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SCORING GUIDELINES (Form B) Questio Let f be the fuctio give by fx ( ) = x x, ad let be the lie y = 8 x, where is taget to the graph of f. Let R be the regio bouded by the graph of f ad the x-axis, ad let S be the regio bouded by the graph of f, the lie, ad the x-axis, as show above. (a) Show that is taget to the graph of y = f() x at the poit x =. Fid the area of S. (c) Fid the volume of the solid geerated whe R is revolved about the x-axis. (a) f ( x) = 8x x ; f () = 7 = f () = 6 7 = 9 Taget lie at x = is y = ( x ) + 9 = x + 8, which is the equatio of lie. : fids f() ad f() fids equatio of taget lie or : shows (,9) is o both the graph of f ad lie fx ( ) = at x = The lie itersects the x-axis at x = 6. Area = ()(9) ( x x ) dx = 7.96 or 7.97 OR : itegral for o-triagular regio : limits : itegrad : area of triagular regio : aswer Area = (( 8 ) ( )) x x x dx + ()(8 ) = 7.96 or 7.97 (c) Volume = ( ) x x dx = 56.8 or 9.8 : limits ad costat : : itegrad : aswer Copyright by College Etrace Examiatio Board. All rights reserved.
The figure above shows the graphs of the circles AP CALCULUS BC SCORING GUIDELINES (Form B) Questio x + y = ad ( x ) + y =. The graphs itersect at the poits (,) ad (, ). Let R be the shaded regio i the first quadrat bouded by the two circles ad the x-axis. (a) Set up a expressio ivolvig oe or more itegrals with respect to x that represets the area of R. Set up a expressio ivolvig oe or more itegrals with respect to y that represets the area of R. (c) The polar equatios of the circles are r = ad r = cos, respectively. Set up a expressio ivolvig oe or more itegrals with respect to the polar agle that represets the area of R. (a) Area = Area = ( ) ( x ) dx + x dx OR + x dx : itegrad for larger circle : itegrad or geometric area : for smaller circle : limits o itegral(s) Note: < > if o additio of terms Area = ( ( )) y y dy : : limits itegrad < > reversal < > algebra error i solvig for x < > add rather tha subtract < > other errors (c) Area = ( ) d + (cos ) d OR Area = ( ) (cos ) d 8 + : itegrad or geometric area for larger circle : : itegrad for smaller circle : limits o itegral(s) Note: < > if o additio of terms Copyright by College Etrace Examiatio Board. All rights reserved.
SCORING GUIDELINES (Form B) Questio A blood vessel is 6 millimeters (mm) log Distace with circular cross sectios of varyig diameter. x (mm) 6 8 6 Diameter The table above gives the measuremets of the B(x) (mm) 8 6 6 diameter of the blood vessel at selected poits alog the legth of the blood vessel, where x represets the distace from oe ed of the blood vessel ad Bx () is a twice-differetiable fuctio that represets the diameter at that poit. (a) Write a itegral expressio i terms of Bx () that represets the average radius, i mm, of the blood vessel betwee x = ad x = 6. Approximate the value of your aswer from part (a) usig the data from the table ad a midpoit Riema sum with three subitervals of equal legth. Show the computatios that lead to your aswer. 75 Bx () (c) Usig correct uits, explai the meaig of dx 5 i terms of the blood vessel. (d) Explai why there must be at least oe value x, for < x < 6, such that B ( x) =. (a) 6 Bx () dx 6 : limits ad costat : itegrad B(6) B(8) B() + + = 6 [ 6( + + )] = 6 : B(6) + B(8) + B() : aswer (c) Bx ( ) Bx ( ) is the radius, so is the area of the cross sectio at x. The expressio is the volume i mm of the blood vessel betwee 5 : volume i mm : betwee x = 5 ad x = 75 mm ad 75 mm from the ed of the vessel. (d) By the MVT, B ( c) = for some c i (6, 8) ad B ( c) = for some c i (, 6). The MVT applied to B ( x) shows that B () x = for some x i the iterval ( c c ),. : explais why there are two values of x where B( x) has the same value : explais why that meas B ( x) = for < x < 6 Copyright by College Etrace Examiatio Board. All rights reserved. Note: max / if oly explais why B ( x) = at some x i (, 6).
SCORING GUIDELINES (Form B) Questio A particle moves i the xy-plae so that the positio of the particle at ay time t is give by ( ) 7 x t e t e t = + ad ( ) t t y t = e e. (a) Fid the velocity vector for the particle i terms of t, ad fid the speed of the particle at time t =. Fid dy dy i terms of t, ad fid lim. dx t dx (c) Fid each value t at which the lie taget to the path of the particle is horizotal, or explai why oe exists. (d) Fid each value t at which the lie taget to the path of the particle is vertical, or explai why oe exists. 7 (a) x () t = 6e t 7e y () t = 9e + e t t t t 7t t t Velocity vector is < 6e 7 e, 9e + e > : : x( t) : y( t) : speed Speed = x() + y() = ( ) + = dy dy dt 9e + e = = dx dx 6e 7e dt t t t 7t dy : i terms of t dx : limit t t dy 9e + e 9 lim = lim = = dx t t 7t 6e 7e 6 t (c) Need y t t () t =, but 9e + e > for all t, so oe exists. : cosiders y( t) = : explais why oe exists (d) Need x () t = ad y() t. t 7t e = e t 7 6 7 e = 6 7 t = l 6 ( ) : cosiders x( t) = : solutio Copyright by College Etrace Examiatio Board. All rights reserved. 5
SCORING GUIDELINES (Form B) Questio 5 Let f be a fuctio defied o the closed iterval [,7]. The graph of f, cosistig of four lie segmets, is show above. Let g be the x fuctio give by gx ( ) = ftdt ( ). (a) Fid g (, ) g ( ), ad g ( ). Fid the average rate of chage of g o the iterval x. (c) For how may values c, where < c <, is g () c equal to the average rate foud i part? Explai your reasoig. (d) Fid the x-coordiate of each poit of iflectio of the graph of g o the iterval < x < 7. Justify your aswer. (a) g() = f( t) dt = ( + ) = g () = f() = g() = f() = = : : g() : g() : g() g() g() = () ftdt 7 ()() + ( + ) = = ( ) : g() g() = f( t) dt : aswer (c) There are two values of c. We eed 7 = g( c) = f( c) The graph of f itersects the lie places betwee ad. 7 y = at two : aswer of : reaso Note: / if aswer is by MVT (d) x = ad x = 5 because g = f chages from icreasig to decreasig at x =, ad from decreasig to icreasig at x = 5. : x = ad x = 5 oly : justificatio (igore discussio at x = ) Copyright by College Etrace Examiatio Board. All rights reserved. 6
SCORING GUIDELINES (Form B) Questio 6 The fuctio f has a Taylor series about x = that coverges to fx ( ) for all x i the iterval of ( +! ) ( ) covergece. The th derivative of f at x = is give by f ( ) = for, ad f ( ) =. (a) Write the first four terms ad the geeral term of the Taylor series for f about x =. Fid the radius of covergece for the Taylor series for f about x =. Show the work that leads to your aswer. (c) Let g be a fuctio satisfyig g ( ) = ad g ( x) = f( x) for all x. Write the first four terms ad the geeral term of the Taylor series for g about x =. (d) Does the Taylor series for g as defied i part (c) coverge at x =? Give a reaso for your aswer.!!! (a) f () = ; f () = ; f () = ; f () =!! fx ( ) = + ( x ) + ( x ) + ( x ) +!! ( + )! ( ) + + x +! = + ( x ) + ( x ) + ( x ) + + ( ) + + x + : ( f ) () : coefficiets i! first four terms : powers of ( x ) i first four terms : geeral term + ( ) + + x + lim = lim x + ( ) x + = x < whe x < The radius of covergece is. : : sets up ratio : limit : applies ratio test to coclude radius of covergece is (c) g () = ; g () = f() ; g() = f() ; g() = f() gx ( ) = + ( x ) + ( x ) + ( x ) + : first four terms : geeral term ( ) x + + + + (d) No, the Taylor series does ot coverge at x = because the geometric series oly coverges o the iterval x <. : aswer with reaso Copyright by College Etrace Examiatio Board. All rights reserved. 7