Curve on Clcultor eperience Fin n ownlo (or type in) progrm on your clcultor tht will fin the re uner curve using given number of rectngles. Mke sure tht the progrm fins LRAM, RRAM, n MRAM. (You nee to show Volpe this on your clcultor to get creit for the curve.) For -, () Evlute the integrl, n (b) check your nswer by ifferentiting. ( ). ( ). ( ).. 6. sin cos 7. sec csc cot 8. sec (sec tn ). 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 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 =. 9. sin csc. Evlute the integrl sin by multiplying the numertor n enomintor by n pproprite epression.. ) Grph some representtive integrl curves of f ( ). b) Fin n eqution for the integrl curve tht psses through the point (, 7).
. () 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 = subintervls. Is tht pproimtion greter thn, less thn, or equl to A? Eplin you resoning, n check you conclusion by clculting the left enpoint pproimtion. 6. () (c) 7. () (b) () (b) cos (c) Sketch the rectngles for the right enpoint pproimtion to the re A using n = subintervls. 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 = subintervls. Is tht pproimtion greter thn, less thn, or equl to A? Eplin you resoning, n check you conclusion by clculting the mipoint pproimtion.. Fin the left enpoint, right enpoint, n mipoint pproimtions of the re uner the curve y over the intervl [, ] using n= subintervls.. Fin the left enpoint, right enpoint, n mipoint pproimtions of the re uner the curve y cos over the intervl [, ] using n= subintervls. In eercises -, use the clcultor progrm tht you now hve 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= subintervls. (c) () 8. Using Riemnn Sums (left), write in epne form (use n subintervls) 6 ( ) In problems 9-, sketch n evlute the integrl. 9..... 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.
. () 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 / ). 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 ( ), n f ( ), evlute. ( f g)( ) b. 7 g ( ) c. f ( ) 7. Wht is the verge vlue of orintes of the semicircle y on [-, ] 8. The verge vlue of f() = ln() on the intervl [,] is 9.Fin using the trpezoi rule with n=. Use three eciml plces.. The verge vlue of cos over the intervl is ectly. The verge vlue of csc over the intervl is ectly 6. Fin t sin. Fin ( t t) t. Fin t. Fin cos 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=.. Fin the lineriztion of f ( ) sin t t 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 subintervls on the intervl to 6. Fin the lineriztion of f ( ) sin t t b. Write out the right Riemnn sum for f() = 7 with n subintervls on the intervl to.. Write out the left Riemnn sum for f() = with n subintervls on the intervl to. b. Write out the right Riemnn sum for f() = with n subintervls on the intervl to.. Write out the left Riemnn sum for f() = + on the intervl - to with n subintervls b. Write out the right Riemnn sum for f() = + on the intervl - to with n subintervls.. Write out the left Riemnn sum for f() = sin on the intervl π to π with n subintervls b. Write out the right Riemnn sum for f() = sin on the intervl π to π with n subintervls. Use the following f(t) s to solve for K in the following integrl K. t t b. t c. sint cost. t t 7. Fin the lineriztion of f ( ) (t ) 8. Fin the lineriztion of 6 f ( ) (t t = 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 ) 8. ( ) 67. e t 68. t 69. ln( t) t 7. cos( t) 7. u 7. u 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. ) 8. ( sin cos 8. ( sec csc cot ) 8. ( sec (sec tn )) 8. ( sin csc ) 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 86.