AP CALCULUS BC 04 SCORING GUIDELINES Questio 6 + The Taylor series for a fuctio f about x = is give by ( ) ( x ) ad coverges to f( x ) for = x < R, where R is the radius of covergece of the Taylor series. (a) Fid the value of R. (b) Fid the first three ozero terms ad the geeral term of the Taylor series for f, the derivative of f, about x =. (c) The Taylor series for f about x =, foud i part (b), is a geometric series. Fid the fuctio f to which the series coverges for x < R. Use this fuctio to determie f for x < R. (a) Let a be the th term of the Taylor series. + + + a+ ( ) ( x ) = a + + ( ) ( x ) ( x ) = + : sets up ratio 3 : : computes limit of ratio : determies radius of covergece lim ( x ) + = x x < x < The radius of covergece is R = (b) The first three ozero terms are 4( x ) + 8( x ).. : first three ozero terms 3 : { : geeral term + The geeral term is ( ) ( x ) for. (c) The commo ratio is ( x. ) f ( x) = = for ( ( x ) ) x f ( x) = dx = l x + C x x < : f ( x) 3 : : atiderivative : f( x) f ( ) = 0 l + C = 0 C = 0 f( x) = l x for x < 04 The College Board.
04 The College Board.
04 The College Board.
04 The College Board.
04 The College Board.
04 The College Board.
04 The College Board.
AP CALCULUS BC 04 SCORING COMMENTARY Questio 6 Overview I this problem studets were give a Taylor series for a fuctio f about x =. I part (a) studets were asked to fid the radius of covergece of this Taylor series. It was expected that studets would use the ratio test to determie that the radius of covergece is. I part (b) studets eeded to differetiate the series term-by-term to fid the first three ozero terms ad the geeral term of the Taylor series for f. I part (c) studets were told that the Taylor series for f is a geometric series. Studets eeded to kow that fidig the sum of that series requires dividig the first term of the series by the differece of ad the commo ratio. This results i f ( x) =. Studets were also asked to fid f. This required itegratig f ( x) to fid x f( x) = l x + C. I order to evaluate the costat of itegratio, studets eeded to use the iitial coditio that f ( ) = 0 which yields f( x) = l x for x <. Sample: 6A Score: 9 The studet eared all 9 poits. I part (c) the studet does ot eed to qualify the closed form expressio for f( x ) with for x < R. Sample: 6B Score: 6 The studet eared 6 poits: 3 poits i part (a), 3 poits i part (b), ad o poits i part (c). I parts (a) ad (b), the studet s work is correct. I part (b) the studet writes the first three ozero terms ad the geeral term of the Taylor series of the origial fuctio f ad the differetiates to fid the required first three terms ad the geeral term of the Taylor series of the derivative of f. Because the questio asks studets to fid the first three ozero terms ad the geeral term of the Taylor series for f, the studet is ot pealized for omittig plus sigs ad a ellipsis i the boxed aswer i part (b). I part (c) the studet misidetifies the costat ratio i the geometric series. Because the first poit was ot eared, the studet is ot eligible for the third poit. The studet does ot atidifferetiate correctly, so the secod poit was ot eared. Sample: 6C Score: 3 The studet eared 3 poits: poit i part (a), poits i part (b), ad o poits i part (c). I part (a) the studet eared poit with a appropriate ratio. The studet s aouced limit ad coclusio are icorrect. I part (b) the studet eared poits for givig the correct first three ozero terms. The studet s geeral term is icorrect. I part (c) the studet does ot provide a expressio for f, so the studet is ot eligible for ay poits. 04 The College Board.