1998 Calculus AB Scoring Guidelines

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2 Velocity (feet per second) v(t) O 1998 Calculus AB Scoring Guidelines Time (seconds) t t v(t) (seconds) (feet per second) The graph of the velocity v(t), in ft/sec, of a car traveling on a straight road, for apple t apple 5, is shown above. A table of values for v(t), at 5 second intervals of time t, is shown to the right of the graph. (a) During what intervals of time is the acceleration of the car positive? Give a reason for your answer. (b) Find the average acceleration of the car, in ft/sec, over the interval apple t apple 5. (c) Find one approximation for the acceleration of the car, in ft/sec, at t = 4. Show the computations you used to arrive at your answer. (d) Approximate Z 5 v(t) dt with a Riemann sum, using the midpoints of five subintervals of equal length. Using correct units, explain the meaning of this integral. (a) Acceleration is positive on (, 5) and (45, 5) because the velocity v(t) is increasing on [, 5] and [45, 5] 8 >< 1: (, 5) 1: (45, 5) >: 1: reason Note: ignore inclusion of endpoints (b) Avg. Acc. = v(5) v() 5 or 1.44 ft/sec = 7 5 = 7 5 1: answer (c) (d) Di erence quotient; e.g. v(45) v(4) = 5 v(4) v(5) = 5 v(45) v(5) = 1 or Slope of tangent line, e.g. Z 5 through (5, 9) and (4, 75): v(t) dt = ft/sec or = 6 5 ft/sec or = 1 1 ft/sec = ft/sec 1[v(5) + v(15) + v(5) + v(5) + v(45)] = 1( ) = 5 feet This integral is the total distance traveled in feet over the time to 5 seconds. ( 1: method 1: answer Note: / if first point not earned 8 1: midpoint Riemann sum >< 1: answer >: 1: meaning of integral Copyright 1998 College Entrance Examination Board. All rights reserved. Advanced Placement Program and AP are registered trademarks 4 of the College Entrance Examination Board.

3 AP CALCULUS AB SCORING GUIDELINES Question The rate of fuel consumption, in gallons per minute, recorded during an airplane flight is given by a twice-differentiable and strictly increasing function R of time t. The graph of R and a table of selected values of R( t ), for the time interval t 9 minutes, are shown above. (a) Use data from the table to find an approximation for R ( 45 ). Show the computations that lead to your answer. Indicate units of measure. (b) The rate of fuel consumption is increasing fastest at time t = 45 minutes. What is the value of R ( 45 )? Explain your reasoning. (c) Approximate the value of 9 Rt () dt using a left Riemann sum with the five subintervals indicated by the data in the table. Is this numerical approximation less than the value of Explain your reasoning. 9 Rt () dt? b (d) For < b 9 minutes, explain the meaning of ( ) R t dt in terms of fuel consumption for the 1 b plane. Explain the meaning of R ( t ) dt b in terms of fuel consumption for the plane. Indicate units of measure in both answers. (a) R(5) R(4) 55 4 R(45) = = 1.5 gal/min : (b) R (45) = since R () t has a maximum at (c) t = Rt () dt ()() + (1)() + (1)(4) + ()(55) + ()(65) = 7 Yes, this approximation is less because the graph of R is increasing on the interval. : : 1 : a difference quotient using numbers from table and interval that contains 45 1 : 1.5 gal/min 1 : R(45) = 1 : reason 1 : value of left Riemann sum 1 : less with reason (d) b Rt () dt is the total amount of fuel in gallons consumed for the first b minutes. 1 b Rt () dt b is the average value of the rate of fuel consumption in gallons/min during the first b minutes. Copyright by College Entrance Examination Board. All rights reserved. 45 Available at apcentral.collegeboard.com. 4 : : meanings 1 : meaning of Rt ( ) dt 1 b 1 : meaning of Rt ( ) dt b < 1 > if no reference to time b 1 : units in both answers b

4 AP CALCULUS AB 4 SCORING GUIDELINES (Form B) Question A test plane flies in a straight line with t (min) positive velocity vt (), in miles per vt ()(mpm) minute at time t minutes, where v is a differentiable function of t. Selected values of vt () for t 4 are shown in the table above. (a) Use a midpoint Riemann sum with four subintervals of equal length and values from the table to 4 approximate vt () dt. Show the computations that lead to your answer. Using correct units, 4 explain the meaning of vt () dtin terms of the plane s flight. (b) Based on the values in the table, what is the smallest number of instances at which the acceleration of the plane could equal zero on the open interval < t < 4? Justify your answer. t 7t (c) The function f, defined by f() t = 6 + cos( ) + sin ( ), is used to model the velocity of the 1 4 plane, in miles per minute, for t 4. According to this model, what is the acceleration of the plane at t =? Indicates units of measure. (d) According to the model f, given in part (c), what is the average velocity of the plane, in miles per minute, over the time interval t 4? (a) Midpoint Riemann sum is 1 [ v( 5) + v( 15) + v( 5) + v( 5) ] = 1 [ ] = 9 The integral gives the total distance in miles that the plane flies during the 4 minutes. : 1 : v( 5) + v( 15) + v( 5) + v( 5) 1 : answer 1 : meaning with units (b) By the Mean Value Theorem, v () t = somewhere in the interval (, 15 ) and somewhere in the interval ( 5, ). Therefore the acceleration will equal for at least two values of t. 1 : two instances : 1 : justification (c) f ( ) =.47 or.48 miles per minute 1 : answer with units 1 4 (d) Average velocity = () 4 f t dt = miles per minute : 1 : limits 1 : integrand 1 : answer Copyright 4 by College Entrance Examination Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and (for AP students and parents). 46 4

5 AP CALCULUS AB 6 SCORING GUIDELINES (Form B) Question 6 t (sec) vt () ( ft sec ) at () ( ft sec ) A car travels on a straight track. During the time interval t 6 seconds, the car s velocity v, measured in feet per second, and acceleration a, measured in feet per second per second, are continuous functions. The table above shows selected values of these functions. (a) Using appropriate units, explain the meaning of vt () dtin terms of the car s motion. Approximate 6 vt () dtusing a trapezoidal approximation with the three subintervals determined by the table. 6 (b) Using appropriate units, explain the meaning of at () dtin terms of the car s motion. Find the exact value of at () dt. (c) For < t < 6, must there be a time t when vt () = 5? Justify your answer. (d) For < t < 6, must there be a time t when at () =? Justify your answer. 6 (a) vt () dtis the distance in feet that the car travels from t = sec to t = 6 sec. Trapezoidal approximation for vt () 6 dt: A = ( ) 5 + ( 1)( 15) + ( 1)( 1) = 185 ft (b) at () dtis the car s change in velocity in ft/sec from t = sec to t = sec. at () dt= v () t dt= v( ) v( ) = 14 ( ) = 6 ft/sec (c) Yes. Since v( 5) = 1 < 5 < = v( 5 ), the IVT guarantees a t in ( 5, 5 ) so that vt () = 5. (d) Yes. Since v( ) = v( 5 ), the MVT guarantees a t in (, 5 ) so that at () = v () t =. Units of ft in (a) and ft/sec in (b) : { 1 : explanation 1 : value : { 1 : explanation 1 : value 1 : v( 5) < 5 < v( 5) : 1 : Yes; refers to IVT or hypotheses 1 : v( ) = v( 5) : 1 : Yes; refers to MVT or hypotheses 1 : units in (a) and (b) 6 The College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and (for AP students and parents). 7 48

6 AP CALCULUS AB 6 SCORING GUIDELINES Question 4 t (seconds) vt () (feet per second) Rocket A has positive velocity vt () after being launched upward from an initial height of feet at time t = seconds. The velocity of the rocket is recorded for selected values of t over the interval t 8 seconds, as shown in the table above. (a) Find the average acceleration of rocket A over the time interval t 8 seconds. Indicate units of measure. 7 (b) Using correct units, explain the meaning of vt () dtin terms of the rocket s flight. Use a midpoint Riemann sum with subintervals of equal length to approximate vt () dt. 1 (c) Rocket B is launched upward with an acceleration of at () = feet per second per second. At time t + 1 t = seconds, the initial height of the rocket is feet, and the initial velocity is feet per second. Which of the two rockets is traveling faster at time t = 8 seconds? Explain your answer. 7 1 (a) Average acceleration of rocket A is 1 : answer v( 8) v( ) ft sec = = 8 8 (b) Since the velocity is positive, vt () dtrepresents the distance, in feet, traveled by rocket A from t = 1 seconds to t = 7 seconds : explanation : 1 : uses v( ), v( 4 ), v( 6) 1 : value A midpoint Riemann sum is [ v( ) + v( 4) + v( 6) ] = [ ] = ft (c) Let vb () t be the velocity of rocket B at time t. vb () t = dt = 6 t C t + 1 = v ( ) = 6 + C B vb () t = 6 t v ( 8) = 5 > 49 = v( 8) B 4 : 1 : 6 t : constant of integration 1 : uses initial condition 1 : finds vb ( 8 ), compares to v( 8 ), and draws a conclusion Rocket B is traveling faster at time t = 8 seconds. Units of ft sec in (a) and ft in (b) 1 : units in (a) and (b) 6 The College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and (for AP students and parents). 5 49

7 AP CALCULUS AB 7 SCORING GUIDELINES Question 5 t (minutes) r () t (feet per minute) The volume of a spherical hot air balloon expands as the air inside the balloon is heated. The radius of the balloon, in feet, is modeled by a twice-differentiable function r of time t, where t is measured in minutes. For < t < 1, the graph of r is concave down. The table above gives selected values of the rate of change, r (), t of the radius of the balloon over the time interval t 1. The radius of the balloon is feet when 4 t = 5. (Note: The volume of a sphere of radius r is given by V = π r. ) (a) Estimate the radius of the balloon when t = 5.4 using the tangent line approximation at t = 5. Is your estimate greater than or less than the true value? Give a reason for your answer. (b) Find the rate of change of the volume of the balloon with respect to time when t = 5. Indicate units of measure. (c) Use a right Riemann sum with the five subintervals indicated by the data in the table to approximate 1 r () t dt. Using correct units, explain the meaning of () r t dt in terms of the radius of the balloon. (d) Is your approximation in part (c) greater than or less than r () t dt? Give a reason for your answer. (a) r( 5.4) r( 5) + r ( 5) t = + (.4) =.8 ft Since the graph of r is concave down on the interval 5 < t < 5.4, this estimate is greater than r ( 5.4 ). 1 1 : { 1 : estimate 1 : conclusion with reason dv dt dv dt 4 π r (b) = ( ) 1 t = 5 dr dt = 4π( ) = 7π ft min (c) r () t dt 4. ( ) +. ( ) + 1. ( ) ( ) ( ) = 19. ft 1 r () t dt is the change in the radius, in feet, from t = to t = 1 minutes. (d) Since r is concave down, r is decreasing on < t < 1. Therefore, this approximation, 19. ft, is less than 1 r () t dt. : dv : dt 1 : answer : { 1 : approximation 1 : explanation 1 : conclusion with reason Units of ft min in part (b) and ft in part (c) 1 : units in (b) and (c) 7 The College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and (for students and parents). 5

12 APÆ CALCULUS AB 1 SCORING GUIDELINES Question The temperature, in degrees Celsius ( C), of the water in a pond is a differentiable function W of time t. The table above shows the water temperature as recorded every days over a 15-day period. (a) Use data from the table to find an approximation for W = (1). Show the computations that lead to your answer. Indicate units of measure. (b) Approximate the average temperature, in degrees Celsius, of the water over the time interval > t > 15 days by using a trapezoidal approximation with subintervals of length t days. ( t /) (c) A student proposes the function P, given by Pt () 1te Г, as a model for the temperature of the water in the pond at time t, where t is measured in days and Pt () is measured in degrees Celsius. Find P= (1). Using appropriate units, explain the meaning of your answer in terms of water temperature. (d) Use the function P defined in part (c) to find the average value, in degrees Celsius, of Pt () over the time interval > t > 15 days. t (days) Wt () ( C) (a) Difference quotient; e.g. W(15) ГW(1) 1 W = (1) N Г C/day or 15 Г 1 W(1) ГW(9) W = (1) N Г C/day or 1 Г 9 : 1 : difference quotient 1 : answer (with units) W(15) ГW(9) 1 W = (1) N Г 15 Г 9 C/day (b) (1) (8) (4) () Average temperature N (76.5) 5.1 C 15 : 1 : trapezoidal method 1 : answer (c) 1 P= (1) 1e Г te Гt/ Гt/ 4 e Г t1 Г Г.549 C/day : 1 : P = (1) (with or without units) 1 : interpretation This means that the temperature is decreasing at the rate of.549 C/day when t = 1 days. (d) Гt / 1te dt C : 1 : integrand 1 : limits and average value constant 1 : answer Copyright 1 by College Entrance Examination Board. All rights reserved. Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board. 4

13 A blood vessel is 6 millimeters (mm) long with circular cross sections of varying diameter. The table above gives the measurements of the diameter of the blood vessel at selected points AP CALCULUS AB SCORING GUIDELINES (Form B) Question along the length of the blood vessel, where x represents the distance from one end of the blood vessel and Bx () is a twice-differentiable function that represents the diameter at that point. (a) Write an integral expression in terms of Bx () that represents the average radius, in mm, of the blood vessel between x = and x = 6. (b) Approximate the value of your answer from part (a) using the data from the table and a midpoint Riemann sum with three subintervals of equal length. Show the computations that lead to your answer. (c) Using correct units, explain the meaning of Distance x (mm) Diameter B(x) (mm) Bx () dx in terms of the blood vessel. (d) Explain why there must be at least one value x, for < x < 6, such that B ( x) =. (a) 1 6 Bx () dx : 6 1 : limits and constant 1 : integrand (b) 1 B(6) B(18) B() = 6 1 [ 6( )] = 14 6 : 1 : B(6) + B(18) + B() 1 : answer (c) Bx ( ) is the radius, so Bx ( ) is the area of the cross section at x. The expression is the volume in mm of the blood vessel between 15 : 1 : volume in mm 1 : between x = 15 and x = 75 mm and 75 mm from the end of the vessel. (d) By the MVT, B ( c1) = for some c 1 in (6, 18) and B ( c) = for some c in (4, 6). The MVT applied to B ( x) shows that B () x = for some x in the interval ( c1 c ),. : : explains why there are two values of x where B( x) has the same value 1 : explains why that means B ( x) = for < x < 6 Copyright by College Entrance Examination Board. All rights reserved. Available at apcentral.collegeboard.com Note: max 1/ if only explains why B ( x) = at some x in (, 6).

14 AP CALCULUS AB 5 SCORING GUIDELINES Question Distance x (cm) Temperature T( x ) ( C) A metal wire of length 8 centimeters (cm) is heated at one end. The table above gives selected values of the temperature T( x ), in degrees Celsius ( C, ) of the wire x cm from the heated end. The function T is decreasing and twice differentiable. (a) Estimate T ( 7. ) Show the work that leads to your answer. Indicate units of measure. (b) Write an integral expression in terms of T( x ) for the average temperature of the wire. Estimate the average temperature of the wire using a trapezoidal sum with the four subintervals indicated by the data in the table. Indicate units of measure. 8 (c) Find T ( x) dx, and indicate units of measure. Explain the meaning of ( ) T x dx in terms of the temperature of the wire. (d) Are the data in the table consistent with the assertion that T ( x) > for every x in the interval < x < 8? Explain your answer. 8 (a) T( 8) T( 6) = = Ccm : answer 1 8 (b) ( ) 8 T x dx Trapezoidal approximation for T( x) dx: A = Average temperature C 8 A = 8 (c) T ( x) dx = T( 8) T( ) = 55 1 = 45 C 8 The temperature drops 45 C from the heated end of the wire to the other end of the wire. 1, 5 is 7 9 = , 6 is 6 7 = T c 1 = 5.75 for some c 1 in the interval ( 1, 5 ) T c = 8 for some c in the interval ( 5, 6 ). It follows that c, c. Therefore T (d) Average rate of change of temperature on [ ] Average rate of change of temperature on [ ] No. By the MVT, ( ) and ( ) T must decrease somewhere in the interval ( 1 ) is not positive for every x in [, 8 ]. : : T( x) dx 8 1 : trapezoidal sum 1 : answer : { 1 : value 1 : meaning 1 : two slopes of secant lines : { 1 : answer with explanation Units of Ccmin (a), and C in (b) and (c) 1 : units in (a), (b), and (c) Copyright 5 by College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and 47 (for AP students and parents). 4

15 AP CALCULUS AB 8 SCORING GUIDELINES (Form B) Distance from the river s edge (feet) Question Depth of the water (feet) 7 8 A scientist measures the depth of the Doe River at Picnic Point. The river is 4 feet wide at this location. The measurements are taken in a straight line perpendicular to the edge of the river. The data are shown in the table above. The velocity of the water at Picnic Point, in feet per minute, is modeled by vt = 16 + sin t+ 1 for t 1 minutes. () ( ) (a) Use a trapezoidal sum with the four subintervals indicated by the data in the table to approximate the area of the cross section of the river at Picnic Point, in square feet. Show the computations that lead to your answer. (b) The volumetric flow at a location along the river is the product of the cross-sectional area and the velocity of the water at that location. Use your approximation from part (a) to estimate the average value of the volumetric flow at Picnic Point, in cubic feet per minute, from t = to t = 1 minutes. π x (c) The scientist proposes the function f, given by f( x) ( ) = 8sin, as a model for the depth of the 4 water, in feet, at Picnic Point x feet from the river s edge. Find the area of the cross section of the river at Picnic Point based on this model. (d) Recall that the volumetric flow is the product of the cross-sectional area and the velocity of the water at a location. To prevent flooding, water must be diverted if the average value of the volumetric flow at Picnic Point exceeds 1 cubic feet per minute for a -minute period. Using your answer from part (c), find the average value of the volumetric flow during the time interval 4 t 6 minutes. Does this value indicate that the water must be diverted? (a) ( + 7) ( 7 + 8) ( 8 + ) ( + ) = 115 ft 1 : trapezoidal approximation 1 1 (b) 115 () 1 vt dt = or ft min 4 (c) ( ) π x 8sin dx = 1. or 1.1 ft : { 1 : integra1 4 (d) Let C be the cross-sectional area approximation from part (c). The average volumetric flow is 1 6 () or ft min. C v t dt = 4 Yes, water must be diverted since the average volumetric flow for this -minute period exceeds 1 ft min. : : 1 : limits and average value constant 1 : integrand 1 : answer 1 : answer 1 : volumetric flow integral 1 : average volumetric flow 1 : answer with reason 8 The College Board. All rights reserved. Visit the College Board on the Web: 51

16 AP CALCULUS AB 8 SCORING GUIDELINES Question t (hours) Lt ()(people) Concert tickets went on sale at noon ( t = ) and were sold out within 9 hours. The number of people waiting in line to purchase tickets at time t is modeled by a twice-differentiable function L for t 9. Values of Lt () at various times t are shown in the table above. (a) Use the data in the table to estimate the rate at which the number of people waiting in line was changing at 5: P.M. ( t = 5.5 ). Show the computations that lead to your answer. Indicate units of measure. (b) Use a trapezoidal sum with three subintervals to estimate the average number of people waiting in line during the first 4 hours that tickets were on sale. (c) For t 9, what is the fewest number of times at which L () t must equal? Give a reason for your answer. (d) The rate at which tickets were sold for t 9 is modeled by rt () = 55te t tickets per hour. Based on the model, how many tickets were sold by P.M. ( t =, ) to the nearest whole number? L( 7) L( 4) (a) L ( 5.5) = = 8 people per hour 7 4 (b) The average number of people waiting in line during the first 4 hours is approximately 1 L( ) + L( 1 ) L() ( ) ( ) ( ) ( 1 ) 1 + L L ( 1) + L 4 ( 4 ) = people (c) L is differentiable on [, 9 ] so the Mean Value Theorem implies L () t > for some t in ( 1, ) and some t in ( 4, 7 ). Similarly, L () t < for some t in (, 4 ) and some t in ( 7, 8 ). Then, since L is continuous on [, 9 ], the Intermediate Value Theorem implies that L () t = for at least three values of t in [, 9 ]. OR The continuity of L on [ 1, 4 ] implies that L attains a maximum value there. Since L( ) > L( 1) and L( ) > L( 4 ), this maximum occurs on ( 1, 4 ). Similarly, L attains a minimum on (, 7 ) and a maximum on ( 4, 8 ). L is differentiable, so L () t = at each relative extreme point on (, 9 ). Therefore L () t = for at least three values of t in [, 9 ]. [Note: There is a function L that satisfies the given conditions with L () t = for exactly three values of t.] (d) rt () dt= There were approximately 97 tickets sold by P.M. : { 1 : estimate 1 : units 1 : trapezoidal sum : { 1 : answer : : 1 : considers change in sign of L 1 : analysis 1 : conclusion OR 1 : considers relative extrema of L on (, 9) 1 : analysis 1 : conclusion : { 1 : integrand 1 : limits and answer 8 The College Board. All rights reserved. Visit the College Board on the Web: 5

AP CALCULUS AB/CALCULUS BC 2015 SCORING GUIDELINES

AP CALCULUS AB/CALCULUS BC 2015 SCORING GUIDELINES AP CALCULUS AB/CALCULUS BC 15 SCORING GUIDELINES Question 3 t (minutes) vt ( ) (meters per minute) 1 4 4 4 15 Johanna jogs along a straight path. For t 4, Johanna s velocity is given by a differentiable