ACCUMULATION. Section 7.5 Calculus AP/Dual, Revised /26/2018 7:27 PM 7.5A: Accumulation 1
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1 ACCUMULATION Secion 7.5 Calculus AP/Dual, Revised /26/2018 7:27 PM 7.5A: Accumulaion 1
2 APPLICATION PROBLEMS A. Undersand he quesion. I is ofen no necessary o as much compuaion as i seems a firs or as all he reading may seem o indicae. B. Use FTC may help differeniaing, F. C. Explain he meaning of a derivaive or definie inegral or is value in erms of he conex of he problem. D. There are problems wih one rae and wih 2 raes (in-ou problems) of change work ogeher E. Max/min and increasing/decreasing analysis. 12/26/2018 7:27 PM 7.5A: Accumulaion 2
3 BASED ON THE 2 ND FTC A. Accumulaion is he ne change using a definie inegral B. Rae of change of a quaniy over an inerval inerpreed as he change of he quaniy over he inerval C. Equaion: f = f a + a f x dx D. Known as: Final Value = Saring Value + Ne Change 12/26/2018 7:27 PM 7.5A: Accumulaion
4 2 ND FTC TO ACCUMULATION f ' x dx f f a a ACCUMULATION FINAL STARTING f ' x dx f f a a f a f a f f a f ' x dx FINAL STARTING ACCUMULATION a 12/26/2018 7:27 PM 7.5A: Accumulaion 4
5 EQUATION By applying he Firs and Second FTC, he equaion follows: f = f a + න f x dx a Final Value = Saring Value + Accumulaed Change න f x dx = f f a a 12/26/2018 7:27 PM 7.5A: Accumulaion 5
6 LABELING ANSWERS N = Noun U = Unis V = Verb A = Answer T = Time 12/26/2018 7:27 PM 7.5A: Accumulaion 6
7 EXAMPLE 1 NON CALC. (2000 AB4) Waer is pumped ino an underground ank a a consan rae of 8 gallons per minue. Waer leaks ou of he ank a a rae of + 1 gallons per minue for minues. A ime = 0, he ank conains 0 gallons of waer. a) How many gallons of waer leak ou of he ank ime from = 0 o = minues? b) How many gallons of waer are in he ank a ime = minues? c) Wrie an expression for A, he oal number of gallons of waer in he ank a ime. d) A wha ime, for 0 < < 120, is he amoun of waer in he ank a maximum? Jusify answer. 12/26/2018 7:27 PM 7.5A: Accumulaion 7
8 EXAMPLE 1A NON CALC (2000 AB4) Waer is pumped ino an underground ank a a consan rae of 8 gallons per minue. Waer leaks ou of he ank a a rae of + 1 gallons per minue for minues. A ime = 0, he ank conains 0 gallons of waer. a) How many gallons of waer leak ou of he ank ime from = 0 o = minues? 2 1/2 u du 1 d u /2 /2 /2 1 2 / /26/2018 7:27 PM 7.5A: Accumulaion 8 0 u 1 du d u u 0 1 2
9 EXAMPLE 1B NON CALC (2000 AB4) Waer is pumped ino an underground ank a a consan rae of 8 gallons per minue. Waer leaks ou of he ank a a rae of + 1 gallons per minue for minues. A ime = 0, he ank conains 0 gallons of waer. b) How many gallons of waer are in he ank a ime = minues? ' f f a f x dx f gallons d f /26/2018 7:27 PM 7.5A: Accumulaion 9 a 148 gallons 0
10 EXAMPLE 1C NON CALC (2000 AB4) Waer is pumped ino an underground ank a a consan rae of 8 gallons per minue. Waer leaks ou of he ank a a rae of + 1 gallons per minue for minues. A ime = 0, he ank conains 0 gallons of waer. c) Wrie an expression for A, he oal number of gallons of waer in he ank a ime. f f a f ' x dx 0 8min. 1 f d f d 0 a 0 12/26/2018 7:27 PM 7.5A: Accumulaion 10
11 EXAMPLE 1D NON CALC (2000 AB4) Waer is pumped ino an underground ank a a consan rae of 8 gallons per minue. Waer leaks ou of he ank a a rae of + 1 gallons per minue for minues. A ime = 0, he ank conains 0 gallons of waer. d) A wha ime, for 0 < < 120, is he amoun of waer in he ank a maximum? Jusify answer. ' f f a f x dx a 120 f 0 8min. 1 d f ' min. A 6min. Since ' is posiive from 0,6 and A' is negaive from 6,120, here mus be a maximum a = 6 minues. 12/26/2018 7:27 PM 7.5A: Accumulaion 11
12 EXAMPLE 1 SCORE SHEET Waer is pumped ino an underground ank a a consan rae of 8 gallons per minue. Waer leaks ou of he ank a a rae of + 1 gallons per minue for minues. A ime = 0, he ank conains 0 gallons of waer. 12/26/2018 7:27 PM 7.5A: Accumulaion 12
13 EXAMPLE 2 CALCULATOR (2006 AB2) A an inersecion in Thomasville, Oregon, cars lef a he rae of L = 60 sin 2 cars per hour over he ime inerval of 0 18 hours. The graph of y = L is shown. a) To he neares whole number, find he oal number of cars urning lef a he inersecion over he ime inerval, 0 18 hours. b) Traffic engineers will consider urn resricions when L() 150 cars per hour. Find all values of for which L() 150 and compue he average value of L over his ime inerval. Indicae he unis of measuremen. c) Traffic engineers will insall a signal if here is any wo-hour ime inerval during which he produc of he oal number of cars urning lef and he oal number of oncoming cars raveling sraigh hrough he inersecion is greaer han 200,000. In every wohour ime inerval, 500 oncoming cars ravel sraigh hrough he inersecion. Does his inersecion require a raffic signal? Explain he reasoning ha leads o he conclusion. 12/26/2018 7:27 PM 7.5A: Accumulaion 1
14 EXAMPLE 2A CALCULATOR (2006 AB2) A an inersecion in Thomasville, Oregon, cars lef a he rae of L = 60 sin 2 cars per hour over he ime inerval of 0 18 hours. The graph of y = L is shown. a) To he neares whole number, find he oal number of cars urning lef a he inersecion over he ime inerval, 0 18 hours. 18 L d cars 12/26/2018 7:27 PM 7.5A: Accumulaion 14
15 EXAMPLE 2B CALCULATOR (2006 AB2) A an inersecion in Thomasville, Oregon, cars lef a he rae of L = 60 sin 2 cars per hour over he ime inerval of 0 18 hours. The graph of y = L is shown. b) Traffic engineers will consider urn resricions when L 150 cars per hour. Find all values of for which L 150 and compue he average value of L over his ime inerval. Indicae he unis of measuremen. A B B 1 B A A L d cars / hour 12/26/2018 7:27 PM 7.5A: Accumulaion 15
16 EXAMPLE 2C CALCULATOR (2006 AB2) A an inersecion in Thomasville, Oregon, cars lef a he rae of L = 60 sin 2 cars per hour over he ime inerval of 0 18 hours. The graph of y = L is shown. c) Traffic engineers will insall a signal if here is any wo-hour ime inerval during which he produc of he oal number of cars urning lef and he oal number of oncoming cars raveling sraigh hrough he inersecion is greaer han 200,000. In every wo-hour ime inerval, 500 oncoming cars ravel sraigh hrough he inersecion. Does his inersecion require a raffic signal? Explain he reasoning ha leads o he conclusion. 200, 000oal 500cars Upper Lower 400cars hr hr /26/2018 7:27 PM 7.5A: Accumulaion 16 L d cars , 000cars During his 2-hour inerval, he amoun of cars urning lef is greaer han 400. Therefore, a raffic signal is required.
17 EXAMPLE 2 SCORE SHEET A an inersecion in Thomasville, Oregon, cars lef a he rae of L = 60 sin 2 hour over he ime inerval of 0 18 hours. The graph of y = L is shown. cars per 12/26/2018 7:27 PM 7.5A: Accumulaion 17
18 EXAMPLE The heigh of he waer in a conical sorage ank, shown above, is modeled by a differenial funcion h, where h is measured in meers and is measured in hours. A = 0, he heigh of he waer in he ank is 25 meers. The heigh is changing a he rae of r = 2 24e meers per hour for (a) (b) (c) When he heigh of he waer in he ank is h meers he volume of waer is V = 1 πh. A wha rae is he volume of he waer changing a ime = 0? Wha is he minimum heigh of he waer during he ime period 0 24? Jusify your answer. The line angen o he graph of h a = 16 is used o approximae he heigh of he waer in he ank. Using he angen line approximaion, a wha ime does he heigh of he waer reurn o 25 meers? 12/26/2018 7:27 PM 7.5A: Accumulaion 18
19 EXAMPLE A The heigh of he waer in a conical sorage ank, shown above, is modeled by a differenial funcion h, where h is measured in meers and is measured in hours. A = 0, he heigh of he waer in he ank is 25 meers. The heigh is changing a he rae of r = 2 24e hour for meers per (a) When he heigh of he waer in he ank is h meers he volume of waer is V = 1 πh. A wha rae is he volume of he waer changing a ime = 0? 1 V dv 1 h d r h 12/26/2018 7:27 PM 7.5A: Accumulaion 19 2 dh d
20 EXAMPLE A The heigh of he waer in a conical sorage ank, shown above, is modeled by a differenial funcion h, where h is measured in meers and is measured in hours. A = 0, he heigh of he waer in he ank is 25 meers. The heigh is changing a he rae of r = 2 24e hour for meers per (a) When he heigh of he waer in he ank is h meers he volume of waer is V = 1 πh. A wha rae is he volume of he waer changing a ime = 0? dv d dv d 1 h 2 h dh d 2 dh d dv d dv d meers hour 0, h 25 m e /26/2018 7:27 PM 7.5A: Accumulaion 20
21 EXAMPLE B The heigh of he waer in a conical sorage ank, shown above, is modeled by a differenial funcion h, where h is measured in meers and is measured in hours. A = 0, he heigh of he waer in he ank is 25 meers. The heigh is changing a he rae of r = 2 24e hour for meers per (b) Wha is he minimum heigh of he waer during he ime period 0 24? Jusify your answer. Minimum Heigh from 0 24 h ' 0 h' 0 when /26/2018 7:27 PM 7.5A: Accumulaion 21
22 EXAMPLE B The heigh of he waer in a conical sorage ank, shown above, is modeled by a differenial funcion h, where h is measured in meers and is measured in hours. A = 0, he heigh of he waer in he ank is 25 meers. The heigh is changing a he rae of r = 2 24e hour for meers per (b) Wha is he minimum heigh of he waer during he ime period 0 24? Jusify your answer. h' 0 when 6.261?? h ' d h?? h 0 0 h ' d h /26/2018 7:27 PM 0 7.5A: Accumulaion h ' d h
23 EXAMPLE B The heigh of he waer in a conical sorage ank, shown above, is modeled by a differenial funcion h, where h is measured in meers and is measured in hours. A = 0, he heigh of he waer in he ank is 25 meers. The heigh is changing a he rae of r = 2 24e hour for meers per (b) Wha is he minimum heigh of he waer during he ime period 0 24? Jusify your answer. h h' 0 when /26/2018 7:27 PM 7.5A: Accumulaion 2 25 h ' d h 0 The minimum heigh is meers a secs.
24 EXAMPLE C The heigh of he waer in a conical sorage ank, shown above, is modeled by a differenial funcion h, where h is measured in meers and is measured in hours. A = 0, he heigh of he waer in he ank is 25 meers. The heigh is changing a he rae of r = 2 24e hour for meers per (c) The line angen o he graph of h a = 16 is used o approximae he heigh of he waer in he ank. Using he angen line approximaion, a wha ime does he heigh of he waer reurn o 25 meers? V dv d 1 1 r h 12/26/2018 7:27 PM 7.5A: Accumulaion 24 h 2 dh d
25 YOUR TURN W/CALC 15 MINUTES (2005 AB2) The ide removes sand from Sandy Poin Beach a a rae modeled by he funcion R, given by: R = 2 + 5sin 4π 25. A pumping saion adds sand o he beach a a rae modeled by he funcion S, given by S = Boh R and S have unis of cubic yards per hour and is measured in hours for 0 6. A ime = 0, he beach conains 2500 cubic yards of sand. A) How much sand will he ide remove from he beach during he 6-hour period? Indicae unis of measure. B) Wrie an expression for Y(), he oal number of cubic yards of sand on he beach a ime,. C) Find he rae a which he oal amoun of sand on he beach is changing a ime = 4. D) For 0 6, a wha ime is he amoun of sand on he beach he minimum? Wha is he minimum value? Jusify. 12/26/2018 7:27 PM 7.5A: Accumulaion 25
26 12/26/2018 7:27 PM 7.5A: Accumulaion 26
27 YOUR TURN CALCULATOR The ide removes sand from Sandy Poin Beach a a rae modeled by he funcion R, given by: R = 2 + 5sin 4π 25. A pumping saion adds sand o he beach a a rae modeled by he funcion S, given by S = Boh R and S have unis of cubic yards per hour and is measured in hours for 0 6. A ime = 0, he beach conains 2500 cubic yards of sand. A) How much sand will he ide remove from he beach during he 6-hour period? Indicae unis of measure. 6 R d yards 12/26/2018 7:27 PM 7.5A: Accumulaion 27
28 YOUR TURN CALCULATOR The ide removes sand from Sandy Poin Beach a a rae modeled by he funcion R, given by: R = 2 + 5sin 4π 25. A pumping saion adds sand o he beach a a rae modeled by he funcion S, given by S = Boh R and S have unis of cubic yards per hour and is measured in hours for 0 6. A ime = 0, he beach conains 2500 cubic yards of sand. B) Wrie an expression for Y, he oal number of cubic yards of sand on he beach a ime, Y S R d 0 12/26/2018 7:27 PM 7.5A: Accumulaion 28
29 YOUR TURN CALCULATOR The ide removes sand from Sandy Poin Beach a a rae modeled by he funcion R, given by: R = 2 + 5sin 4π 25. A pumping saion adds sand o he beach a a rae modeled by he funcion S, given by S = Boh R and S have unis of cubic yards per hour and is measured in hours for 0 6. A ime = 0, he beach conains 2500 cubic yards of sand. C) Find he rae a which he oal amoun of sand on he beach is changing a ime = 4. Rae of Change Y S 4 R 4d Y ' 4 S 4 R /26/2018 7:27 PM 7.5A: Accumulaion yards / hr
30 YOUR TURN CALCULATOR The ide removes sand from Sandy Poin Beach a a rae modeled by he funcion R, given by: R = 2 + 5sin 4π 25. A pumping saion adds sand o he beach a a rae modeled by he funcion S, given by S = Boh R and S have unis of cubic yards per hour and is measured in hours for 0 6. A ime = 0, he beach conains 2500 cubic yards of sand. D) For 0 6, a wha ime is he amoun of sand on he beach he minimum? Wha is he minimum value? Jusify Y() Y() Y 2500 S R d 0 Min.: Y ' S R yards. The amoun of sand is a a minimum when = hours and he minimum value is 12/26/2018 7:27 PM 7.5A: Accumulaion 0
31 YOUR TURN SCORE SHEET 12/26/2018 7:27 PM 7.5A: Accumulaion 1
32 ASSIGNMENT Workshee 12/26/2018 7:27 PM 7.5A: Accumulaion 2
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