Ch AP Problems

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Daniela Chambers
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1 Ch AP Prolems. Willy nd his friends decided to rce ech other one fternoon. Willy volunteered to rce first. His position is descried y the function f(t). Joe, his friend from school, rced ginst him, nd his position is represented y g(t), the line tngent to f(t) t t=. They strted t t= nd rced for 8 seconds. Whoever rn the furthest t the end of 8 seconds ws the winner. Who won the rce nd why?. Speing of, it lwys plys second fiddle to the more fmous e. If the re ounded y oth functions, y =, =, nd = is clculted seprtely, how mny times lrger is the re under the more fmous curve thn its less renowned counterprt? A).6 B).8 C) 6. D). E).8. A prticle moves left nd right long the -is. If its position (with respect to the origin) is given y s t t t t (where t is in seconds nd s(t) is in inches), () How fst is the prticle moving when t = seconds? Where is prticle t tht time? () At wht time(s) is the prticle t rest? (c) When is the prticle moving to the left?. A prticle sits defintly t position + meters on the numer line. It follows the velocity eqution, v(t) = t - over the time intervl [, ]. How fr hs our rrognt prticle trveled during this time intervl? A. 6 meters B. meters C. meter D. meters E. meters. The region ounded y the grph f() = ln(), y =, =, nd = is first rotted out the -is nd then the originl region is rotted out the y-is. Which rottion cretes the lrger volume nd y how much? A. -rottion y.7 units B. y-rottion y. units C. -rottion y. units D. y-rottion y.6 units E. Both rottions produce the sme volumes 6. Find the re of the region ound y the y-is, the line y =, nd the curve y. A.. B. 8.7 C..8 D..7 E.. 7. A prticle moves long the -is ccording to the t position eqution ( t ). Wht is the prticle's ccelertion when the prticle hs velocity of? (A).6 (B) (C). (D).88 (E). 8. Assume tht prticle (ll hopped up on cffeine) is moving long the y-is, nd its velocity (in ft/sec) t ny time t (in seconds) is given y v ( t ) t t t. () If its position t time t = is two units elow the origin, find the position eqution (t). () Wht is the prticle's verge ccelertion on the intervl t = [,]? Include units. (c) Wht is the prticle's verge position on the sme intervl? (d) How mny times does the prticle chnge direction on the t intervl [,]? (e) When, during the t intervl [,], is the prticle moving upwrd? (f) Wht is the totl distnce trveled y the prticle on [,]?. Upon the relese of the new Pc-Mn movie, Pc- Mn: Integrl Adventure II, Midwy hs decided to crete Pc-mn doll. The doll hs volume equl to the volume of the solid generted y rotting the grph ounded y y,y,y 6 out the -is from to, where the volume is in cuic meters.. Grph Pc-Mn's re ounded y y, y, y 6 from to.. Find the volume of the Pc-Mn doll. c. If Midwy hs only llotted cuic meters of stuffing per Pc-Mn doll, will they e le to me the doll or should they order more filling?. Leon lives on lnd loced in etween the lines y =, =, = nd the nturl logrithmic curve y = ln(). Leon would love to hve less lnd. So let's lessen his lot y lots y legislting tht ll lots must e lessened y 8%. Leving the left ( = ) lone, ly down new line (to replce = ) tht lessens Leon's lot y the legislted mount. The line tht would lessen it is: A) =.6 B) =. C) =. D) =.7 E) =.. A solid is formed y revolving the grph of the line y cos. etween = nd = out the -is. Clculte the volume of wht would e the

2 upper hlf of the formed hourglss if it were positioned verticlly.. The Settle Arch ws designed y grphing the following two prols: y nd 6 y. The re etween the two curves represents two-dimensionl view of the structure. It hs uniform width of feet nd the nd y used in ech eqution re lso mesured in feet.. Wht is the verticl height of the structure?. Wht is the volume of the structure? Show how you clculted this using clculus! c. Fresh concrete weighs out pounds per cuic yrd. How much will the new structure weigh?. The re in (in squre units) under the curve y ounded y the lines y =, =, nd = is: (A) (B) ln (C) (D) ln (E) ln. Wht is the volume (in cuic units) of the enclosed region from question # when it is rotted out the y-is? (A).7 (B).78 (C) 7.7 (D). (E).. The se of solid is circle of rdius, nd every plne section perpendiculr to -is to dimeter is squre. The solid hs volume A. 8 B. C. D. 6 E The se of solid is the region ounded y the prol 8y nd the line y, nd ech plne section perpendiculr to the y-is is n equilterl tringle. The volume of the solid is 6 A. B. 6 C. D. E. none of these 7. The se of solid is the region ounded y y e, the -is, the y-is, nd the line =. Ech cross section perpendiculr to the -is is squre. The volume of the solid is e A. E. e B. e C. D. e e 8. If the curves of f() nd g() intersect for = nd = nd if f ( ) g( ) for ll on (, ), then the volume otined when the region ounded y the curves is rotted out the -is is equl to A. f ( )d g ( )d B. f () g() d C. f() g() d D. f () g () d E. none of these. Let f sec nd g 6 ounded y f nd g. Find:. Let the region. the re. the volume of the solid generted y revolving out y c. the volume of the solid whose cross sections perpendiculr to the -is re: i. semi-circles ii. equilterl tringles iii. squres. Let R e the region ounded y the -is, the grph of y, nd the line =.. Find the re of the region R. Find the vlue of h such tht the verticl line = h divides the region R into two regions of equl re. c. Find the volume of the solid generted when R is revolved out the -is d. The verticl line = divides the region R into two regions such tht when these two regions re revolved out the -is, they generte solids with equl volumes. Find the vlue of.. Let R e the region in the first qudrnt under the grph of y for.. Find the re of R. If the line = divides the region R into two regions of equl re, wht is the vlue of? c. Find the volume of the solid whose se is the region R nd whose cross sections cut y plnes perpendiculr to the -is re squres.. Petey Bo Betey just put on n eting ehiition for his friends. He devoured succulent oneless hm in its entirety. The hm hd the ect shpe of the region formed etween the curve f ( ), the -is, nd the lines =. nd =., when it is rotted out the -is. Assuming the coordinte system is in feet,

3 wht is the volume of the hm tht Petey scrfed down? A). cuic feet B) 7.6 cuic feet C).7 cuic feet D).8 cuic feet E) 8. cuic feet. A prticle moves long the -is so tht t time t > its position is given y t t t 7t. At wht time t is the prticle t rest? (A) t = only (B) t = only (C) t = 7 only (D) t = nd t = 7 (E) t = nd t =. A prticle moves long stright line. The grph of the prticle s position (t) t time t is shown elow for < t < 6. The grph hs horizontl tngents t t = nd t = nd point of inflection t t =. For wht vlues of t is the velocity of the prticle incresing? (A) < t < (B) < t < (C) < t < 6 (D) < t < (E) < t < nd < t < 6. A prticle moves long the -is with velocity given y vt t 6t for time t. If the prticle is t position = t time t =, wht is the position t time t =? (A) (B) 6 (C) (D) (E). A prticle moves long the -is so tht t ny time t vt.cos.t., its velocity is given y Wht is the ccelertion of the prticle t time t =? A. -.6 B C..6 D..8 E..78. A prticle moves long the -is so tht t ny time t t ln t. If the >, its ccelertion is given y velocity of the prticle is t time t =, then the velocity of the prticle t time t = is A..6 B..6 C.. D..886 E..6. The velocity, in ft/sec, of prticle moving long the - t t is is given y the function vt e te. Wht is the verge velocity of the prticle from time t = to time t =? A..86 ft/sec B. 6.7 ft/sec C..8 ft/sec D..67 ft/sec E. 7. ft/sec. The tle gives selected vlues of the velocity, vt, of prticle moving long the -is. At time t =, the prticle is t the origin. Which of the following could e the grph of the position, t, of the prticle for. t v(t) (A) (B) (C) 6. The velocity of prticle moving on line t time t is vt / t / meters per second. How mny meters did the prticle trvel from t = to t =? (A) (B) (C) 6 (D) 8 (E) 8 7. If the position of prticle on the -is t time t is t, then the verge velocity of the prticle for t is (A) (B) (C) (D) (E) 8. A prticle trvels in stright line with constnt ccelertion of meters per second per second. If the velocity of the prticle is meters per second t time seconds, how fr does the prticle trvel during the time intervl when its velocity increses from meters per second to meters per second? A. m B. m C. 7 m D. 6 m E. m (D) (E)

4 Answers. Since Joe's position is represented y the tngent to f(t), his velocity t t = is the sme s Willy's velocity t t = (since the tngent nd the function hve the sme slope t the point of tngency) which is f. But looing t the grph of f '(t) on the intervl [,8], you cn see tht Willy strted out running in negtive direction, turned round nd rn in positive direction nd then turned round once gin nd rn in negtive direction. Joe's speed nd direction were constnt nd he ws the furthest from the strting point fter 8 seconds.. B. () To find the position of the prticle, plug into s(t). To find the velocity, plug into v(t). s(t) v(t) t t v( ), therefore, the speed is in/sec. s( ) ( ) ( ), therefore, the prticle is inches to the right of the origin. () The prticle will e t rest (i.e. come to stop) whenever its velocity is. v( t ) t t ( t )(t ) t, Therefore, the prticle stops fter / of second nd second. (c) Construct wiggle grph to determine the sign of the velocity. If the velocity is negtive, the prticle is moving left. (d) The prticle my chnge direction when it stops (when velocity is ). However, you must verify tht the direction ctully chnges, which you cn do y emining the sign of the velocity (in this prolem, positive velocity indictes upwrd movement, nd negtive velocity, downwrd movement). All of this cn e ccomplished with wiggle grph of velocity. Begin y finding criticl points for velocity: t t t. Uh oh. This eqution does not hve ny solutions, nd therefore, the velocity never equls, nd the prticle never chnges direction! Thus, the prticle only goes in one direction the entire time. Plug ny point on t = [,] into the derivtive to determine the sign (nd hence the direction of the prticle on tht intervl). The prticle lwys moves upwrd. (e) We just nswered tht question. (f) Since the prticle doesn't chnge direction, the totl distnce trveled is () - () = 8 - (-) = feet... The grph of Pc-Mn's re ounded y y, y, y 6 from to loos lie this. Therefore, the prticle moves to the left on (/,).. B. D 6. E 7. E 8. () The ntiderivtive of v is position, so ( t ) v( t )dt t t t 6 t C Now, you now tht () = - from the given info, so ( ) C C ( t ) t t t 6 t () The verge ccelertion is given y the slope of the secnt line to velocity on t = [, ]. v( ) v( ) 78. ft/s (You could lso hve found the derivtive of velocity, which is ccelertion, nd used the verge vlue of function formul.) (c) To clculte verge position, find the verge vlue of the position eqution. (t)dt t t t tdt 6.. The volume of the Pc-Mn doll cn e otined y using the wsher nd dis method V r dr d 6 d. 86m c. Midwy will e le to crete the Pc-Mn doll with out. cuic meters left over.. B... V (cos( ). ) d. 6 units Becuse the structure is mde from the re etween two prols tht re oth symmetricl to the y-is, the height of the structure will e the verticl distnce from the verte of the "upper" function (the one tht is ove the other) to the y-vlue of the point of intersection of the two prols. The upper function is the second one listed nd hs verte t y=. The y-vlue of the points of intersection (found on your clcultor or on pper) is -. Therefore, the structure is 8 feet high. **Note: The loction of the -is hs no ering on the structure!. The volume will e found y ting the se re (represented y the re etween the two curves) multiplied y the distnce etween the two ses (the uniform width of feet). It cn e clculted using the following integrl:

5 6 d 7. 8 ft c. Convert the nswer from prt into cuic yrds nd then multiply y : 7.8 ft =.8 yd.8() = 6.6 pounds =.8 tons. D. D. D 6. B 7. E 8. D... Are ( 6 sec ) d c. i V ( sec) ( ( )) d.. 6 sec V d V 6 sec d. 8 ii... iii. A h V 6 sec d. 8. d d h h c. V d 8 d. 6 d 8.. A d. d c. V d. 8. C. E. A. B 6. D 7. C 8. B. C. E. A. C

7.1 Integral as Net Change Calculus. What is the total distance traveled? What is the total displacement?

7.1 Integral as Net Change Calculus. What is the total distance traveled? What is the total displacement? 7.1 Integrl s Net Chnge Clculus 7.1 INTEGRAL AS NET CHANGE Distnce versus Displcement We hve lredy seen how the position of n oject cn e found y finding the integrl of the velocity function. The chnge