Chapter 1: Review Exercises

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Natalia Lipińska
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1 Chpter : Review Eercises Chpter : Review Eercises - Evlute the following integrls: ( + ) ( + ). ( ) (csc + + ) sin tn 6. ( )( + ) 7. ) ( + )( ). ( ) 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

2 Chpter : Review Eercises - 6 Evlute the following integrls: ( + ). p +. p ( + ) + +. p ( ) ( + )( + ). sin 6. sin 9. sin sin ( + ) 6. csc sec ( + ) sin sin cos + sin sin + 9. sin sin sin ( + ) sin cos ( + ) sin ( + ) sec tn

3 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 + )

4 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 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 =

5 Chpter : Review Eercises 9 - Evlute the following integrls: ( ) 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: ( + ) ( + ). ( + ). < 7. f () =, [, ] 8. f () = +, [, ] + p Find the verge vlue of the function f on the given intervl.. f () =, [, ] 6. f () = 9, [, ]

6 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 () =, [, ] Find the derivtive of the following functions: sin t dt 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 the error. 7. t + dt t + dt 66. p, n= 7. 7., e, n = n= 7. p +, n= 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 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 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: 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) The vlue of the integrl is equl to + by using + + by using

8 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 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 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 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 ) )

10 Chpter : Review Eercises - Find the derivtives of the following functions:. y = (tn )tn 8. y = 6. y = 7. y = y = sin. y = (ln )tn Evlute the following integrls: + 6. sin ln. e e + e e e eln(sin sin 6. ) etn sec ln 6. + cos (ln ) ( + ) (ln ) sin sin + 6. e ( e ) where > 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 )

11 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 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

12 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 e 6 sinh cosh.. tnh sech. esinh cosh... e csch cosh y = sinh (tnh ). lim cosh sech tnh sech. tnh

13 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

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