Multiple-Choice Answer Key!!!!! "#$$%&'! "#$$%&'!!,#-! ()*+%$,#-! ()*+%$!!!!!! "!!!!! "!! 5!! 6! 7!! 8! 7! 9!!! 5:!!!!! 5! (!!!! 5! "! 5!!! 5!! 8! (!! 56! "! :!!! 59!!!!! 5! 7!!!! 5!!!!! 55! "! 6! "!! 58!! 9! 7!! 8:!!! 7!! 8!!!!! 8! "! 5!! 8!! :! 7!! 7!! (!!! 6!! 9! 7!! "!! 7! 5! (! -6-
Question #! t $ The rate at which raw sewage enters a treatment tank is given by Et!"% 85 & 75cos' ) 9 ( * gallons per hour for + t + hours Treated sewage is removed from the tank at the constant rate of 65 gallons per hour The treatment tank is empty at time t % (a) How many gallons of sewage enter the treatment tank during the time interval + t +? Round your answer to the nearest gallon (b) For + t +, at what time t is the amount of sewage in the treatment tank greatest? To the nearest gallon, what is the maximum amount of sewage in the tank? Justify your answers (c) For t, 5, t dollars per gallon To the nearest dollar, what is the total cost of treating all the sewage that enters the tank during the time interval + t +? + + the cost of treating the raw sewage that enters the tank at time t is! " gallons : : integral (a) E!" t - 98 : answer (b) Let St!" be the amount of sewage in the treatment tank at time t Then S! t" % E! t", 65 and S!" t % when Et!"% 65 On the interval + t +, Et!"% 65 when t % 9 and t % 559 t (hours) amount of sewage in treatment tank!" : sets Et % 65 : identifies t % 9 as : a candidate : amount of sewage at t % 9 : conclusion 9 9! "! " 559 E t, 65 9 % 6778 559! "! " E t, 65 559 % 85 98, 65() % The amount of sewage in the treatment tank is greatest at t % 9 hours At that time, the amount of sewage in the tank, rounded to the nearest gallon, is 67 gallons (c) Total cost! "! " % 5, t E t % 7 The total cost of treating the sewage that enters the tank during the time interval + t +, to the nearest dollar, is $7 : : integrand : limits : answer -7-
Question A particle moving along a curve in the xy-plane has position!! ",! "" 6! " At time t %, the particle is at the point! 5, " (a) Find the speed of the particle at time t % dy, t %, and % te t x t y t at time t 5, where (b) Write an equation for the line tangent to the path of the particle at time t % (c) Is there a time t at which the particle is farthest to the right? If yes, explain why and give the value of t and the x-coordinate of the position of the particle at that time If no, explain why not (d) Describe the behavior of the path of the particle as t increases without bound! "!!"" (a) Speed % x!" & y % : answer (b) At t %,! "!", dy y e % % %, x, e %, 9 : slope : : equation of tangent line An equation for the line tangent to the path at the point! 5, " is y, %,! x, 5 " e (c) Since 5 for 6 t 6 and 6 for t 5, the particle is farthest to the right at t % The x-coordinate of the position is! " 7 6 8! " x % 5 &, % 579 or 579 9 t : : yes with explanation : t % : integral for x! " : : x-coordinate dy %, and lim %, t: t: the particle will continue moving farther to the left while approaching a horizontal asymptote (d) Since lim! " : behavior of x : : behavior of y -8-
Question t (minutes) 8 6 Ht!"! C" The temperature, in degrees Celsius! " < 65 68 7 8 9 < C, of an oven being heated is modeled by an increasing differentiable function H of time t, where t is measured in minutes The table above gives the temperature as recorded every minutes over a 6-minute period (a) Use the data in the table to estimate the instantaneous rate at which the temperature of the oven is changing at time t % Show the computations that lead to your answer Indicate units of measure (b) Write an integral expression in terms of H for the average temperature of the oven between time t % and time t % 6 Estimate the average temperature of the oven using a left Riemann sum with four subintervals of equal length Show the computations that lead to your answer (c) Is your approximation in part (b) an underestimate or an overestimate of the average temperature? Give a reason for your answer (d) Are the data in the table consistent with or do they contradict the claim that the temperature of the oven is increasing at an increasing rate? Give a reason for your answer (a) H! "! " H! " H, 8 8, 7 7 - % % < C min, 8 6 (b) Average temperature is!" 6 H t 6! "! 65 68 7 8" H t - = & & & Average temperature = 86 - % 75< C 6 (c) The left Riemann sum approximation is an underestimate of the integral because the graph of H is increasing Dividing by 6 will not change the inequality, so 75< C is an underestimate of the average temperature (d) If a continuous function is increasing at an increasing rate, then the slopes of the secant lines of the graph of the function are increasing The slopes of the secant lines for the four intervals in the table are 5 7,,, and, respectively Since the slopes are increasing, the data are consistent with the claim OR By the Mean Value Theorem, the slopes are also the values of H! c k " for some times c 6 c 6 c 6 c, respectively Since these derivative values are positive and increasing, the data are consistent with the claim : : : difference quotient : answer with units : 6 6!" H t : left Riemann sum : answer : answer with reason : : considers slopes of four secant lines : explanation : conclusion consistent with explanation -9-
Question Let f be the function given by f! x"! x"! x" x! 6 + The function g is defined by g! x"! " (a) Find g!" and g!" % ln sin The figure above shows the graph of f for x % f t for 6 x +! (b) On what intervals, if any, is g increasing? Justify your answer (c) For 6 x +!, find the value of x at which g has an absolute minimum Justify your answer (d) For 6 x 6!, is there a value of x at which the graph of g is tangent to the x-axis? Explain why or why not (a) g! " % f! t" % and g! " f! " % % :!"!" : g : g (b) Since g! x" % f! x", g is increasing on the interval + x +! because! " (c) For x!, f x 5 for 6 x 6! 6 6 g! x" % f! x" % when x %,! g % f changes from negative to positive only at x % The absolute minimum must occur at x % or at the right endpoint Since g!" % and!!!! "! "! "! " g! % f t % f t & f t 6 by comparison of the two areas, the absolute minimum occurs at x %! (d) Yes, the graph of g is tangent to the x-axis at x % since g!" % and g!" %! : : interval : reason : : : identifies and! as candidates - or - indicates that the graph of g decreases, increases, then decreases : justifies g 6 : answer!!" g! " : answer of yes with x % : explanation --
Question 5 Let f be the function satisfying f! x" % x, x f! x" for all real numbers x, with! " lim f! x" % x: (a) Find the value of!,! "" x x f x Show the work that leads to your answer f % 5 and (b) Use Euler s method to approximate f!,, " starting at x %, with two steps of equal size (c) Find the particular solution y f! x" condition f! " % 5 dy % to the differential equation % x, xy with the initial (a)!,! "" x x f x b b % f! x" lim f! x" lim > f! x"? % b: % b: @ B AC % lim! f! b", f! " " %, 5 %, b: : : use of FTC : answer from limiting process (b) f! " f! " f! "! "! ", - & =, % 5 & =, % 5 7 f, - f, & f, =, % & =, %! "! "! "! " 5! " : Euler s method with two steps : : Euler s approximation to f,! " (c) dy % x, y, ln, y % x & A ln, y %, x & B y, % Ce,x C % 6 y % & e, x for all real numbers x 5 : : separates variables : antiderivatives : constant of integration : uses initial condition : solves for y Note: max 5 D---- E if no constant of integration --
Question 6 The function g is continuous for all real numbers x and is defined by cos! x", gx! " % for x F x (a) Use L Hospital s Rule to find the value of g! " Show the work that leads to your answer (b) Let f be the function given by f! x"! x" % cos Write the first four nonzero terms and the general term of the Taylor series for f about x % (c) Use your answer from part (b) to write the first three nonzero terms and the general term of the Taylor series for g about x % (d) Determine whether g has a relative minimum, a relative maximum, or neither at x % Justify your answer (a) Since g is continuous, cos! x", g! " % lim g! x" % lim x: x: x, sin! x" % lim x: x, cos! x" % lim %, x: : : uses L Hospital s Rule correctly at least once : answer (b)! x"! "! "! " 6 n x x x x cos %, &, &!&, &!!! 6!! n! "! "! n" n 6 n 6 6 %, x & x, x &!&, x &!!! 6!! n! "! n " : : first two terms : next two terms : general term n 6 6 n n, : first three terms g x %, & x, x &!&, x &! :!! 6!! : general term (c)! "! "! n " = 6 = 6! 6! (d) g! x" % x, x &!, so! " = 6 = = 6 g x x! 6! g %! " %, &!, so! " g 5 : : answer : justification Therefore, g has a relative minimum at x % by the Second Derivative Test --