Net Change In 1-3, write a complete sentence including appropriate units of measure.

Transcription

1 AP Calculus CHAPTER WORKSHEET INTEGRALS Name Sea # Ne Change In 1-3, wrie a complee senence including appropriae unis of measure. 1. If 1 w' is he rae of growh of a child in pounds per year, wha does w ' d represen?. If oil leaks from a ank a a rae of Dae R 1 gallons per minue, wha does R d represen? n' bees per week. Wha 3. A honey bee populaion sars wih 1 bees and increases a a rae of does 1 1 n ' d represen? 4. If he unis for x are fee and he unis for m x are pounds per foo, wha are he unis for Wha unis does 8 x m dx have? dm? dx ESSAY FROM 4B AB EXAM: YOU MAY USE YOUR CALCULATOR. For, he rae of change of he number of mosquioes on Tropical Island a ime days is modeled by M cos mosquioes per day. There are 1 mosquioes on Tropical Island a ime =. a) Show ha he number of mosquioes on Tropical Island is increasing a ime = 6. b) Se up an inegral o deermine he ne change in he mosquio populaion on he island for. Round your answer o he neares whole number. c) How many mosquioes will be on he island a ime =? 6. The linear densiy of a rod of lengh 4 meers is given by Dx x, measured in kilograms per meer, where x is measured in meers from one end of he rod. Se up an inegral o find oal mass of he rod. Use he FTC o evaluae your inegral wihou using your calculaor. SEE OTHER SIDE

2 ESSAY FROM 1 AB EXAM: 7. The rae a which waer flows ou of a pipe, in gallons per hour, is given by a differeniable funcion R of ime. The able o he righ shows he rae as measured every 3 hours for a 4-hour period. a) Use a midpoin Riemann sum wih 4 subdivisions of equal lengh o approximae 4 R d. Using correc unis, explain he meaning of your answer in erms of waer flow. b) The waer flows ou of he pipe and ino a waer ank ha originally conained 11 gallons of waer. If he oal capaciy of he ank is 3 gallons, will he ank be able o sore all he waer over he 4-hour period? 8. The marginal cos of manufacuring x yards of a cerain fabric is C' x 3.1x.6x, in dollars per yard. Se up an inegral o deermine he increase in cos if he producion level is raised from yards o 4 yards. (Noe: noice ha your inegral could be evaluaed by paper and pencil, using he FTC Bu you may use your calculaor o evaluae he inegral!) ESSAY FROM AB EXAM: YOU MAY USE YOUR CALCULATOR TO EVALUATE ALL INTEGRALS. The rae a which people ener an amusemen park on a given day is modeled by he funcion E 16 defined by E. The rae a which people leave he same amusemen park on he same day is modeled by he funcion L defined by L. Boh E and L are measured in people per hour and ime is measured in hours afer midnigh. These funcions are valid for 3, he hours during which he park is open. A ime =, here are no people in he park. a) Se up an inegral and find how many people enered he park by : P.M. ( =.) Round your answer o he neares whole number. b) The price of admission o he park is $1 unil : P.M. ( =.) Afer : P.M., he price of admission o he park is $11. Se up wo inegrals and find how many dollars were colleced by he park on he given day. Round your answer o he neares whole number. c) Is he number of people in he park increasing or decreasing a : P.M. ( = )? Jusify your answer. d) In he conex of his problem, wha does L d 37 E represen?

3 ESSAY FROM 3 AB EXAM: 1. The rae of fuel consumpion, in gallons per minue, recorded during an airplane fligh is given by a differeniable and sricly increasing funcion R of ime. The graph of R and a seleced able of values of R, for he ime inerval minues, are shown below. a) Use daa from he able o find an approximaion for R ' 4 o your answer and indicae unis of measure. b) Approximae he value of R d indicaed by he daa in he able. Explain he meaning of consumpion for he airplane. c) Is your approximaion for R d Jusify your answer. 1 d) Food for hough Explain he meaning of R d he airplane.. Show he compuaions ha lead using a lef Riemann sum wih he five subinervals R d in erms of fuel from par (b) and overesimae or an underesimae? in erms of fuel consumpion for

4 AP Calculus CHAPTER WORKSHEET INTEGRALS Ne Change ANSWER KEY 1 1. w ' d is he ne change in he child s weigh measured in pounds from o 1 years of age. 1. R d is he amoun of oil leaked from he ank measured in gallons from o 1 minues n ' d is he bee populaion measured in bees 1 weeks laer. 4. Unis for Unis for dm are pounds per foo per foo or pounds per foo dx 8 x m dx are pounds.. 6 M, he number of mosquioes is increasing a = 6. a) Since 6 6 cos M. The mosquio populaion decrease by 36 individuals. b) d cos d c) 1 d M. There are 64 mosquioes a = xdx x 14 D dx 46 kilograms a) d R gallons. This is he oal amoun of waer ha flowed from he pipe in he 4-hour period. b) The iniial amoun in he ank added o he amoun ha flowed from he pipe yield 4 d R gallons. This is more han he capaciy of he ank, so ha ank will no be able o sore all he waer.

5 xdx 3.1x.6x dx 8, C ' dollars. Increasing producion from yards o 4 yards means an added cos of $8,.. 16 a) E d d So 64 people enered he park by PM b) 1 E d 11 E d 1 d 11 d 14, Therefore, he amoun colleced was $14,48. OR 3 E E 16 d d d d , 64 people enered he park by PM and, 171 people enered he park beween PM and 11 PM. Therefore, he amoun colleced was $14, 41 E L 38. people per hour. Since his rae is negaive, he amoun of people in he park is decreasing (people are leaving he park faser han enering in a =.) c) 81 d) L d 37 E is he ne change in he number of people in he park since i opened unil PM. Also, since when he park opened here were no people in i, his is also he number of people in he park a PM.. 1. a) ' 4 R4 R R 1. gallons/minue 4 b) R d gallons of fuel consumed by he airplane during he minues of fligh. c) The approximaion of 37 gallons is less han he acual value of R d. This is he oal number of, because R is sricly increasing on. 1 d) R d is he average of he rae of fuel consumpion measured in gallons/minue during he minues fligh.