Calculus BC Fial Review Name: Revised 7 EXAM Date: Tuesday, May 9 Remiders:. Put ew batteries i your calculator. Make sure your calculator is i RADIAN mode.. Get a good ight s sleep. Eat breakfast. Brig: TI-8, pecils, eraser, watch, ad jacket. Work these o otebook paper. Do ot use your calculator uless the problem tells you to. cos cos ) Evaluate lim 5 ) Evaluate: lim ) Evaluate: h hh a, ) Fid a ad b so that f is differetiable everywhere, f b,, 5) f Evaluate : f, ) Let R be the regio bouded by the graphs of y = ad y = +. (a) Fid the area of R. (b) Write, but do ot evaluate, a itegral epressio for the volume of the solid geerated whe R is rotated about the horizotal lie, y = 9. (c) Write, but do ot evaluate, a itegral epressio for the volume of the solid geerated whe R is rotated about the horizotal lie, y =. (d) Write, but do ot evaluate, a itegral epressio for the volume of the solid geerated whe R is rotated about the vertical lie, = 5. 7) Let R be the regio bouded by the graphs of y ad y. (a) The regio R is the base of a solid. For this solid, the cross sectios perpedicular to the -ais are rectagles whose heights are three times the legths of the bases. Write, but do ot evaluate, a itegral epressio for the volume of this solid. (b) The regio R is the base of a solid. For this solid, the cross sectios perpedicular to the y-ais are semicircles. Write, but do ot evaluate, a itegral epressio that could be used to fid the volume of this solid. f 5, f, f 8, ad f. The fuctio 8) Let f be a differetiable fuctio such that g is differetiable ad g() = f () for all. What is the value of g? 9) (Calc.) The first derivative of the fuctio f is defied by f si for. O what itervals is f icreasig? f cos. How may poits of iflectio ) (Calc.) The derivative of the fuctio f is give by does the graph of f have o the ope iterval,? (A) Oe (B) Two (C) Three (D) Four (E) Five
) (Calc.) A particle moves alog the -ais so that at ay time t >, its acceleratio is give by l t a t. If the velocity of particle is at time t =, the velocity of the particle at time t = is: (A). (B).9 (C).555 (D).88 (E). ) (Calc.) A particle moves alog the -ais so that at ay time t, its velocity is give by v t l t t. The total distace traveled by the particle from t = to t = is (A).7 (B).7 (C).5 (D).7 (E).9 ) ( AB, Calc.) A particle moves alog the y-ais so that its velocity at time t is give by t ta e v t. At time t =, the particle is at y =. (Note: (a) Fid the acceleratio of the particle at time t =. ta arcta ) (b) Is the speed of the particle icreasig or decreasig at time t =? Give a reaso for your aswer. (c) Fid the time t at which the particle reaches its highest poit. Justify your aswer. (d) Fid the positio of the particle at time t =. Is the particle movig toward the origi or away from the origi at time t =? Justify your aswer. ) Let f be a fuctio that is twice-differetiable for all real umbers. The table above gives values of f for selected poits i the closed iterval. Evaluate 5) If icreases at a costat rate of rad/mi, at what rate is icreasig i uits/mi whe = uits? 5 8 f 5 f. ) ( AB 5)
7) ( AB 5/BC 5) Match the slope fields with their differetial equatios. (A) (B) (C) (D) dy 8) si 9) dy y ) dy y ) dy 9) Fid the particular solutio y f to the differetial equatio dy y where f 7. dy ) Which of the followig is the solutio to the differetial equatio, where y? y (A) y for > (B) y for (C) y (D) y for (E) y for.5 for ) ( BC ) Show above is a slope field for which of the followig differetial equatios? (A) dy (B) y dy (C) y dy (D) y dy dy (E) y y
) (5BAB) (Calc.) A water tak at Camp Newto holds gallos of water at time t =. Durig the t time iterval t 8 hours, water is pumped ito tak at the rate W t 95 t si gallos per t hour. Durig same time iterval, water is removed from tak at rate Rt 75si gallos per hour. (a) Is the amout of water i the tak icreasig at time t = 5? Why or why ot? (b) To the earest whole umber, how may gallos of water are i the tak at time t = 8? (c) At what time t, for t 8, is the amout of water i the tak at a absolute miimum? Show the work that leads to your coclusio. (d) For t > 8, o water is pumped ito the tak, but water cotiues to be removed at the rate Rt util the tak becomes empty. Let k be the time at which the tak becomes empty. Write, but do ot solve, a equatio ivolvig a itegral epressio that ca be used to fid the value of k. ) ( AB 5) 7) Cosider the curve give by (a) Show that dy = y. 8y (b) Show that there is a poit P with -coordiate at which the lie taget to the curve at P is horizotal. Fid the y-coordiate of P. d y (c) Fid the value of at the poit P foud i part (b). Does the curve have a local maimum, a local miimum, or either at the poit P? Justify your aswer. 8) Let R be the regio i the first quadrat eclosed by the graphs of f() = 8 ad g() = si(π) as show i the figure o the right. (a) Write a equatio for the lie taget to the graph of f at =. (b) Fid the area of R. (c) Write, but do ot evaluate, a itegral epressio for the volume of the solid geerated whe R is rotated about the horizotal lie, y =. (d) The regio R is the base of a solid. For this solid, the cross sectios perpedicular to the -ais are squares. Write, but do ot evaluate, a itegral epressio for the volume of this solid.
9) Give below, the graphs of the fuctios f ad g are show. The value of lim f(g()) =. ) 7si lim si ) If f si l, the f ' ) Three graphs labeled I, II, ad III are show above. Oe is the graph of f, oe is the graph of ad oe is the graph of Which of the followig correctly idetifies each of the three graphs? ) The local liear approimatio to the fuctio g at = is y = +. What is the value of g ( ) + g ( )? ) Write the followig itegral epressios is equal to lim k k with a rage of [,]. 5) The right triagle show i the figure above represets the boudary of a tow that is bordered by a highway. The populatio desity of the tow at a distace of miles from the highway is modeled by D where D() is measured i thousads of people per square mile. Accordig to the model, which of the followig epressios gives the total populatio, i thousads, of the tow? (A) (B) 8 (C) (D) ) f() 5 7 f () 5 The table above gives selected values of a differetiable ad decreasig fuctio f ad its derivative. If g is the iverse fuctio of f, what is the value of g ()?
7) (calc) The secod derivative of a fuctio g is give by g () = + cos +. For 5 < < 5, o what ope itervals is the graph of g cocave up? 8) (calc) Let h be the fuctio defied by h() = the value of g()? 5 +. If g is a atiderivative of h ad g() =, what is 9) (Calculator) Let R be the regio i the first quadrat bouded by the graph of g, ad let S be the regio i the first quadrat betwee the graphs for f ad g, as show i the figure above. The regio i the first quadrat bouded by the graph of f ad the coordiate ais has a area of.. The fuctio g is give by g cos, ad the fuctio f is ot eplicitly give. The graphs itersect at the poit, (,). 8 (a) Fid the area of S. (b) A solid is geerated whe S is revolved about the horizotal lie, y = 5. Write, but do ot evaluate, a epressio ivolvig oe or more itegrals that gives the volume. (c) Regio R is the base of the art sculpture. All of the poits of R at a distace of from the y-ais, the height of the sculpture is give by h() =. Fid the volume of the sculpture. ) t (miutes) 5 9 r(t) Rotatios per miute 7 95 77 5 Rochelle rode a statioary bicycle. The umber of rotatios per miute of the wheel of the statioary bicycle at time t miutes durig Rochelle s ride is modeled by a differetiable fuctio r for t 9 miutes. Values of r(t) for the selected values of t are show o the table above. a) Estimate r (). Show the computatios that lead to your aswer. b) Is there a time t, for t 5, at which r(t) is rotatios per miute? Justify your aswer? c) Use a left Riema s Sum with four subitervals idicated by the data i the table to approimate 9 9. Use the correct uits, eplai the meaig of r t dt r t dt i the cotet of the problem. d) Sarah also rode a statioary bicycle. The umber of rotatios per miute of the wheel of the statioary bicycle at time t miutes durig Sarah s ride is modeled by the fuctio s, defied by t st si 8 for t 9 miutes. Fid the average umber of rotatios per miute of the wheel of the statioary bicycle for t 9 miutes. ) Let f be a cotiuous fuctio defied o the closed iterval [,]. The graph of f, cosistig of three lie segmets, is show. Let g be the fuctio defied by g 5 (a) Fid g(). f t dt. (b) O what itervals is g icreasig? Justify your aswer. (c) O the closed iterval fid the absolute miimum value of g ad fid the absolute maimum value of g. Justify your aswers. (d) Let h() = g(). Fid h ().
) 8 ) If f f with f 7 ad lim f, fid f. ) si 5) (8 BC ) The table gives values of f, f, g, ad g for selected values of. If f g 5, the f g (A) (B) (C) (D) 7 (E) 5 ) Which of the followig itegrals represets the area eclosed by the smaller loop of the graph of r si? (A) si d (B) sid (C) si 7 7 (D) si d (E) 7 si d 7 7 d 7) (998 BC 9) The area of the regio iside the polar curve r si ad outside the polar curve r is give by: (A) si 5 d (B) si d (C) si d 5 (D) si d (E) si d 8) Give the polar curves r cos ad r =, write a itegral epressio which gives the commo iterior of r cos ad r = 9) (5 BC -calculator) The curve o the right is draw i the y-plae ad is described by the equatio i r si for, where r is measured i meters ad polar coordiates is measured i radias. The derivative of r with respect to is give by dr cos d. (a) Fid the area bouded by the curve ad the -ais. (b) Fid the agle that correspods to the poit o the curve with -coordiate,. dr (c) For, is egative. What does this fact say about r? What does this fact say about the curve? d (d) Fid the value of i the iterval that correspods to the poit o the curve i the first quadrat with greatest distace from the origi. Justify your aswers. 5) Give the polar curve r cos. Fid the slope of the curve at the poit where.
5) (Calculator) The rate of chage, dp, of the umber of people at a dace who have heard a rumor is dt modeled by a logistic differetial equatio. There are people at the dace. At 9PM, the umber of people who have heard the rumor is ad is icreasig at a rate of 5 people per hour. Write a differetial equatio to model the situatio. dp P P dt.? Is the solutio curve icreasig or decreasig? Justify aswer. 5) The populatio Pt of fish i a lake satisfies the logistic differetial equatio (a) If (b) If (c) If (d) If P, what is lim Pt t P,, what is lim Pt t P,, what is lim Pt t? Is the solutio curve icreasig or decreasig? Justify.? Is the solutio curve icreasig or decreasig? Justify. P, what is the populatio whe it is growig the fastest? Justify your aswer. dy y 5) Give. Let f be the particular solutio to the give differetial equatio with iitial coditio. f.. f Use Euler s method startig at =, with a step size of., to approimate dy 5) ( BC-) (Calculator) Give cos t ad si t for t. dt dt At time t =, the object is at positio (, 5). (a) Fid the speed ad the acceleratio vector at time t =. (b) Fid the total distace traveled by the object over the time iterval t. (c) Fid the positio of the object at time t =. 55) ( BC-) A fuctio is defied by f...... for all i the iterval of covergece of the give power series. f (a) Fid lim. (b) Write the first three ozero terms ad the geeral term for the ifiite series that represets f (c) Fid the sum of the series foud i part (b). f si 7, the coefficiet of i the Maclauri series for f is? 5) If f is a fuctio such that 57) The coefficiet of i the Taylor series for e about = is? 58) What are all values of for which the series coverge? cos 59) (998 BC 8) Which of the followig coverge? I. II. III. (A) Noe (B) II oly (C) III oly (D) I ad II oly (E) I ad III oly
) (Calculator) The fuctio f has derivatives of all orders for all real umbers. Assume that f, f, f, ad f for all i [.57, ]. f 5, (a) Write the third-degree Taylor polyomial for f about =. (b) Use your aswer to (a) to approimate f(.57). Give your aswer correct to five decimal places. (c) Use the Lagrage error boud o the approimatio of f(.57) to eplai why f(.57).8. ) The Taylor series about = 5 for a certai fuctio f coverges to f() for all i its iterval of covergece. The th derivative of f at = 5 is give by f the first four terms of the Taylor polyomial for f about = 5 approimates! 5 ad 5. Show that f f with a error less tha. Justify your aswer. 5 5 ) A fuctio f has Maclauri series give by....... Which of the followig is a!!!! (A) epressio for f()? si (B) cos (C) cos (D) e (E) e ) What is the value of? ) What is the value of 5 cos if? si 5) Fid the sum of si si... si...!!! ) Give: f cetered at = (a) Fid the power series. Write the first four ozero terms ad the geeral term. For what values of does this series coverge? (b) Fid a power series for f. Write the first four ozero terms ad the geeral term. (c) Use your aswer to (b) to fid the sum of...... if possible. If it is ot 7 possible, eplai. 5 (d) Use your aswer to (b) to fid the sum of...... if possible. If it is ot possible, eplai. 7) Give: f cetered at = (a) Fid the power series. Write the first four ozero terms ad the geeral term. (b) Fid a power series for f t dt. Write the first four ozero terms ad the geeral term. (c) Use your aswer to (b) to fid the sum of...... if possible.
8) Cosider the differetial equatio, dy = A + y, where A is a costat. Let y = f() be the particular solutio to the differetial equatio with the iitial coditio f() =. Euler s method, startig at with a step size of, is used to approimate f(). Steps from this approimatio are show i the table. What is the value of A? 9) Which of the followig series caot be show to coverge usig the limit compariso test with the series,? 5 (A) (B) (C) si 5 (D) l 7) What is the iterval of covergece for the power series,? 7) (calc) For time t secods, the positio of a object travelig alog a curve i the y-plae is give by the parametric equatios (t) ad y(t) where = dt t + ad dy = dt t + t. At what time t is the speed of the object uits per secod? 7) (calc) A particle movig i the y-plae has velocity vector give by v(t) = e si t, 5t for time t. What is the magitude of the displacemet of the particle betwee time t = ad t =? 7) Cosider the series the series coverges? a, where a > for all. Which of the followig coditios guaratees that (A) lim a = a (B) lim + < (C) a a + < a for all (D) f() coverges for all 7) Let r be the fuctio give by r(θ) = θ si θ for θ π. The graph of r i polar coordiates cosists of two loops, as show i the figure. Poit P is o the graph of r ad the y-ais. (A) Fid the rate of chage of the -coordiate with respect to at the poit P. (B) Fid the area of the regio betwee the ier ad outer loops of the graph. (C) The fuctio r satisfies dr = si θ + θ cos θ. For θ π, fid the value of dt that gives the poit o the graph that is farthest from the origi. Justify your aswer. 75) Cosider the fuctio f give by f() = e for all (a) Fid lim f() (b) Fid the maimum value of f for. Justify your aswer. (c) Evaluate f() or show that the itegral diverges.