AP Calculus BC 2004 Free-Response Questions Form B

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1 AP Calculus BC 200 Free-Response Quesions Form B The maerials included in hese files are inended for noncommercial use by AP eachers for course and exam preparaion; permission for any oher use mus be sough from he Advanced Placemen Program. Teachers may reproduce hem, in whole or in par, in limied quaniies, for face-o-face eaching purposes bu may no mass disribue he maerials, elecronically or oherwise. This permission does no apply o any hird-pary copyrighs conained herein. These maerials and any copies made of hem may no be resold, and he copyrigh noices mus be reained as hey appear here. The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in 1900, he associaion is composed of more han,500 schools, colleges, universiies, and oher educaional organizaions. Each year, he College Board serves over hree million sudens and heir parens, 23,000 high schools, and 3,500 colleges hrough major programs and services in college admissions, guidance, assessmen, financial aid, enrollmen, and eaching and learning. Among is bes-known programs are he SAT, he PSAT/NMSQT, and he Advanced Placemen Program (AP ). The College Board is commied o he principles of excellence and equiy, and ha commimen is embodied in all of is programs, services, aciviies, and concerns. For furher informaion, visi Copyrigh 200 College Enrance Examinaion Board. All righs reserved. College Board, Advanced Placemen Program, AP, AP Cenral, AP Verical Teams, APCD, Paceseer, Pre-AP, SAT, Suden Search Service, and he acorn logo are regisered rademarks of he College Enrance Examinaion Board. PSAT/NMSQT is a regisered rademark joinly owned by he College Enrance Examinaion Board and he Naional Meri Scholarship Corporaion. Educaional Tesing Service and ETS are regisered rademarks of Educaional Tesing Service. Oher producs and services may be rademarks of heir respecive owners. For he College Board s online home for AP professionals, visi AP Cenral a apcenral.collegeboard.com.

2 CALCULUS BC SECTION II, Par A Time 5 minues Number of problems 3 A graphing calculaor is required for some problems or pars of problems. 1. A paricle moving along a curve in he plane has posiion ax ( ), y ( ) f a ime, where dx d = + 9 and dy d = 2e + 5 e for all real values of. A ime = 0, he paricle is a he poin a1, f. (a) Find he speed of he paricle and is acceleraion vecor a ime = 0. (b) Find an equaion of he line angen o he pah of he paricle a ime = 0. (c) Find he oal disance raveled by he paricle over he ime inerval 0 3. (d) Find he x-coordinae of he posiion of he paricle a ime = Le f be a funcion having derivaives of all orders for all real numbers. The hird-degree Taylor polynomial for f abou x = 2 is given by (a) Find f ( 2) and f ( 2. ) 2 3. Tx ( ) = 7-9( x - 2) - 3( x - 2) (b) Is here enough informaion given o deermine wheher f has a criical poin a x = 2? If no, explain why no. If so, deermine wheher f ( 2) is a relaive maximum, a relaive minimum, or neiher, and jusify your answer. (c) Use Taxf o find an approximaion for f a0f. Is here enough informaion given o deermine wheher f has a criical poin a x = 0? If no, explain why no. If so, deermine wheher f ( 0) is a relaive maximum, a relaive minimum, or neiher, and jusify your answer. (d) The fourh derivaive of f saisfies he inequaliy f ( ) ( x) 6 for all x in he closed inerval 0, 2. Use he Lagrange error bound on he approximaion o f ( 0) found in par (c) o explain why f ( 0) is negaive. 2 GO ON TO THE NEXT PAGE.

3 (minues) v () (miles per minue) A es plane flies in a sraigh line wih posiive velociy v a f, in miles per minue a ime minues, where v is a differeniable funcion of. Seleced values of v a f for 0 0 are shown in he able above. (a) Use a midpoin Riemann sum wih four subinervals of equal lengh and values from he able o approximae meaning of 0 z 0 0 z 0 v ( ) d. Show he compuaions ha lead o your answer. Using correc unis, explain he v ( ) d in erms of he plane s fligh. (b) Based on he values in he able, wha is he smalles number of insances a which he acceleraion of he plane could equal zero on he open inerval 0 < < 0? Jusify your answer. (c) The funcion f, defined by faf= + F I H K + F 7 6 cos 3 sin I 10 H 0 K, is used o model he velociy of he plane, in miles per minue, for 0 0. According o his model, wha is he acceleraion of he plane a = 23? Indicae unis of measure. (d) According o he model f, given in par (c), wha is he average velociy of he plane, in miles per minue, over he ime inerval 0 0? END OF PART A OF SECTION II 3

4 CALCULUS BC SECTION II, Par B Time 5 minues Number of problems 3 No calculaor is allowed for hese problems.. The figure above shows he graph of f, he derivaive of he funcion f, on he closed inerval - 1 x 5. The graph of f has horizonal angen lines a x = 1 and x = 3. The funcion f is wice differeniable wih f ( 2) = 6. (a) Find he x-coordinae of each of he poins of inflecion of he graph of f. Give a reason for your answer. (b) A wha value of x does f aain is absolue minimum value on he closed inerval - 1 x 5? A wha value of x does f aain is absolue maximum value on he closed inerval - 1 x 5? Show he analysis ha leads o your answers. (c) Le g be he funcion defined by g( x) = xf( x). Find an equaion for he line angen o he graph of g a x = 2. GO ON TO THE NEXT PAGE.

5 5. Le g be he funcion given by g( x) = 1. x (a) Find he average value of g on he closed inerval 1,. (b) Le S be he solid generaed when he region bounded by he graph of y = g( x), he verical lines x = 1 and x =, and he x-axis is revolved abou he x-axis. Find he volume of S. (c) For he solid S, given in par (b), find he average value of he areas of he cross secions perpendicular o he x-axis. b L (d) The average value of a funcion f on he unbounded inerval a, f is defined o be lim f x dx O ( ) Mz a P. Show bffi NM b - a QP ha he improper inegral gx ( ) dxis divergen, bu he average value of g on he inerval, f is finie. z 6. Le l be he line angen o he graph of y = x n a he poin a11, f, where n > 1, as shown above. 1 (a) Find z x n dx in erms of n. 0 (b) Le T be he riangular region bounded by l, he x-axis, and he line x = 1. Show ha he area of T is 1 2n. (c) Le S be he region bounded by he graph of y = x n, he line l, and he x-axis. Express he area of S in erms of n and deermine he value of n ha maximizes he area of S. END OF EXAMINATION 5

AP CALCULUS AB 2003 SCORING GUIDELINES (Form B)

AP CALCULUS AB 2003 SCORING GUIDELINES (Form B) SCORING GUIDELINES (Form B) Quesion A blood vessel is 6 millimeers (mm) long Disance wih circular cross secions of varying diameer. x (mm) 6 8 4 6 Diameer The able above gives he measuremens of he B(x)