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Copyright 5-6 by Skylight Publishig Chapter. Aotated Solutios to Past Free-Respose Questios This material is provided to you as a supplemet to the book Be Prepared for the AP Calculus Exam. You are ot authorized to publish or distribute it i ay form without our permissio. However, you may prit out oe copy of this chapter for persoal use ad for face-to-face teachig for each copy of the Be Prepared book that you ow or receive from your school. Skylight Publishig 9 Bartlet Street, Suite 7 Adover, MA 8 web: e-mail: http://www.skylit.com sales@skylit.com support@skylit.com
6 AB AP Calculus Free-Respose Solutios ad Notes Questio AB- (a) R() R() 95 9 R liters per hour per hour. (b) The left Riema sum approximatio is 49 9574 85 liters. This is a overestimate, sice R t is decreasig for t 8. The amout of water pumped ito the tak from t to t 8 is 8 W t dt 786.95. So the amout of water i the tak at t 8 is approximately 5 786.95 85 49786 liters. (d) Yes. The fuctio f t W t R t is cotiuous. 4 f 88.54 7. By the Itermediate Value Theorem, there is a time t, Notes: t 8 whe. f t. At that time, W t R t f ad. Use the give ames for the fuctios i your aswers, rather tha their formulas, to avoid trascriptio errors.
4 FREE-RESPONSE SOLUTIONS ~ 6 AB Questio AB- (a) v4.979 ad 4 v.64. Sice the velocity is positive ad decreasig at t = 4, the particle is slowig dow. (b) The particle chages directio oly at t.77 o the iterval t, sice the velocity chages sig at that time ad at o other time o that iterval. (d) 4 x() vtdt.85. vt dt 5.. Questio AB- (a) Neither. gx f x ad f x does ot chage sig at x =. (b) Yes, because gx f x chages from icreasig to decreasig at x = 4. The oly poits where g could have a absolute maximum or absolute miimum are x 4, x, x 6, ad 4 4 8 g 6 8, x. g, g, g 4. The absolute miimum is 8 ad the absolute maximum is 8. g x o 4 x ad o x. (d) Notes:. The cadidate test is the easiest way to justify absolute extrema for a cotiuous fuctio o a closed iterval.. g() ad for x gx ( ), because the itegrad is positive ad we are itegratig from right to left. Itegratig from x to x 4 oly icreases g by 4, ot eough to make it positive. For x >, g (), ad g decreases o the iterval [, ].
FREE-RESPONSE SOLUTIONS ~ 6 AB 5 Questio AB-4 (a) y x (b) dy 9. A equatio for the taget lie is y9x. dx x, y f. 9..9. dx dx Separatig variables, we get y dy y dy x x l x C. Usig the iitial coditio, l C C. So y l x l x y y. Sice the iitial coditio is give at x =, we ca remove the absolute value: f( x). l x Notes:. Be careful ot to make the claim that f..9. The use of = i approximatios of this kid is usually pealized.. The atural domai of this solutio is x e, but it is ot ecessary to iclude the domai uless it is specifically asked for i the problem.
6 FREE-RESPONSE SOLUTIONS ~ 6 AB Questio AB-5 h h dh h dh h (a) iches. (b) Volume = 4 r dh h dh h h dh. 9 6 4 5 h 9hh 9 4 5 4 cubic iches. dr dh h. At h =, this is dt dt h dh dh dh dt dt 5 dt iches per secod. Notes:. You ca leave the aswer usimplified. For the record, it is 9 6.. Agai, the usimplified aswer is sufficiet to receive full credit. This is 9. 4 Questio AB-6 (a) k f g f 6 4 ad k fg g f g A equatio for the taget lie is y4x. 6 5. (b) f f g g f 48 6 h. 6 7. f xdx f x f 6 f 5
6 BC AP Calculus Free-Respose Solutios ad Notes Questio BC- See AB Questio. Questio BC- (a) x x 5 si x t dt t t dt 4.77 y. At t =. The particle is at 4.77,.. From the graph, (b) The slope is dy dy dx /.5. dx dt dt 9 si 7 t dy dx dt dt 9.969 The speed is 9 si7 (d) Distace traveled = t si t dt t si t dt 4.5 Notes:. Or 9.968, if trucated at the third decimal place.. Or 4.49, if trucated at the third decimal place... Questio BC- See AB Questio. 7
8 FREE-RESPONSE SOLUTIONS ~ 6 BC Questio BC-4 (a) d y dy x x x y x x y. dx dx 4 (b) At, 8, 8 dy dx dx solutio has a local maximum at, 8. d y 4 8 4, so the particular 4 Both the umerator ad the deomiator approach, so we ca apply L Hospital s gx gx Rule: lim lim x x x 6 x. Agai, both the umerator ad the deomiator approach, so we ca apply L Hospital s Rule a secod time: gx gx lim lim x 6x. From Part (a), g( ) x 6 g x lim. x 6 6 (d) At, the slope is. 4 4. 5 y. 4 y.5. At, the slope is Questio BC-5 See AB Questio 5.
FREE-RESPONSE SOLUTIONS ~ 6 BC 9 Questio BC-6 (a) x x x ( )!... x... 4! 8!! x x x... x.... 8 4 (b) At x =, the series is............, which 8 4 is the harmoic series with a additioal term, so diverges. At x =, the series is............, which 8 4 coverges, sice it is the alteratig harmoic series with a additioal term. So the iterval of covergece is x. f.... 8 (d) The magitude of the error is bouded by the magitude of the first omitted term, which is... 4 Notes:. Leave it at this to save time ad avoid arithmetic mistakes. (This is equal to.95.)