AP Calculus BC 1999 Scoring Guidelines

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1 AP Calculus BC Scoring Guidelines The materials included in these files are intended for non-commercial use by AP teachers for course and eam preparation; permission for any other use must be sought from the Advanced Placement Program. Teachers may reproduce them, in whole or in part, in limited quantities, for face-to-face teaching purposes but may not mass distribute the materials, electronically or otherwise. These materials and any copies made of them may not be resold, and the copyright notices must be retained as they appear here. This permission does not apply to any third-party copyrights contained herein. These materials were produced by Educational Testing Service (ETS), which develops and administers the eaminations of the Advanced Placement Program for the College Board. The College Board and Educational Testing Service (ETS) are dedicated to the principle of equal opportunity, and their programs, services, and employment policies are guided by that principle. The College Board is a national nonprofit membership association dedicated to preparing, inspiring, and connecting students to college and opportunity. Founded in, the association is composed of more than, schools, colleges, universities, and other educational organizations. Each year, the College Board serves over three million students and their parents,, high schools, and,5 colleges, through major programs and services in college admission, guidance, assessment, financial aid, enrollment, and teaching and learning. Among its best-known programs are the SAT, the PSAT/NMSQT, the Advanced Placement Program (AP ), and Pacesetter. The College Board is committed to the principles of equity and ecellence, and that commitment is embodied in all of its programs, services, activities, and concerns. Copyright by College Entrance Eamination Board. All rights reserved. College Board, Advanced Placement Program, AP, and the acorn logo are registered trademarks of the College Entrance Eamination Board.

2 BC{. A particle moves in the y{plane so that its position at any time t, t, isgiven by (t) = t ; ln( + t) andy(t) = sin t. (a) Sketch the path of the particle in the y{plane below. Indicate the direction of motion along the path. (b) At what time t, t, does(t) attain its minimum value? What is the position ((t) y(t)) of the particle at this time? (c) At what time t, <t<, is the particle on the y{ais? Find the speed and the acceleration vector of the particle at this time. (a) y ( : graph : direction O (b) (t) =t ; +t = t + t ; = t = ;+p 5 or t =:6 in [ ] (:6) = ;: y(:6) = :7 (c) (t) = t ; ln( + t) = t =:5 or :6 (t) =t ; +t y (t) = cos t 4 : (t) = : solution for t : position : (t) = : solution for t : speed : acceleration vector speed = p ( (:6)) +(y (:6)) =:6 (t) =+ ( + t) y (t) =; sin t acceleration vector = < (:6) y (:6) > = <: ;:7 >

3 AB{ / BC{. The shaded region, R, is bounded by the graph of y = and the line y =4,asshown in the gure above. (a) Find the area of R. (b) Find the volume of the solid generated by revolving R about the {ais. (c) There eists a number k, k>4, such that when R is revolved about the line y = k, the resulting solid has the same volume as the solid in part (b). Write, but do not solve, an equation involving an integral epression that can be used to nd the value of k. O y y y 4 (a) Area = Z ; = Z (4 ; ) d (4 ; ) d ( : integral : answer = = 4 ; =:666 or :667 (b) Volume = Z ;4 ; ( ) d = Z (6 ; 4 ) d > < : limits and constant : integrand : answer = 6 ; 5 5 = 56 5 = 6:4 or 6:5 (c) Z ; (k ; ) ; (k ; 4) d = > < : limits and constant : integrand <; > each error : equation

4 AB{ / BC{. The rate at which water ows out of a pipe, in gallons per hour, is given by a dierentiable function R of time t. The table above shows the rate as measured every hours for a 4{hour period. (a) Use a midpoint Riemann sum with 4 subdivisions of equal length to approimate Z 4 R(t) dt. Using correct units, eplain the meaning of your answer in terms of water ow. (b) Is there some time t, <t<4, such that R (t) =? Justify your answer. (c) The rate of water ow R(t) can be approimated by Q(t) = ; 76 + t ; t. Use Q(t) toapproimate the 7 average rate of water ow during the 4{hour time period. Indicate units of measure. t R(t) (hours) (gallons per hour) (a) Z 4 R(t) dt 6[R() + R() + R(5) + R()] =6[:4+:+:+:] = 5.6 gallons This is an approimation to the total ow in gallons of water from the pipe in the 4{hour period. : R() + R() + R(5) + R() : answer : eplanation (b) Yes Since R() = R(4) = :6, the Mean Value Theorem guarantees that there is a t, <t<4, such thatr (t) =. ( : answer : MVT or equivalent (c) Average rate of ow average value of Q(t) = Z (76 + t ; t ) dt = :75 gal/hr or.74 gal/hr : limits and average value constant : Q(t) asintegrand : answer (units) Gallons in part (a) and gallons/hr in part (c), or equivalent. : units

5 BC{4 4. The function f has derivatives of all orders for all real numbers. Assume f() = ;, f () = 5, f () =, and f () = ;. (a) Write the third{degree Taylor polynomial for f about = and use it to approimate f(:5). (b) The fourth derivative off satises the inequality jf (4) ()j for all in the closed interval [:5 ]. Use the Lagrange error bound on the approimation to f(:5) found in part (a) to eplain why f(:5) 6= ;5. (c) Write the fourth{degree Taylor polynomial, P (), for g() = f( +) about =. Use P to eplain why g must have a relative minimum at =. (a) T (f )() =;+5( ; ) + ( ; ) ; ( ; ) 6 f(:5) T (f )(:5) = ;+5(;:5) + (;:5) ; 4 (;:5) = ;4:5=;4:5 4 : T (f )() < ; > each error : approimation of f(:5) (b) Lagrange Error Bound = 4! j:5 ; j4 =:75 f(:5) > ;4:5 ; :75 = ;4:66 > ;5 ( : value of Lagrange Error Bound : eplanation Therefore, f(:5) 6= ;5. (c) P () =T 4 (g )() = T (f )( +)=; The coecient of in P () isg (). This coecient is, so g ()=. : T 4 (g )() < ; > each incorrect, missing, or etra term : eplanation The coecient of in P () is. This coecient! is 5, so g () = which is greater than. g () Therefore, g has a relative minimum at =. Note: < ; > ma for improper use of + ::: or equality

6 AB{5 / BC{5 5. The graph of the functionz f, consisting of three line segments, is given above. Let g() = f (t) dt. (a) Compute g(4) and g(;). (b) Find the instantaneous rate of change of g, with respect to, at =. (c) Find the absolute minimum value of g on the closed interval [; 4]. Justify your answer. (d) The second derivative ofg is not dened at = and =. How many of these values are {coordinates of points of inection of the graph of g? Justify your answer. 4 (, 4) (, ) O 4 (4, ) Z 4 (a) g(4) = f (t) dt = ++ ; = 5 g(;) = Z ; f (t) dt = ; () = ;6 ( : g(4) : g(;) (b) g () = f () = 4 : answer (c) g is increasing on [; ] and decreasing on [ 4]. Therefore, g has absolute minimum at an endpoint of[; 4]. : interior analysis : endpoint analysis : answer Since g(;) = ;6 and g(4) = 5, the absolute minimum value is ;6. (d) One = On (; ), g () =f () > On ( ), g () =f () < : choice of = only : show ( g()) is a point of inection : show ( g()) is not a point of inection On ( 4), g () =f () < Therefore ( g()) is a point of inection and ( g()) is not.

7 BC{6 6. Let f be the function whose graph goes through the point ( 6) and whose derivative is given by f () = +e. (a) Write an equation of the line tangent to the graph of f at = and use it to approimate f(:). (b) Use Euler's method, starting at =with a step size of.5, to approimate f(:). Use f to eplain why this approimation is less than f(:). (c) Use Z : f () d to evaluate f(:). (a) f () = +e y ; 6= +e ( ; ) =:4 or :4 : f () : equation : approimation of f(:) y =6+ +e ( ; ) f(:) 6+ +e (:) = 6:4 (b) f(:5) f() + f ()(:5) = 6+:74 = 6:74 f(:) 6:74 + f (:5)(:5) = 6:74 + (:775)(:5) = 6:6 f () = e ; ( + e ) 4 = ( ; )e ; 4 : Euler's method equations or equivalent table : Euler approimation to f(:) (not eligible without rst point) : f () : reason For, f () > e ; > and the graph of f is concave upward on ( :). Therefore, the Euler approimation lines at and.5 lie below the graph. (c) f(:) ; f() = Z : +e d f(:) = 6 + :7 = 6:7 or 6: Z : +e : d = f(:) ; f() : answer

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