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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 = 3. - 2. 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.
(minues) v () (miles per minue) 0 5 10 15 20 25 30 35 0 7.0 9.2 9.5 7.0.5 2. 2..3 7.3 3. 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
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. 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