then f () is equl to () (b). The vlue of the integrl (b) +. The derivtive of the integrl (b) tn () + tn 6. If G() = () e lnt e (b) () d tn t dt is equl to dt sec + sec dt, then G (e) is equl to e e
9 Chpter : Review Eercises Chpter : Review Eercises -6 Solve for in the following equtions:. = eln. ln = ln + ln. ln = ln( + ) + ln( ). ln =. ln = ln ln 8 7 - Find the following limits: 7. lim ln 6. e + e 8 =. lim ln e 8. lim. lim log + e 9. lim e +. lim+ ln sin +ln - Find the derivtives of the following functions:. f () = ln 9. f () = e+. f () = ln( + + ). f () = esin. f () = ln. f () = esec 6. f () = ln sin 7. f () = ln + 8. f () = ln( ). f () = sin(e. f () = e + 9. f () = sin ln. f () = e ln. f () = ln( sin ) +. f () = ln + ln( ). f () = (ln ). f () = ln( + ). f () = e sec. f () = eln + 6. f () = e+ sin +. f () = e+ 6. f () = e tn 7. f () = e ln 8. f () = e 9. f () = πcos. f () = sin. f () =. f () = tn(sin ) 7. f () = e +. f () = log ( 6+ ) 8. f () = ln(sin e ). f () = log(ln ) )
Chpter : Review Eercises - Find the derivtives of the following functions:. y = (tn )tn 8. y = 6. y = 7. y = - 7.. 9. y = sin. y = (ln )tn Evlute the following integrls: + 6. sin 6. +. + ln. e e + e e e eln(sin sin 6. ) etn sec 6.. + 66. ln 6. + cos (ln ) ( + ) (ln ) 8. 9. + 68. 69. 7. sin sin + 6. e ( e ) 7. 6. 67. 7. + 7. + where > + 7-89 Choose the correct nswer: 7. If f () = log =, then is equl to () (b) 7. The vlue of the integrl () ln (b) ln is equl to ln 7. If f () = +, then f () is equl to () ( + + ln )+ (b) (ln + )+ ln ( + ln )+ ( + + ln )
Chpter : Review Eercises +e 76. lim e+e is equl to () (b) None of these R 77. The integrl tn is equl to () (b) sec + c ln sec (b) sin + c () ln() sin + c 79. The integrl e (+e) sec + c ln ln(sin ) is equl to 78. The integrl () e is equl to (e + ) (b) 8. If f () = ln then f (e) is equl to () (b) e sin + c ln + c (+e) e 8. lim+ sin ln is equl to () (b) 8. If f () = ln(ln ) then f () is equl to () ln (b) ln (ln) 8. The integrl ln sin is equl to () sin + c (b) (ln )sin + c tn is equl to sec () ln sec + tn + sin + c ln sec + tn sin + c sin ln sin ln + c 8. The integrl 8. The vlue of the integrl () ln (b) ln is equl to 86. If f () =, then f () is equl to () (b) e 87. The vlue of the integrl () e (7)7 is equl to ln 7 (b) ln sec + tn + c ln sec sin + c (b) ln 7 9 ln 7 88. If F() =, the F (e) is equl to () (b) e e e 89. If log =, then is equl to () (b) ee e 7 ln 7
Chpter : Review Eercises Chpter : Review Eercises - 8 Find y in the following:. y = sin ( + ). y = cos. y = tn (sinh ). y = esech cosh (cosh ) sinh cosh. y = tn. y =. y = sec. y = sech. y = sinh ( + ). y = coth 6. y = cosh e. y = cosh 7. y = tnh 8. y = e cosh 9. y = sinh + cosh 9 - Find the following limits: 9. lim sinh 6. y = e tnh - 8. y = tnh ( + ). lim e tnh. lim esech Evlute the following integrls: e + 8 6 + 9 8 6 + 9 e 6 sinh cosh.. tnh sech. esinh cosh... e csch cosh 7. 6. 8. 7. y = sinh (tnh ). lim cosh sech tnh sech. tnh 6. 7. 8. 9.. +. 9....
Chpter : Review Eercises - Choose the correct nswer:. The derivtive of the function f () = tn (sinh ) is equl to () sech (b) csch tnh sech. The vlue of the integrl sinh is equl to (b) e e e. If f () = cosh, then f () is equl to () (b) + () None of these, is equl to + c (b) cosh sec + c sec None of these sec 6. The integrl () cosh cosh 7. If f () = tnh (), then f () is equl to sin () csc (b) csc os, is equl to + sin (b) tn (sin ) + c 8. The integrl () +sin is 6 cos (b) 6 + c + tnh (sin ) + c 9. The vlue of the integrl () cos 6 is + (b) sinh + c sin sin. The vlue of the integrl () sin + c sinh sin cosh is equl to sinh () tn (sinh ) + c (b) tn (sinh ) + c osh tnh (sinh ) + c. The integrl. If F() = tn + tn ( ) where 6=, then F () is equl to () + (b) + +. The derivtive of the function f () = tn (sinh ) is equl to (b) sec (sinh ) cosh () +sinh is equl to + (b) sin + c sinh cosh sinh. The vlue of the integrl () sinh. The vlue of the integrl cosh is equl to () (b) e e e e sin