AP CALCULUS Test #6: Unit #6 Basic Integration and Applications

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1 AP CALCULUS Test #6: Unit #6 Bsi Integrtion nd Applitions A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS IN THIS PART OF THE EXAMINATION. () The ext numeril vlue of the orret nswer does not lwys pper mong the hoies given. When this hppens, selet from mong the hoies the numer tht est pproximtes the ext numeril vlue. () Unless otherwise speified, the domin of funtion f is ssumed to e the set of ll rel numers x for whih f(x) is rel numer. Multiple Choie. If k is onstnt, find its vlue if kx x dx 8. k A. 9 B. C. D. 9 E. 8. After eing poured into up, offee ools so tht its temperture, T(t), is represented y the funtion t T( t) 7 e, where t is mesured in minutes nd T(t) is mesured in degrees Fhrenheit. Wht is the verge temperture of the offee during the first four minutes fter eing poured? A F B F C F D F E F Dily Lessons nd Assessments for AP* Clulus AB, A Complete Course Pge 549

2 . Whih of the following integrls is/re true if f(x) is differentile funtion on the open intervl (, ), is on the open intervl (, ), f ( x) dx 6 nd f ( x) dx. I. f ( x) dx 4 II. f ( x) dx f ( x) dx III. f ( x) dx 9 A. I nd II only B. II only C. I only D. II nd III only E. III only ( ) t 4. At time t = wter egins leking from tnk t the rte of L(t) = 5e gllons per minute, where t is mesured in minutes. How muh wter hs leked out of the tnk fter 5 minutes? A..6 gllons B..676 gllons C..5 gllons D.. gllons E gllons 5. The grph to the right represents the rte t whih people rrive t n musement prk ride throughout the dy, where t is mesured in hours from the time the ride egins opertion. If there were 75 people in line when the ride egn opertion, How mny people hve wited in line for the ride fter 4 hours? A. 8 B. 675 C. 4 D. 475 E. 6 Dily Lessons nd Assessments for AP* Clulus AB, A Complete Course Pge 55

3 6. Using right Riemnn sum over the given intervls, estimte 5 F ( t) dt. 5 A. 7 B. 66 C. 564 D. 474 E At.m. the temperture t ski resort egins to inrese using the snow to egin to melt t rte defined y the eqution t M ( t) 8os. If there re 78 ui yrds of snow t tht point, how muh snow remins t 5 p.m. if no dditionl snow hs een dded nd the temperture hs ontinully inresed throughout the dy? A ui yrds B ui yrds C ui yrds D ui yrds E. 7.8 ui yrds Free Response The rte t whih people enter n musement prk on given dy is modeled y the funtion E defined y 56 E ( t). t 4t 6 The rte t whih people leve the sme musement prt on the sme dy is modeled y the funtion L defined y 989 L ( t). t 8t 7 Both E(t) nd L(t) re mesured in people per hour nd time t is mesured in numer of hours fter midnight. These funtions re vlid for 9 < t <, the hours during whih the prk is open. At t = 9, there re no people in the prk.. The prie of dmission to the prk is $5 until 5: p.m. After 5: p.m., the prie of dmission to the prk is $. How mny dollrs re olleted from dmissions to the prk on the given dy? Round your nswer to the nerest whole numer. ( t dt for 9 < t <. Find the vlues of H(7) nd H '(7) nd explin, using 9 orret units, the mening of oth vlues in the ontext of the prk.. Let H t) E( t) L( t). During the hours tht the prk is open, 9 < t <, wht is the mximum numer of people in the prk t ny given moment? Show your work nd justify your nswer. Dily Lessons nd Assessments for AP* Clulus AB, A Complete Course Pge 55

4 AP CALCULUS Test #6: Unit #6 Bsi Integrtion nd Applitions A GRAPHING CALCULATOR IS NOT ALLOWED FOR THIS SECTION OF THE EXAM. () The ext numeril vlue of the orret nswer does not lwys pper mong the hoies given. When this hppens, selet from mong the hoies the numer tht est pproximtes the ext numeril vlue. () Unless otherwise speified, the domin of funtion f is ssumed to e the set of ll rel numers x for whih f(x) is rel numer. Multiple Choie 5. If f ( x) dx 6 nd ( ) 4 f x dx, then 5 ( f ( x)) dx A. B. C. D. 5 E. 5. If g(x) = x x + 4 nd f(x) = g '( x), then f ( x) dx = A. B. C. D. 4 E. 4 4 Dily Lessons nd Assessments for AP* Clulus AB, A Complete Course Pge 559

5 . Given to the right is the grph of the derivtive of funtion, y = f '( x) f '( x). If f() = 9, wht is the vlue of f()? A. + π B π C. 7 + π D. + 4π E. + π 4. A left Riemnn sum, right Riemnn sum, nd trpezoidl sum re used to pproximte the vlue of f ( x) dx, eh using the sme numer of suintervls. The grph of the funtion f is shown in the figure to the right. Whih of the sums give(s) n underestimte of the vlue of f ( x) dx? I. Left Sum II. Right Sum III. Trpezoidl Sum A. I only B. II only C. III only D. I nd III only E. II nd III only Dily Lessons nd Assessments for AP* Clulus AB, A Complete Course Pge 56

6 5. A tnk ontins 5 liters of oil t time t = 4 hours. Oil is eing pumped into the tnk t rte of R(t), where R(t) is mesured in liters per hour, nd t is mesured in hours. Seleted vlues of R(t) re given in the tle ove. Using right Riemnn sum with three suintervls nd dt from the tle, wht is the pproximtion of the numer of liters of oil tht re in the tnk t time t = 5 hours? A B. 68. C. 4.9 D. 6.6 E The funtion f is ontinuous on the losed intervl [, 6] nd hs the vlues given in the tle ove. The trpezoidl pproximtion for 6 f ( x) dx, found with suintervls of equl length is 5. Wht is the vlue of k? A. B. 6 C. 7 D. E. 4 Dily Lessons nd Assessments for AP* Clulus AB, A Complete Course Pge 56

7 Use the tle of vlues elow to nswer question 7. f(x) is twie-differentile funtion for whih vlues of f, f ', nd f '' re given. x f(x) 7 7 f (x) 5 5 f (x) 7. Find the vlue of '( x) f "( x) A. B. C. D. 9 E. 4 f dx. Free Response A prtile is moving long stright pth. The veloity of the prtile for < t < is shown in the tle elow for seleted vlues of t nd veloity is t mximum t t = se. Answer the questions tht follow. t v(t) m/se Using the midpoints of five suintervls of equl length, pproximte the vlue of v ( t) dt. Using orret units, explin the mening of the vlue of v ( t) dt.. Find the verge elertion of the prtile over the intervl 6 < t < 8. Express your nswer using orret units.. Find n pproximtion of v '(6). Using orret units, explin wht this vlue represents nd stte, providing justifition, if the speed of the prtile is inresing or deresing t t = 6? d. During wht intervl(s) of time is the elertion negtive? Justify your nswer. Dily Lessons nd Assessments for AP* Clulus AB, A Complete Course Pge 56

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