x ) dx dx x sec x over the interval (, ).

Transcription

1 Curve on 6 For -, () Evlute the integrl, n (b) check your nswer by ifferentiting. ( ). ( ). ( ).. 6. sin cos 7. sec csccot 8. sec (sec tn ) 9. sin csc. Evlute the integrl sin by multiplying the numertor n enomintor by n pproprite epression.. ) Grph some representtive integrl curves of f ( ). (Hint Wht is n integrl of the function? It coul be +, right? Or + or lots of nswers write? Sketch bunch of the nswers tht it coul be. b) Fin n eqution for the integrl curve tht psses through the point (, 7).. Use grphing utility to generte some representtive integrl curves of the function f ( ) sec over the intervl (, ).. Suppose tht point moves long curve y f () in the y-plne in such wy tht t ech point (, y) on the curve the tngent line hs slope ( sin ). Fin n eqution for the curve, given tht it psses through the point (, ). For 6-7 solve the initil-vlue problem. y 6. (), y() y (b), y() y (c), y( ) t t 7. Fin the generl form of function whose secon erivtive is [Hint: Solve the eqution f ( ) for f () by integrting both sies twice. For 8-9 fin n eqution of the curve tht stisfies the given conitions. 8. At ech point (, y) on the curve the slope is + ; the curve psses through the point (-, ). 9. At ech point (, y) on the curve, y stisfies the conition y 6; the line y = is tngent to the curve t the point where =.. () Use n pproprite geometric formul to fin the ect re A uner the line + y = over the intervl [, ]. (b) Sketch the rectngles for the left enpoint pproimtion to the re A using n =. Is tht pproimtion greter thn, less thn, or equl to A? Eplin you resoning, n check you conclusion by clculting the left enpoint pproimtion. (c) Sketch the rectngles for the right enpoint pproimtion to the re A using n =. Is tht pproimtion greter thn, less thn, or equl to A? Eplin you resoning, n check you conclusion by clculting the right enpoint pproimtion. () Sketch the rectngles for the mipoint pproimtion to the re A using n =. Is tht pproimtion greter thn, less thn, or equl to A? Eplin you resoning, n check you conclusion by clculting the mipoint pproimtion.

2 . Fin the left enpoint, right enpoint, n mipoint pproimtions of the re uner the curve y over the intervl [, ] using n=.. Fin the left enpoint, right enpoint, n mipoint pproimtions of the re uner the curve y cos over the intervl [, ] using n=. In eercises -, use clcultor progrm to fin the left enpoint, right enpoint, n mipoint pproimtions to the re uner the curve y f () over the stte intervl using n =, n =, n n=. You must show Volpe this ownloe progrm on your clcultor n you must show her tht you cn ctully use the progrm. (This prt of the curve requires you to go to the TI web site to lern how to ownlo progrms, then ctully o it. Volpe will sk to see these ll on the sme y in clss). y ;[, ]. y sin ;[, ]. y ln ;[, ] For 6-7, sketch the region whose signe re is represente by the efinite integrl, n evlute the integrl using n pproprite formul from geometry, where neee. 6. () (c) 7. () (c) (b) () () (b) cos 8. Using Riemnn Sums (left), write in epne form (use n ) 6 ( ) In problems 9-, sketch n evlute the integrl () Let f be n o function; tht is, f(-) = - f(). Invent theorem tht mkes sttement bout the vlue of n integrl of the form f ( ) (b) Confirm tht your theorem works for the integrls n sin( ) (c) Let f be n even function; tht is, f(-) = f(). Invent theorem tht mkes sttement bout the reltionship between the integrls f ( ) n f ( ) () Confirm tht your theorem works for the integrls / n cos( ), Define F() by / F ( ) ( t ) t Fin F ()

3 . If n is positive integer, then lim n cn be n n n n n epresse s. b. c.. e. 6. Suppose f ( ) 6, g ( ), f ( ), g ( ) evlute. ( f g)( ) b. 7 g ( ) c. f ( ) (Hint: by the time you o this pcket, we ll hve covere substitution techniclly, this shoul be substitution problem for the Chpter 6 curve) 7. Wht is the verge vlue of the semicircle y on [-, ] (Use re) 8. The verge vlue of f() = ln() on the intervl [,] is (clcultor. Show wht you type in. eciml plces) 9.Fin Use three eciml plces. using the trpezoi rule with n=.. The verge vlue of cos over the intervl is ectly. The verge vlue of csc over the intervl is ectly 6. Fin tt sin. Fin ( t t) t tt. Fin costt. Fin 6. Fin the re uner the curve of y = - from = to = with n= inscribe rectngles. 7. Fin the re uner the curve of y = - from = to = with n= circumscribe rectngles. 8. Fin the re uner the curve of y = - from = to = using the trpezoi rule with n=. 9. Fin the re uner the curve of y = - from = to = using the mipoint formul with n=.. Write out the left Riemnn sum for f() = 7 with n on the intervl to b. Write out the right Riemnn sum for f() = 7 with n on the intervl to.. Write out the left Riemnn sum for f() = with n on the intervl to. b. Write out the right Riemnn sum for f() = with n on the intervl to.. Write out the left Riemnn sum for

4 f() = + on the intervl - to with n b. Write out the right Riemnn sum for f() = + on the intervl - to with n.. Write out the left Riemnn sum for f() = sin on the intervl π to π with n b. Write out the right Riemnn sum for f() = sin on the intervl π to π with n. Use the following f(t) s to solve for K in the following integrl f ( t) t K. f ( t) t (Hint: Actully fin the integrl on the left from to then K. Fin the integrl of the function gin from to. You hve two equtions set = to ech other solve for K ) f ( t) t t b. f ( t) t c. f ( t) sint cost. f ( t) t t For 9-6 you re given the eqution for velocity, nswer the following questions:. Wht is the totl istnce trvele? b. Wht is the isplcement? c. Wht is the position of the prticle if the initil position is =. Show sketch for ech of the equtions. 9. y' sin from = to 6. y' sin from to 6. y' from = to 6. y ' sin from = to 7 (use the clcultor for intersections n bsic mth) 6. y' from = - to 6. y' ln from ½ to 6. y' ( ) from = to 66. sin( t ) t. Fin the lineriztion of f ( ) sin 6. Fin the lineriztion of f ( ) sin tt tt t t 67. e t t t 68. t 69. ln( t) t 7. Fin the lineriztion of f ( ) (t ) t 8. Fin the lineriztion of 6 f ( ) (t t = t) t t t =. 7. u u tt cos( t)

5 7 7. Fin the re uner the curve y over the intervl [, ]. Mke sketch of the region. 7. Fin the re below the intervl [-, -], but bove the curve y =. Mke sketch of the region. 76. Fin the totl re between the curve y n the intervl [-, 8]. Mke sketch of the region 77. ( ( )) 78. ( ) 79. ( ( ) ) ( ) ) 8. ( sin cos 86. Approimte the error USING THE ERROR FORMULA FOR THE TRAPEZOID RULE () n= (b) n= (c) () (e) n=6 n= n= (f) cos n= 87. Use the trpezoi rule to fin the re uner the curve for ll of the equtions in question For the integrls in 86, fin n for ech if you wnt the error to be less thn.. 8. ( sec csc cot ) 8. ( sec (sec tn )) 8. ( sin csc )