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5 SCORING GUIDELINES Questio x Let f ad g be the fuctios give by f ( x) = + si ( x) ad gx ( ) = 4. Let 4 R be the shaded regio i the first quadrat eclosed by the y-axis ad the graphs of f ad g, ad let S be the shaded regio i the first quadrat eclosed by the graphs of f ad g, as show i the figure above. (a) Fid the area of R. (b) Fid the area of S. (c) Fid the volume of the solid geerated whe S is revolved about the horizotal lie y =. x f ( x) = g( x) whe + si ( x) = 4. 4 f ad g itersect whe x =.7 ad whe x =. Let a =.7. a (a) ( g ( x ) f ( x )) =.64 or.65 : : limits : itegrad : aswer (b) ( f ( x ) g ( x )) =.4 a : : limits : itegrad : aswer ( ) (c) ( f( x) + ) ( g( x) + ) = 4.55 or 4.559 a : { : itegrad : limits, costat, ad aswer Copyright 5 by College Board. All rights reserved.
5 SCORING GUIDELINES Questio The curve above is draw i the xy-plae ad is described by the equatio i polar coordiates r = θ + si ( θ) for θ, where r is measured i meters ad θ is measured i radias. The derivative of r with respect to θ is dr give by cos( θ ). dθ = + (a) Fid the area bouded by the curve ad the x-axis. (b) Fid the agle θ that correspods to the poit o the curve with x-coordiate. (c) For < θ < dr, is egative. What does this fact say about r? What does this fact say about the curve? dθ (d) Fid the value of θ i the iterval θ that correspods to the poit o the curve i the first quadrat with greatest distace from the origi. Justify your aswer. (a) Area = r dθ = ( θ + si( θ) ) dθ = 4. : : limits ad costat : itegrad : aswer (b) = r cos( θ) = ( θ + si( θ) ) cos( θ) θ =.76 : { : equatio : aswer dr (c) Sice dθ < for < θ <, r is decreasig o this iterval. This meas the curve is gettig closer to the origi. : iformatio about r : { : iformatio about the curve (d) The oly value i, where dr dθ = is θ =. θ r.9.57 : : θ = or.47 : aswer with justificatio The greatest distace occurs whe θ =. Copyright 5 by College Board. All rights reserved.
5 SCORING GUIDELINES Questio Distace x (cm) Temperature T( x ) ( C) 5 6 9 7 6 55 A metal wire of legth cetimeters (cm) is heated at oe ed. The table above gives selected values of the temperature T( x ), i degrees Celsius ( C, ) of the wire x cm from the heated ed. The fuctio T is decreasig ad twice differetiable. (a) Estimate T ( 7. ) Show the work that leads to your aswer. Idicate uits of measure. (b) Write a itegral expressio i terms of T( x ) for the average temperature of the wire. Estimate the average temperature of the wire usig a trapezoidal sum with the four subitervals idicated by the data i the table. Idicate uits of measure. (c) Fid T ( x), ad idicate uits of measure. Explai the meaig of ( ) T x i terms of the temperature of the wire. (d) Are the data i the table cosistet with the assertio that T ( x) > for every x i the iterval < x <? Explai your aswer. (a) T( ) T( 6) 55 6 7 = = Ccm 6 : aswer (b) ( ) T x Trapezoidal approximatio for T( x) : + 9 9+ 7 7 + 6 6 + 55 A = + 4 + + Average temperature 75.675 C A = (c) T ( x) = T( ) T( ) = 55 = 45 C The temperature drops 45 C from the heated ed of the wire to the other ed of the wire., 5 is 7 9 = 5.75. 5 5, 6 is 6 7 =. 6 5 T c = 5.75 for some c i the iterval (, 5 ) T c = for some c i the iterval ( 5, 6 ). It follows that c, c. Therefore T (d) Average rate of chage of temperature o [ ] Average rate of chage of temperature o [ ] No. By the MVT, ( ) ad ( ) T must decrease somewhere i the iterval ( ) is ot positive for every x i [, ]. : : T( x) : trapezoidal sum : aswer : { : value : meaig : two slopes of secat lies : { : aswer with explaatio Uits of Ccmi (a), ad C i (b) ad (c) : uits i (a), (b), ad (c) Copyright 5 by College Board. All rights reserved. 4
5 SCORING GUIDELINES Questio 4 dy Cosider the differetial equatio x y. = (a) O the axes provided, sketch a slope field for the give differetial equatio at the twelve poits idicated, ad sketch the solutio curve that passes through the poit (, ). (Note: Use the axes provided i the pik test booklet.), has a local miimum at ( ) (b) The solutio curve that passes through the poit ( ) x = l. What is the y-coordiate of this local miimum? (c) Let y = f ( x) be the particular solutio to the give differetial equatio with the iitial coditio f ( ) =. Use Euler s method, startig at x = with two steps of equal size, to approximate f (.4 ). Show the work that leads to your aswer. d y (d) Fid i terms of x ad y. Determie whether the approximatio foud i part (c) is less tha or greater tha f (.4 ). Explai your reasoig. (a) : : zero slopes : ozero slopes : curve through (, ) dy (b) = whe x y = The y-coordiate is ( ) l. dy : sets = : : aswer (c) f (.) f ( ) + f ( )(.) = + ( )(.) =. f(.4) f(.) + f (.)(.). + (.6)(.) =.5 : : Euler's method with two steps : Euler approximatio to f (.4) (d) d y dy = = x + y d y is positive i quadrat II because x < ad y >..5 < f (.4) sice all solutio curves i quadrat II are cocave up. : d y : : aswer with reaso Copyright 5 by College Board. All rights reserved. 5
5 SCORING GUIDELINES Questio 5 A car is travelig o a straight road. For t 4 secods, the car s velocity vt (), i meters per secod, is modeled by the piecewise-liear fuctio defied by the graph above. 4 4 (a) Fid vt () dt. Usig correct uits, explai the meaig of () vt dt. (b) For each of v ( 4) ad v ( ), fid the value or explai why it does ot exist. Idicate uits of measure. (c) Let at () be the car s acceleratio at time t, i meters per secod per secod. For < t < 4, write a piecewise-defied fuctio for at (). (d) Fid the average rate of chage of v over the iterval t. Does the Mea Value Theorem guaratee a value of c, for < c <, such that v ( c) is equal to this average rate of chage? Why or why ot? 4 (a) vt () dt= ( 4)( ) + ( )( ) + ( )( ) = 6 The car travels 6 meters i these 4 secods. : { : value : meaig with uits (b) v ( 4) does ot exist because vt () v( 4) vt () v( 4) lim = 5 = lim. t 4 t 4 + t 4 t 4 5 v ( ) = = m sec 6 4 : v ( 4 ) does ot exist, with explaatio : : v ( ) : uits (c) 5 if < t < 4 at () = if 4 < t < 6 5 if 6 < t < 4 at () does ot exist at t = 4 ad t = 6. : 5 : fids the values 5,, : idetifies costats with correct itervals (d) The average rate of chage of v o [, ] is v( ) v( ) 5 = m sec. 6 No, the Mea Value Theorem does ot apply to v o [, ] because v is ot differetiable at t = 6. : : average rate of chage of v o [, ] : aswer with explaatio Copyright 5 by College Board. All rights reserved. 6
5 SCORING GUIDELINES Questio 6 Let f be a fuctio with derivatives of all orders ad for which f ( ) = 7. Whe is odd, the th derivative ( of f at x = is. Whe is eve ad, the th derivative of f at x = is give by ) (! ) f ( ) =. (a) Write the sixth-degree Taylor polyomial for f about x =. (b) I the Taylor series for f about x =, what is the coefficiet of ( x ) for? (c) Fid the iterval of covergece of the Taylor series for f about x =. Show the work that leads to your aswer. (a) P 6( x!! 5! ) = 7 + ( ) ( ) ( )! 4 4! 6 x + x + 6! x 4 6 : polyomial about x = : P6( x) : each icorrect term max for all extra terms, +, misuse of equality (b) ( )! = ( )! : coefficiet ( ) (c) The Taylor series for f about x = is f ( x ) 7 ( ) = +. x = ( ) ( ) ( ) ( + x ) + + L = lim ( x ) ( x ) = lim ( x ) = ( + ) 9 L < whe x <. Thus, the series coverges whe < x < 5. Whe x = 5, the series is 7 + = 7 +, = = which diverges, because, the harmoic series, diverges. = Whe x =, the series is ( ) 7 + = 7 +, = = which diverges, because, the harmoic series, diverges. = The iterval of covergece is (, 5 ). 5 : : sets up ratio : computes limit of ratio : idetifies iterior of iterval of covergece : cosiders both edpoits : aalysis/coclusio for both edpoits Copyright 5 by College Board. All rights reserved. 7