# AP Calculus AB 2006 Scoring Guidelines

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2 Question 1 Let R be the shaded region bounded by the graph of y = ln x and the line y = x, as shown above. (a) Find the area of R. (b) Find the volume of the solid generated when R is rotated about the horizontal line y = 3. (c) Write, but do not evaluate, an integral expression that can be used to find the volume of the solid generated when R is rotated about the y-axis. ln = x when x = and Let S = and T = T (a) Area of R = ( ln ( x ) ) dx = S 3 : 1 : integrand 1 : limits 1 : answer T ( ) (b) Volume = π ( ln + 3 ) ( x + 3 ) dx S = or : { : integrand 1 : limits, constant, and answer ( ) T (c) Volume = y π ( + ) ( ) S y e dy 3 : { : integrand 1 : limits and constant

3 Question At an intersection in Thomasville, Oregon, cars turn t left at the rate Lt () = 60 tsin ( 3) cars per hour over the time interval 0 t 18 hours. The graph of y = L() t is shown above. (a) To the nearest whole number, find the total number of cars turning left at the intersection over the time interval 0 t 18 hours. (b) Traffic engineers will consider turn restrictions when Lt () 150 cars per hour. Find all values of t for which Lt () 150 and compute the average value of L over this time interval. Indicate units of measure. (c) Traffic engineers will install a signal if there is any two-hour time interval during which the product of the total number of cars turning left and the total number of oncoming cars traveling straight through the intersection is greater than 00,000. In every two-hour time interval, 500 oncoming cars travel straight through the intersection. Does this intersection require a traffic signal? Explain the reasoning that leads to your conclusion (a) Lt () dt 1658 cars (b) Lt () = 150 when t = , Let R = and S = Lt () 150 for t in the interval [ R, S ] 1 S Lt () dt= S R cars per hour R (c) For the product to exceed 00,000, the number of cars turning left in a two-hour interval must be greater than OR Lt () dt= > 400 The number of cars turning left will be greater than 400 on a two-hour interval if Lt () 00 on that interval. Lt () 00 on any two-hour subinterval of [ , ]. : { 1 : setup 1 : answer 3 : 1 : t-interval when L() t : average value integral 1 : answer with units 1 : considers 400 cars 1 : valid interval [ h, h ] + h+ 1 : value of Lt () dt h 1 : answer and explanation OR 1 : considers 00 cars per hour 1 : solves Lt () 00 1 : discusses hour interval 1 : answer and explanation Yes, a traffic signal is required. 3

4 The graph of the function f shown above consists of six line segments. Let g be the function given by x g = f( t) dt. 0 (a) Find g ( 4, ) g ( 4, ) and g ( 4. ) (b) Does g have a relative minimum, a relative maximum, or neither at x = 1? Justify your answer. AP CALCULUS AB Question 3 (c) Suppose that f is defined for all real numbers x and is periodic with a period of length 5. The graph above shows two periods of f. Given that g ( 5) =, find g ( 10) and write an equation for the line tangent to the graph of g at x = 108. (a) g( 4) = f( t) dt = g ( 4) = f( 4) = 0 g ( 4) = f ( 4) = 3 : 1 : g( 4) 1 : g ( 4) 1 : g ( 4) (b) g has a relative minimum at x = 1 because g = f changes from negative to positive at x = 1. : { 1 : answer 1 : reason (c) g = 0 and the function values of g increase by for every increase of 5 in x. g( 10) = g( 5) = g( 108) = f() t dt + f() t dt = 1g( 5) + g( 3) = 44 1 : g( 10) 1 : g( 108) 3 : 1 : g ( 108) 1 : equation of tangent line g ( 108) = f( 108) = f( 3) = An equation for the line tangent to the graph of g at x = 108 is y 44 = ( x 108 ). 4

5 Question 4 t (seconds) vt () (feet per second) Rocket A has positive velocity vt () after being launched upward from an initial height of 0 feet at time t = 0 seconds. The velocity of the rocket is recorded for selected values of t over the interval 0 t 80 seconds, as shown in the table above. (a) Find the average acceleration of rocket A over the time interval 0 t 80 seconds. Indicate units of measure. 70 (b) Using correct units, explain the meaning of vt () dt in terms of the rocket s flight. Use a midpoint Riemann sum with 3 subintervals of equal length to approximate vt () dt (c) Rocket B is launched upward with an acceleration of at () = feet per second per second. At time t + 1 t = 0 seconds, the initial height of the rocket is 0 feet, and the initial velocity is feet per second. Which of the two rockets is traveling faster at time t = 80 seconds? Explain your answer (a) Average acceleration of rocket A is 1 : answer v( 80) v ft sec = = (b) Since the velocity is positive, vt () dtrepresents the distance, in feet, traveled by rocket A from t = 10 seconds to t = 70 seconds : explanation 3 : 1 : uses v( 0 ), v( 40 ), v( 60) 1 : value A midpoint Riemann sum is 0[ v( 0) + v( 40) + v( 60) ] = 0[ ] = 00 ft (c) Let vb () t be the velocity of rocket B at time t. 3 vb () t = dt = 6 t C t + 1 = vb = 6 + C vb () t = 6 t v ( 80) = 50 > 49 = v( 80) B 1 : 6 t : constant of integration 1 : uses initial condition 1 : finds vb ( 80 ), compares to v( 80 ), and draws a conclusion Rocket B is traveling faster at time t = 80 seconds. Units of ft sec in (a) and ft in (b) 1 : units in (a) and (b) 5

6 Question 5 dy 1 + y Consider the differential equation =, where x 0. dx x (a) On the axes provided, sketch a slope field for the given differential equation at the eight points indicated. (Note: Use the axes provided in the pink exam booklet.) (b) Find the particular solution y = f to the differential equation with the initial condition f ( 1) = 1 and state its domain. (a) : sign of slope at each point and relative steepness of slope lines in rows and columns (b) 1 1 dy = dx 1 + y x ln 1 + y = ln x + K ln x 1 + y = e 1 + y = C x = C 1+ y = x + K y = x 1 and x < 0 or y = x 1 and x < 0 7 : 1 : separates variables : antiderivatives 6 : 1 : constant of integration 1 : uses initial condition 1 : solves for y Note: max 3 6 [ ] if no constant of integration Note: 0 6 if no separation of variables 1 : domain 6

7 Question 6 The twice-differentiable function f is defined for all real numbers and satisfies the following conditions: f =, f = 4, and f = 3. ax (a) The function g is given by gx ( ) = e + f for all real numbers, where a is a constant. Find g and g in terms of a. Show the work that leads to your answers. (b) The function h is given by hx ( ) = cos( kx) f for all real numbers, where k is a constant. Find h and write an equation for the line tangent to the graph of h at x = 0. (a) g = ae ax + f g = a 4 ax g = a e + f g = a : g 1 : g 1 : g 1 : g (b) h = f cos( kx) ksin ( kx) f h = f cos ksin f = f = 4 h = cos f = The equation of the tangent line is y = 4x +. 5 : : h 1 : h 3 : 1 : h 1 : equation of tangent line 7

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