AP * Calculus Review

Transcription

1 AP * Calculus Review Posiion, Velociy, and Acceleraion Teacher Packe AP* is a rademark of he College Enrance Examinaion Board. The College Enrance Examinaion Board was no involved in he producion of his maerial.

2 Posiion, Velociy, and Acceleraion Page 1 of 15 Session Noes Suppose an objec is moving along a sraigh line, such as he x-axis, so ha is posiion x, as a funcion of ime, on ha line is given by y = x (). Average velociy of he objec over he ime inerval x( +Δ) x( ) change in posiion, or. Δ change in ime o + Δ is given by Insananeous velociy of he objec is he derivaive of he posiion funcion x( ) wih respec o ime. v ( ) = x ( ) dx Speed is he absolue value of he velociy. Speed = v() =. d Acceleraion is he derivaive of velociy wih respec o ime. a () = v () = x () vd () = x () + c, ad () = v () + c Toal disance raveled from ime = 1 o = 2 is given by 2 TDT = v() d. 1

3 Posiion, Velociy, and Acceleraion Page 2 of 15 Speeding Up or Slowing Down If he velociy and acceleraion have he same sign (boh posiive or boh negaive), hen speed is increasing. If an objec s velociy is 40 miles per hour and he objec acceleraes 10 miles per hour per hour, he objec is speeding up. If he velociy and acceleraion are opposie in sign (one is posiive and he oher is negaive), hen speed is decreasing. If an objec s velociy is 40 miles per hour and he objec acceleraes 10 miles per hour per hour, he objec is slowing down. Sign Convenion: When he objec is moving in he righ direcion or moving upward hen he velociy is posiive (Graph of velociy vs. ime is above he axis). When he objec is moving in he lef direcion or moving downward hen he velociy is negaive. A graph of velociy vs. ime is shown below. V < < v( ), a( ) < < v( ), a( ) < < v( ), a( ) < < v( ), a( ) Objec is slowing down Objec is speeding up + Objec is slowing down + + Objec is speeding up

4 Posiion, Velociy, and Acceleraion Page 3 of 15 Wha you need o know abou moion along he x-axis: When you see Think Iniially = 0 A res v ( ) = 0 A he origin x ( ) = 0 Velociy is posiive Velociy is negaive Average velociy (Given x( ) ) Average velociy (Given v ( ) ) Insananeous velociy Posiive acceleraion Negaive acceleraion Paricle is moving righ (or up) Paricle is moving lef (or down) Change in posiion divided by change in ime 1 b v () d b a (The average value a of he velociy funcion.) Velociy a an exac momen Velociy is increasing Velociy is decreasing Speed v () b Toal Disance Traveled v () d a

5 Posiion, Velociy, and Acceleraion Page 4 of 15 Muliple Choice 3 1. The velociy of a paricle moving on a line a ime is v= How many meers did he paricle ravel from = 1 o = 8? 2 (A) 10 (B) 224 (C) (D) 282 (E) The posiion of he paricle is given as x( ) = cos(3 ) sin(4 ). Find he acceleraion a = 0. (A) -9 (B) 0 (C) 1 (D) 2 (E) 16

6 Posiion, Velociy, and Acceleraion Page 5 of A paricle moves along he x-axis wih acceleraion a any ime given as 2 a () = If he paricle s iniial velociy is 10 and is iniial posiion is 2, wha is he posiion funcion? (A) (C) (E) x () = (B) x() = (D) 1 2 x () = x = () x () = If he posiion of an an raveling along a horizonal pah a ime is , wha is he an s average velociy from = 1 o = 6? (A) 1 21 (B) 6 (C) (D) 21 (E) 220

7 Posiion, Velociy, and Acceleraion Page 6 of 15 The following informaion applies o problems 5, 6 and 7. A bole rocke is sho upward from a 10 foo sand wih velociy v ( ) = GC Wha is he posiion of he bole rocke afer 2 seconds? (A) 46.8 f (B) 56.8 f (C) 96.8 f (D) f (E) f 6.GC When will he bole rocke hi he ground? (A) = 0 (B) = 8.66 (C) = (D) = 62.5 (E) = GC Afer 3 seconds he rocke is (A) falling a an increasing rae (B) rising a an increasing rae (C) rising a a decreasing rae (D) falling a a decreasing rae (E) rising a consan rae

8 Posiion, Velociy, and Acceleraion Page 7 of The posiion of he paricle raveling along a sraigh line is 3 2 x() = On he inerval = 0 o = 10, when is he paricle farhes o he lef? (A) = 0 (B) = 1 (C) = 3 (D) = 5 (E) = 10 9.GC Choose he inegral expression ha would resul in he oal disance raveled on he inerval [0, 3] if he velociy is given by v ( ) = e 6. (A) (B) (C) (D) (E) ln 6 3 ( e 6) d + ( e 6) d 0 ln6 ln 6 0 ( e 6) d ( e 6) d 3 ln6 3 ln6 ( e 6) d ( e 6) d ln 6 0 ln 6 3 ( e 6) d ( e 6) d 0 ln6 3 ( e 6) d 0 10.GC When wo paricles sar a he origin wih velociies v () = 4cos and v () = 4sin, how many imes in he inerval [0, 2π] will heir speeds be equal? (A) 2 (B) 3 (C) 4 (D) 5 (E) 6

9 Free Response 1 No Calculaor Posiion, Velociy, and Acceleraion Page 8 of 15 The graph given above is y = v ( ), he velociy of an objec moving on a line over he ime inerval [0, 8]. A = 0 he posiion of he objec is 5. (a) When is he objec a res? (b) Evaluae 1 6 v () d. Explain he meaning of he resul. (c) Wha is he posiion of he objec a = 5? (d) Find he oal disance raveled over [0, 8]. (e) A = 2, is he objec speeding up or slowing down? Explain your answer.

10 Free Response 2 Calculaor allowed Posiion, Velociy, and Acceleraion Page 9 of 15 (hours) v (miles per hour) The able represens daa colleced in an experimen on a new ype of elecric engine for a small neighborhood vehicle (i.e., one ha is licensed for ravel on roads wih speed limis of 35 mph or less). The readings represen velociy, in miles per hour, aken in 15-minue inervals on a 2 hour rip. (a) Wha is he average acceleraion over [0.25, 0.75]? Include unis. (b) Wha is he meaning of your answer. 1 b () b a v d if a = 0.75 and b = 2? Include unis in a (c) Use a midpoin Riemann sum wih four subinervals o approximae 2 v () d. 0 (d) A he end of he wo hours he vehicle is 35 miles from a source for recharging he baery. Assuming ha he vehicle can ravel 75 miles on a single charge, can he vehicle ge back o he source (wihou being owed or pushed)? Explain your answer.

11 Free Response 3 No Calculaor Posiion, Velociy, and Acceleraion Page 10 of 15 1 π Le v () = + sin3 represen he velociy of an objec moving on a line. A =, he π 3 posiion is 4. (a) Wrie he acceleraion funcion. (b) Wrie he posiion funcion. (c) A π = is he objec speeding up or slowing down? Explain your answer. 4 π (d) On he inerval, π 2, wha is he velociy when he acceleraion is 3?

12 Posiion, Velociy, and Acceleraion Page 11 of 15 KEY 1. D 2. A 3. B 4. D 5. E 6. E 7. C 8. D 9. C 10. C

13 Free Response 1 No Calculaor Posiion, Velociy, and Acceleraion Page 12 of 15 The graph given above is y = v(), he velociy of an objec moving on a line over he ime inerval [0, 8]. A = 0 he posiion of he objec is 5. (a) When is he objec a res? 6 (b) Evaluae v () d. Explain he meaning of he resul. 1 (c) Wha is he posiion of he objec a = 5? (d) Find he oal disance raveled over [0, 8]. (e) A = 2, is he objec speeding up or slowing down? Explain your answer. (a) The objec is a res when v ( ) = 0. 1 p for correc imes This occurs a = 5, 8. (b) The displacemen of he objec over 1 p for correc answer he ime inerval = 1 o = 6 is 4 unis. 1 p for displacemen (c) v ( ) d= p for iniial value (5) The posiion a = 5 is 11 unis. 1 p for correc answer (d) 8 TDT = v ( ) d 1 p for correc inegral 0 The oal disance raveled over he inerval [ 0, 8 ] Is 10 unis. 1 p for correc answer (e) The objec is speeding up because 1 p for speeding up v (2) > 0 and a (2) > 0. 1 p for v(2) and a(2) posiive

14 Free Response 2 Calculaor allowed Posiion, Velociy, and Acceleraion Page 13 of 15 (hours) v (miles per hour) The able represens daa colleced in an experimen on a new ype of elecric engine for a small neighborhood vehicle (i.e., one ha is licensed for ravel on roads wih speed limis of 35 mph or less). The readings represen velociy, in miles per hour, aken in 15- minue inervals on a 2 hour rip. (a) (b) (c) (d) Wha is he average acceleraion over [0.25, 0.75]? Include unis. 1 b Wha is he meaning of () b a v d if a =0.75 and b = 2? Include unis in your answer. a 2 Use a midpoin Riemann sum wih four subinervals o approximae v () d. 0 A he end of he wo hours he vehicle is 35 miles from a source for recharging he baery. Assuming ha he vehicle can ravel 75 miles on a single charge, can he vehicle ge back o he source (wihou being owed or pushed)? Explain your answer. (a) v(0.75) v(0.25) = The average acceleraion over he inerval is 5 miles per hour per hour. 1 p se up 1 p answer (b) The average velociy in miles per hour per hour on he inerval = 0.75 o = 2. 1 p meaning (c) 0.5( ) 2 ps Riemann sum miles 1 p answer (d) Yes, he car has raveled approximaely 1 p answer miles. I should be able o reurn he 1 p explanaion he 75 mile range. 1 p unis in (a) and (b)

15 Free Response #3 No Calculaor Posiion, Velociy, and Acceleraion Page 14 of 15 1 π Le v () = + sin3 represen he velociy of an objec moving on a line. A =, he posiion is 4. π 3 (a) Wrie he acceleraion funcion. (b) Wrie he posiion funcion. (c) π A = is he objec speeding up or slowing down? Explain your answer. 4 (d) π On he inerval, π 2, wha is he velociy when he acceleraion is 3? (a) a () = v () = 3cos3 1 p for a () (b) x() = v() d 1 x() = + sin3 d π 1 x() = cos3+ C 1 p for correc inegraion π = cosπ + C C = 3 1 p for C 1 10 x () = cos3+ π p for x( ) (c) The objec is slowing down. 1 3 v π sin π = + > 0 4 π 4 1 p correc answer 1 p for v π is posiive 4 3 a π = 3cos π < p for a π 4 is negaive (d) 3= 3cos3 1 p for equaion 1= cos3 3 = 0, 2 π... 2π A =, he acceleraion is π 1 v = p 1 or π π

16 Posiion, Velociy, and Acceleraion Page 15 of 15 AP Calculus Exam Connecions The lis below idenifies free response quesions ha have been previously asked on he opic of Posiion, Velociy, and Acceleraion. These quesions are available from he CollegeBoard and can be downloaded free of charge from AP Cenral. hp://apcenral.collegeboard.com. Free Response Quesions 2004 AB Quesion Form B AB Quesion AB Quesion 4