Multiple Choice Solutions 1. E (2003 AB25) () xt t t t 2. A (2008 AB21/BC21) 3. B (2008 AB7) Using Fundamental Theorem of Calculus: 1

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1 Solutions We have intentionally included more material than can be covered in most Student Study Sessions to account for groups that are able to answer the questions at a faster rate. Use your own judgment, based on the group of students, to determine the order and selection of questions to work in the session. Be sure to include a variety of types of questions (multiple choice, free response, calculator, and non-calculator) in the time allotted. Multiple Choice Solutions. E ( AB5) () 7 xt t t t vt () x() t 6t 4t7 6 t 7t t( t4) t,4. A (8 AB/BC) V is increasing when v() t a() t which occurs when x() t is concave up, so t.. B (8 AB7) Using Fundamental Theorem of Calculus: x() x() (t 6 t) dt () ( ) t t x t t x() (4) 6 Alternatively: vt () t 6t xt () t t c x() 6( ) c c xt () t t x() 6 4. D (985 AB4) vt ( ) for all t therefore, 4 4 x() t v() t dt t 5t dt t4 5 t t t meters Copyright 4 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at

2 5. C (985 AB8) Average velocity of the particle is s t 5() 5() B (988 BC appropriate for AB) vt () dttcand v() () C 4 C Distance traveled from v() 4 and v() x() t (t4) dt t t 4t t 68 4 meters 7. C (8 AB86) v() x(), so x() t has a horizontal tangent at t ; therefore, the only possible graphs are C and E. From the table, v() x(), so x() t is increasing at t, so the answer is C. 8. C ( AB76) Using the derivative function on the calculator: v( t) a( t) a(4).6 9. E ( AB9/BC9) Using the Fundamental Theorem of Calculus and the integral function on the calculator: v() v() ln t dt t v() ln dt.46. A ( AB8) Average velocity of a function on [, ]: t t feet ( e te ) dt.86 second Copyright 4 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at

3 Free Response. ( AB/BC) (a) Runner A: velocity m or sec : velocity for Runner A : velocity for Runner B Runner B : 48 m v() sec (b) Runner A: acceleration. meters / sec 7 Runner B: a() v() (t ) t meters / sec 49 (c) Runner A: distance ()() 7() 85 meters Runner B: distance 4t dt 8.6 meters t 4 : acceleration for Runner A : acceleration for Runner B : distance for Runner A : method : answer : distance for Runner B : integral : answer Copyright 4 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at

4 . (999 AB) (a) v(.5).5sin(.5 ).67 Up, because v(.5) (b) at () v() t sint t cost a(.5) v(.5).48 or.49 No, v is decreasing at.5 because v(.5) (c) y() t v() t dt cost tsin t dt C 7 y() C C 7 yt () cost 7 y() cos 4.86 or.87 (d) distance = vt ( ) dt.7 or vt () tsint t or t.77 y () ; y 4; y ().86 or.87 y y() y y().7 or.74 : answer and reason : a (.5) : conclusion and reason : y( t) v( t) dt : yt () cost C : y () : limits of and on an integral of vt () or vt ( ) or uses y () and y () to compute distance : handles change of direction at student s turning point : answer / if incorrect turning point Copyright 4 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at

5 . (5 Form B AB) (a) 5 a(4) v(4) : answer 7 (b) vt ( ) t t t t ( t)( t) t, vt ( ) for t vt ( ) for t vt ( ) for t 5 : sets vt ( ) : direction change at t, : interval with reason (c) t () () ln( ) st s u u du () 8 ln( ) s u u du 8.68 or 8.69 : ln( u u ) du : handles initial condition : answer (d) vt ( ) dt.7 or.7 : integral : answer Copyright 4 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at

6 4. (8 AB4/BC4) (a) Since vt () for t and 5 t 6, and vt () for t 5, we consider t and t 6. x() v( t) dt 8 6 x(6) v( t) dt 8 9 Therefore, the particle is farthest left at time t when its position is x() (b) The particle moves continuously and monotonically from x() to x(). Similarly, the particle moves continuously and monotonically from x() to x(5) 7and also from x(5) 7 to x(6) 9. By the Intermediate Value Theorem, there are three values of t for which the particle is at xt ( ) 8. (c) The speed is decreasing on the interval t since on this interval v and v is increasing. (d) The acceleration is negative on the intervals t and 4t 6 since velocity is decreasing on these intervals. : identifies t as a candidate 6 : considers vt () dt : conclusion : position at t, t 5, and t 6 : description of motion : conclusion : answer with reason : answer : justification Copyright 4 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at

7 5. (B AB/BC5) v v. (a) a5. meters/sec (b) 6 vt dtis the total distance, in meters, that Ben rides over the 6-second intervalt tot 6. : answer : meaning of integral : approximation 6 vt dt meters B B (c) Because, the Mean Value 6 4 Theorem implies there is a time t, 4 t 6, such that vt. : difference quotient : conclusion with justification Copyright 4 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at