4 x 4, and. where x is Town Square
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1 Accumulation and Population Dnsity E. A city locatd along a straight highway has a population whos dnsity can b approimatd by th function p 5 4 th distanc from th town squar, masurd in mils, whr 4 4, and p whr is Town Squar is masurd in popl pr mil. Th town tnds for four mils on ithr sid of th town squar. Find th population of th city. E. A city can b approimatd by a circl of radius 8 mils. Moving away from th cntr of th city along a, whr radius, th population dnsity can b approimatd by th function.5 masurd in popl pr squar mil and r is masurd in mils.. (a) Writ an prssion to rprsnt th numbr of popl in th shadd ring. p r r p r is (c) Find th population of th city..t E. Watr is pumpd out of a holding tank at a rat of 4 4 litrs/minut, whr t is in minuts sinc th pump is startd. If th holding tank contains litrs of watr whn th pump is startd, how much watr dos th tank hold minuts latr? Homwork: Worksht and AP Rviw 9
2 Logistic Growth Day In ponntial growth (or dcay), w assum that th rat of incras (or dcras) of a population at any tim t is dirctly proportional to dp th population P. In othr words, kp. Howvr, in many dt situations population growth lvls off and approachs a iting numbr L (th carrying capacity) bcaus of itd rsourcs. In this situation th rat of incras (or dcras) is dirctly proportional to both P and L P. This typ of growth is calld logistic growth. dp It is modld by th diffrntial quation kp L P dt. If w find d P dt, w can find out an important fact about th tim whn logistic P is growing th fastst. W will do this in th ampl blow. E. Th population Pt of fish in a lak satisfis th logistic diffrntial quation dp P P, whr t is masurd in yars, and dt 6 P 4. ponntial (a) Pt t (b) What is th rang of th solution curv? (c) For what valus of P is th solution curv incrasing? Dcrasing? Justify your answr. (d) For what valus of P is th solution curv concav up? Concav down? Justify your answr. () Dos th solution curv hav an inflction point? Justify your answr. (f) Us th information you found to sktch th graph of Pt.
3 E. Th population (a) Pt of fish in a lak satisfis th logistic diffrntial quation dp P P, whr t is masurd in yars, and dt 6 P,. Pt t (b) What is th rang of th solution curv? (c) For what valus of P is th solution curv incrasing? Dcrasing? Justify your answr. (d) For what valus of P is th solution curv concav up? Concav down? Justify your answr. () Dos th solution curv hav an inflction point? Justify your answr. (f) Us th information you found to sktch th graph of E. Th population Pt Pt of fish in a lak satisfis th logistic diffrntial quation dp P P, whr t is masurd in yars, and dt 6 P,. (a) Pt t (b) What is th rang of th solution curv? (c) For what valus of P is th solution curv incrasing? Dcrasing? Justify your answr.. (d) For what valus of P is th solution curv concav up? Concav down? Justify your answr. () Dos th solution curv hav an inflction point? Justify your answr. (f) Us th information you found to sktch th graph of Pt. Homwork: Worksht on Logistic Growth and AP Rviw 9
4 Logistic Growth Day E. Th rat at which th flu sprads through a community is modld by th logistic diffrntial dp.p P, whr t is masurd in days. (a) If quation dt P 5, solv for P as a function of t. All work must b shown. (b) Us your solution to (a) and your graphing calculator to find th siz of th population whn t = days. (c) Us your solution to (a) and your graphing calculator to find th numbr of days that hav passd whn 4 popl hav contractd th flu. Homwork: Worksht on Logistic Growth and AP Rviw -
5 8.7 Indtrminat Forms and L'Hopital's Rul In this sction w will b valuating its which may giv rsults that may surpris you. For th following it problms, mak a guss as to what you think th it will b, and thn valuat th givn prssion for th valus givn, and s if you think your guss was corrct. )? Guss: To s if your guss is corrct, valuat th following: Now what do you think th original it is? ) ln? Guss: To s if your guss is corrct, valuat th following: ln ln ln... Now what do you think th original it is? )? Guss: To s if your guss is corrct, valuat th following:, Now what do you think th original it is?
6 Indtrminat Forms and L Hopital s Rul Earlir this yar w valuatd som indtrminat forms by using algbraic tchniqus. E. E. E. 7 7 Not all indtrminat forms can b valuatd by algbraic tchniqus. Anothr mthod is to valuat thm by L Hopital s Rul. L Hopital s Rul: If f a g f f a g a g rsults in th indtrminat form or, providd that th lattr it ists or is infinit., thn E. E. Somtims th problm must first b convrtd into or. E. E. ln
7 Somtims logs ar ndd to valuat th it. E. Homwork: Worksht on L Hopital s Rul and AP Rviw 5
8 8.8 Impropr Intgrals Day Evaluat by using your calculator. ) ) ) d d = =,, d = Do you think d convrgs or divrgs? If you think it convrgs, to what valu dos it convrg? 4) 5) d d = = Do you think d convrgs or divrgs? If you think it convrgs, to what valu dos it convrg? Sktch th graphs of f and f, and shad th rgions rprsntd by d and d. Why would ths two intgrals giv such diffrnt rsults? Hint: On your graphing calculator, start with th following window: :, 5, y:,. Thn chang your window to: :, 5, y:.,..
9 Impropr Intgrals Day Intgrals such as,, and a f d f d f d a ar calld impropr intgrals. - Thy ar valuatd by rwriting th intgral as a propr intgral and thn using its: b f d f d a b a b f d f d, and a c c b f d f d f d f d f d c a a b E. d b a c E. d E. 4d E. Lt f, and lt R b th rgion in th first quadrant btwn th graph of f and th -ais. Find th volum of th solid formd gnratd whn R is rvolvd about th -ais. Homwork: P. 576: 6, 4,, 46, 5 P. 587:, 9 5 odd, 77, 79 AP Rviw 6 8
10 8.8 Impropr Intgrals Day Evaluat by using your calculator. ) ) ). d = d =. d =. Do you think convrgs or divrgs? If you think it convrgs, to what valu dos it convrg? 4) 5) 6) d. d. d. = = = Do you think d convrgs or divrgs? If you think it convrgs, to what valu dos it convrg? Sktch th graphs of f and f, and shad th rgions rprsntd by and d. Why would ths two intgrals giv such diffrnt rsults? Hint: On your graphing calculator, start with th following window: y Thn chang your window to: :.,, y:., 5. :.,, :.,.
11 Impropr Intgrals Day If f has an infinit discontinuity at a or at b or at som c in a, b, thn th intgral b f a d is an impropr intgral. Ths impropr intgrals ar valuatd by rwriting th intgral as a propr intgral and thn using its. E. d E. d 7 d E. 7 d E. E. Lt ln f, and lt R b th unboundd rgion in th fourth quadrant btwn th graph of f and th -ais. Find th ara of R. Homwork: P. 587: - odd, 4,,, 7-49 odd, 8 AP Rviw 9
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