A Curriculum Module for AP Calculus BC Curriculum Module

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1 Vecors: A Curriculum Module for AP Calculus BC 00 Curriculum Module

2 The College Board The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in 900, he College Board is composed of more han 5,700 schools, colleges, universiies and oher educaional organizaions. Each year, he College Board serves seven million sudens and heir parens,,000 high schools, and,800 colleges hrough major programs and services in college readiness, college admission, guidance, assessmen, financial aid and enrollmen. Among is widely recognized programs are he SAT, he PSAT/NMSQT, he Advanced Placemen Program (AP ), SpringBoard and ACCUPLACER. The College Board is commied o he principles of excellence and equiy, and ha commimen is embodied in all of is programs, services, aciviies and concerns. For furher informaion, visi The College Board wishes o acknowledge all he hird-pary sources and conen ha have been included in hese maerials. Sources no included in he capions or bo of he ex are lised here. We have made every effor o idenify each source and o race he copyrigh holders of all maerials. However, if we have incorrecly aribued a source or overlooked a publisher, please conac us and we will make he necessary correcions. 00 The College Board. College Board, ACCUPLACER, Advanced Placemen Program, AP, AP Cenral, Pre-AP, SpringBoard and he acorn logo are regisered rademarks of he College Board. inspiring minds is a rademark owned by he College Board. PSAT/NMSQT is a regisered rademark of he College Board and Naional Meri Scholarship Corporaion. All oher producs and services may be rademarks of heir respecive owners. Visi he College Board on he Web:

3 Conens Inroducion... 4 Day : Graphing Parameric Equaions and Eliminaing he Parameer... 6 Day : Parameric Equaions and Calculus... 4 Day : Review of Moion Along a Line... Day 4: Moion Along a Curve Vecors... 7 Day 5: Moion Along a Curve Vecors (coninued)... 5 Day 6: Moion Along a Curve Vecors (coninued)... 9 Abou he Auhor... 4

4 Vecors Vecors in AP Calculus BC Nancy Sephenson Clemens High School Sugar Land, Texas Inroducion According o he AP Calculus BC Course Descripion, sudens in Calculus BC are required o know: Analysis of planar curves given in parameric form and vecor form, including velociy and acceleraion vecors Derivaives of parameric and vecor funcions The lengh of a curve, including a curve given in parameric form Wha does his mean? For parameric equaions x f and y g, sudens should be able o:. Skech he curve defined by he parameric equaions and eliminae he parameer.. Find d y and and evaluae hem for a given value of.. Wrie an equaion for he angen line o he curve for a given value of. 4. Find he poins of horizonal and verical angency. 5. Find he lengh of an arc of a curve given by parameric equaions. For vecors describing paricle moion along a curve in erms of a ime variable, sudens should be able o:. Find he velociy and acceleraion vecors when given he posiion vecor.. Given he componens of he velociy vecor and he posiion of he paricle a a paricular value of, find he posiion a anoher value of.. Given he componens of he acceleraion vecor and he velociy of he paricle a a paricular value of, find he velociy a anoher value of. 4. Find he slope of he pah of he paricle for a given value of. 5. Wrie an equaion for he angen line o he curve for a given value of. 6. Find he values of a which he line angen o he pah of he paricle is horizonal or verical The College Board.

5 Vecors 7. Find he speed of he paricle (someimes asked as he magniude of he velociy vecor) a a given value of. 8. Find he disance raveled by he paricle for a given inerval of ime. I like o sar his uni wih parameric equaions, eaching he sudens he five ypes of parameric problems lised above. Then I ake a day o review he concep of moion along a horizonal or verical line, which hey learned earlier in he year, as a bridge o moion along a curve. The uni on parameric equaions and vecors akes me six days o cover (see he following schedule), no including a es day. I each on a radiional seven-period day, wih 50 minues in each class period. Day Graphing parameric equaions and eliminaing he parameer Day Calculus of parameric equaions: Finding d y and and evaluaing hem for a given value of, finding poins of horizonal and verical angency, finding he lengh of an arc of a curve Day Review of moion along a horizonal and verical line. (The sudens have sudied his opic earlier in he year.) Days 4, 5 and 6 Paricle moion along a curve (vecors): Finding he velociy and acceleraion vecors when given he posiion vecor; Given he componens of he velociy vecor and he posiion of he paricle a one value of, finding he posiion of he paricle a a differen value of ; Finding he slope of he pah of he paricle for a given value of ; Wriing an equaion for he angen line o he curve for a given value of ; Finding he values of a which he line angen o he pah of he paricle is horizonal or verical; Finding he speed of he paricle; and Finding he disance raveled by he paricle for a given inerval of ime The College Board.

6 Vecors Day : Graphing Parameric Equaions and Eliminaing he Parameer My sudens have sudied parameric equaions and vecors in heir precalculus course, so his lesson is a review for hem. Many of hem have also sudied parameric equaions and vecors in heir physics course. If your exbook conains his maerial, you migh wan o follow your book here. Direcions: Make a able of values and skech he curve, indicaing he direcion of your graph. Then eliminae he parameer. (a) x and y Soluion: Firs make a able using various values of, including negaive numbers, posiive numbers and zero, and deermine he x and y values ha correspond o hese values. 0 x 5 5 y 0 Plo he ordered pairs ( x, y), drawing an arrow on he graph o indicae is direcion as increases. To eliminae he parameer, solve x subsiue x + for x + or x +. Then in place of in he equaion y o ge y x + Look a he graph of he parameric equaions o see if his equaion maches he graph, and observe ha i does The College Board.

7 Vecors (b) x, y + Soluion: Since x, we can use only nonnegaive values for x 0 y 5 0 To eliminae he parameer, solve x for x. Then subsiue x ino y s equaion so ha y x +. To make his equaion mach he graph, we mus resric x so ha i is greaer han or equal o 0. The soluion is y x +, x 0. (c) x and y, Soluion: Firs make a able using values ha lie beween and, and deermine he x and y values ha correspond o hese values. 0 x 7 y 0 To eliminae he parameer, solve y for o find ha y, y. Then subsiue y in place of in he oher equaion so ha x 4y. To make his 7 00 The College Board.

8 Vecors equaion mach he graph, we mus resric y so ha i lies beween and soluion is x 4y, y. (d) x + cos, y + sin. The 0 π π π π x 5 5 y 4 Soluion: To eliminae he parameer, solve for cos in x s equaion o ge x cos y and sin in y s equaion o ge sin +. Subsiue ino he ( x ) ( y + ) rigonomeric ideniy cos + sin o ge +. Discuss 4 9 wih he sudens he fac ha his is an ellipse cenered a he poin (, ) wih a horizonal axis of lengh 4 and a verical axis of lengh 6. Day Homework Make a able of values and skech he curve, indicaing he direcion of your graph. Then eliminae he parameer. Do no use your calculaor.. x + and y. x and y,. x and y 4. x and y 5. x and y 8 00 The College Board.

9 Vecors 6. x and y 7. x and y 8. x cos and y sin + 9. x sin and y cos + 0. x sec and y an Answers o Day Homework. x + and y 0 x 5 y 0 To eliminae he parameer, solve for x. Subsiue ino y s equaion o ge y x.. x and y, 0 x 0 4 y The College Board.

10 Vecors To eliminae he parameer, solve for x. Subsiue ino y s equaion o ge x y, x 4. (Noe: The resricion on x is needed for he graph of y 4 o mach he parameric graph.) x 4. x and y 0 x y 0 To eliminae he parameer, noice ha y. Subsiue ino x s equaion o ge x y. 4. x and y x 0 y 6 To eliminae he parameer, solve for x. Subsiue ino y s equaion o ge y x, 0. (Noe: The resricion on x is needed for he graph of y x o mach he parameric graph.) 0 00 The College Board.

11 Vecors 5. x and y x 7 y 0 To eliminae he parameer, solve for x +, x (since 0). Subsiue ino y s equaion o ge y x x and y 0 x y 0 To eliminae he parameer, solve for x x y or y. x. Subsiue ino y s equaion o ge 00 The College Board.

12 Vecors 7. x and y x y und. 4 4 To eliminae he parameer, noice ha x. Subsiue ino y s equaion o ge y. x 8. x cos and y sin + 0 π π π π x y 4 - To eliminae he parameer, solve for cos in x s equaion and sin in y s equaion. Subsiue ino he rigonomeric ideniy cos + sin o ge ( x + ) ( y ) The College Board.

13 Vecors 9. x sin and y cos + 0 π π π 4 π x y To eliminae he parameer, solve for cos in y s equaion and sin in x s equaion. Subsiue ino he rigonomeric ideniy cos + sin o ge ( x + ) 4 + y. 0. x sec and y an 0 π 4 π π 4 π 5π π 7π 4 4 π x und. und. y 0 und. 0 und. 0 To eliminae he parameer, subsiue ino he rigonomeric ideniy + an sec o ge + y x or x y. 00 The College Board.

14 Vecors Day : Parameric Equaions and Calculus If your exbook conains his maerial, you migh wan o follow your book here. Formulas o Know If a smooh curve C is given by he equaions x f() and y g(), hen he slope of C a he poin (x, y) is given by, where 0, and he second derivaive is given by d y d d. The reason ha, where 0, is he Chain Rule: Since y is a funcion of x, and x is a funcion of, he Chain Rule gives, hence, where 0. Applying his formula o and x raher han o y and d x, we have d y d. Or, applying he Chain Rule: Since we have d d d y, hence d y is a funcion of x, and x is a funcion of, d The College Board.

15 Vecors Example (no calculaor): Given he parameric equaions x and y, find and d y. Soluion: To find, we mus differeniae boh of he parameric equaions wih respec o. Since d d 6 and 6 6., hen d y To find d y d, we mus differeniae d 6 wih respec o and divide by : Example (no calculaor): Given he parameric equaions x 4cos and y sin, wrie an equaion of he angen line o he curve a he poin where π. 4 Soluion: d sin cos co d 4sin 4. 4cos π When ( ) x,, 4 and y Therefore he angen line equaion is: y x +. 4 (Remind sudens ha hey may leave heir angen line equaions in poin-slope form.) 5 00 The College Board.

16 Vecors Example (no calculaor): Find all poins of horizonal and verical angency given he parameric equaions x +, y + 5. Soluion: d + 5 d + +. A horizonal angen will occur when 0, which happens when 0 (and + 0 ), so so a a horizonal angen occurs a. Subsiuing equaions, we find ha a horizonal angen will occur a will occur when 5 4 ino he given, 4. A verical angen is undefined, which happens when + 0 (and 0), so a verical angen occurs a. Subsiuing ino he given equaions, we find ha a verical angen will occur a 7, 4 4. Example 4 (no calculaor): Se up an inegral expression for he arc lengh of he curve given by he parameric equaions x +, y 4, 0. Do no evaluae. Soluion: For parameric equaions x f() and y g(), a b, he formula for arc lengh is: s b a + For our problem,. d d + and 4, so he arc 4 lengh is given by he inegral expression s + ( ) The College Board.

17 Vecors Day Homework Do no use your calculaor on he following problems. On problems 5, find. x, y x +, y and d y.. x, y + 4. x ln, y + 5. x sin +, y 4cos 6. A curve C is defined by he parameric equaions x + y,. Find: (a) in erms of. (b) an equaion of he angen line o C a he poin where. 7. A curve C is defined by he parameric equaions x cos, y sin. Find: (a) in erms of. (b) an equaion of he angen line o C a he poin where π 4. On problems 8 0, find: (a) in erms of. (b) all poins of horizonal and verical angency. 8. x + 5, y 4 9. x +, y 0. x + cos, y + 4sin, 0 < π On problems, a curve C is defined by he parameric equaions given. For each problem, wrie an inegral expression ha represens he lengh of he arc of he curve over he given inerval.. x, y, 0. x e +, y, 7 00 The College Board.

18 Vecors Answers o Day Homework. Since d d and, hen To find d y, we mus differeniae wih respec o and divide by : d y d d +.. d d 6 and + 6 d d d y.. d d and d d + 4 d y The College Board.

19 Vecors d d ln + and + + d d d y d d cos 4sin sin and + cos 4sin 4 an cos d d 4 d y an 4 4 sec sec 4 sec cos 9 6. (a). + (b) When, 8, x 5 and y 4, so he angen line equaion is y 4 ( x 5 ) The College Board.

20 Vecors 7. (a) cos co. sin (b) When π 4, equaion is y x π co, x and y 4., so he angen line 8. (a) 4 wih 4 and. (b) A horizonal angen occurs when 0 and 0, so a horizonal angen occurs when 4 0, or a. When, x 7 and y 4, so a horizonal angen occurs a he poin (7, 4). A verical angen occurs when 0 and 0. Since 0, here is no poin of verical angency on his curve. 9. (a). (b) A horizonal angen occurs when 0 and 0, so a horizonal angen occurs when 0, or when ±. When, x and y, and when, x and y, so horizonal angens occur a he poins (, ) wih and (, ). A verical angen occurs when and 0 and 0 so a verical angen occurs when 0, or. When, x 4 y and, so a verical 8 angen occurs a he poin, The College Board.

21 Vecors 0. (a) 4cos co wih sin (b) A horizonal angen occurs when 4cos and sin. 0 and 0, so a horizonal angen π π occurs when 4cos 0, or a and. When π, x and y, and when π, x and y 5, so horizonal angens occur a he poins ( ), and, 5. A verical angen occurs when 0 and 0, so a verical angen occurs when sin 0, or when 0 and π. When 0, x 5 and y, and when π, x and y, so verical angens ( ) occur a he poins 5, and,.. s b a s b a + + e 4e The College Board.

22 Vecors Day : Review of Moion Along a Line My sudens su moion along a line early in he year, so his assignmen is a review for hem. I like o spend a day on moion along a line as a segue ino moion along a curve. For an excellen inroducion o moion along a line, see he Curriculum Module on moion by Dixie Ross a AP Cenral. (hp://apcenral.collegeboard.com/apc/public/ reposiory/ap_curricmodcalculusmoion.pdf) Day Homework The following problems are from old AP Exams and he sample muliple-choice problems in he Course Descripion, available a AP Cenral (hp://apcenral.collegeboard.com/apc/public/courses/eachers_corner/8.hml). Muliple-Choice Iems:. 00 AP Calculus AB Exam, Iem 5 (no calculaor): A paricle moves along he x-axis so ha a ime 0 is posiion is given by x 7 5. A wha ime is he paricle a res? + (A) only (B) only (C) 7 only (D) and 7 (E) and AP Calculus AB Exam, Iem 4 (no calculaor): The maximum acceleraion aained on he inerval 0 by he paricle whose + + velociy is given by v 4 is (A) 9 (B) (C) 4 (D) (E) The College Board.

23 Vecors. AP Calculus AB, sample muliple-choice Iem 9 (no calculaor): The posiion of a paricle moving along a line is given by s for For wha values of is he speed of he paricle increasing? (A) < < 4 only (B) > 4 only (C) > 5 only (D) 0 < < and > 5 (E) < < 4 and > AP Calculus AB Exam, Iem 76 (calculaor): A paricle moves along he x-axis so ha a any ime 0, is velociy is given by v 4. cos( 0. 9). Wha is he acceleraion of he paricle a ime 4? + (A).06 (B) (C).6 (D).84 (E) AP Calculus AB Exam, Iem 9 (calculaor): A paricle moves along he x-axis so ha a any ime > 0, is acceleraion is given by a ln ( + ). If he velociy of he paricle is a ime, hen he velociy of he paricle a ime is (A) 0.46 (B).609 (C).555 (D).886 (E) The College Board.

24 Vecors 6. AP Calculus AB, sample muliple-choice Iem 9 (calculaor): Two paricles sar a he origin and move along he x-axis. For 0 0, heir respecive posiion funcions are given by x sin and x e. For how many values of do he paricles have he same velociy? (A) None (B) One (C) Two (D) Three (E) Four 7. AP Calculus AB, sample muliple-choice Iem 5 (calculaor): A paricle ravels along a sraigh line wih a velociy of v e sin meers per second. Wha is he oal disance raveled by he paricle during he ime inerval 0 seconds? (A) 0.85 (B).850 (C).055 (D).6 (E) 7.05 Free-Response Quesions: AP Calculus AB Exam, FRQ (calculaor): A paricle moves along he y-axis so ha is velociy a ime 0 is given by v an e. A ime 0, he paricle is a y. (Noe: an x arcan x.) (a) Find he acceleraion of he paricle a ime. (b) Is he speed of he paricle increasing or decreasing a ime? Give a reason for your answer. (c) Find he ime 0 a which he paricle reaches is highes poin. Jusify your answer. (d) Find he posiion of he paricle a ime. Is he paricle moving oward he origin or away from he origin a ime? Jusify your answer The College Board.

25 Vecors AP Calculus AB/BC Exams, Iem 4 (no calculaor): (seconds) v() (fee per second) Rocke A has posiive velociy v afer being launched upward from an iniial heigh of 0 fee a ime 0 seconds. The velociy of he rocke is recorded for seleced values of over he inerval 0 80 seconds, as shown in he able above. (a) Find he average acceleraion of rocke A over he ime inerval 0 80 seconds. Indicae unis of measure. (b) Using correc unis, explain he meaning of 0 70 v in erms of he rocke s fligh. Use a midpoin Riemann sum wih subinervals of equal lengh o approximae 0 70 v. (c) Rocke B is launched upward wih an acceleraion of a fee per + second. A ime 0 seconds, he iniial heigh of he rocke is 0 fee, and he iniial velociy is fee per second. Which of he wo rockes is raveling faser a ime 80 seconds? Explain your answer. Answers o Day Homework. Since x'() ( 7 + ) 6( )( 4) 0 when and when 4, he answer is E.. Noe ha a() 6 +, so ha a'() when. Compuing he acceleraion a he criical number and a he endpoins of he inerval, we have a(0), a() 9, and a(). The maximum acceleraion is, so he answer is D.. Noe ha v() ( )( 5) and a() 48 ( 4). The speed is increasing on < < 4, where he velociy and he acceleraion are boh negaive, and also for > 5, where he velociy and he acceleraion are boh posiive, so he answer is E. 4. Since d cos.., he answer is C Since v ln. 46, he answer is E The College Board.

26 Vecors 6. Firs find d d sin cos e e and. Then graph y cos x and y e x in funcion mode wih an x-window of [0, 0] and a y- window of [, ]. The wo graphs inersec a hree poins, so he answer is D Disance v e sin. 6, so he answer is D (a) a() v'() 0. or 0.. (b) v() Since a() < 0, and v() < 0, he speed is increasing. (c) Noe ha v() 0 when an (e ). The only criical number for y is ln(an) Since v() > 0 for 0 < ln(an ) and v() < 0 for > ln(an), y() has an absolue maximum a (d) y + 0 v. 60 or. 6. Since v() < 0 and y() < 0, he paricle is moving away from he origin. 9. (a) Average acceleraion of rocke A is v 80 v f / sec (b) Since he velociy is posiive, 70 0 v represens he disance, in fee, raveled by rocke A from 0 seconds o 70 seconds. A midpoin Riemann sum is + + f. 0 v 0 v 40 v (c) Le v v B be he velociy of rocke B a ime. Then C B Since v ( 0) 6 + C B + B Hence, v ( 80) 50 > 49 v( 80) and Rocke B is v B raveling faser a ime 80 seconds., hen C 4 and 6 00 The College Board.

27 Vecors Day 4: Moion Along a Curve Vecors I give my sudens he following lis of erms and formulas o know. Parameric Equaions, Vecors, and Calculus Terms and Formulas o Know: If a smooh curve C is given by he equaions x f and y g, hen he slope of C a he poin (x, y) is given by where 0, and he second derivaive is given d by d y d. The derivaive also may be inerpreed as he slope of he angen line o he curve C, or as he slope of he pah of a paricle raveling along he curve C, or as he rae of change of y wih respec o x. The second derivaive d y respec o x. is he rae of change of he slope of he curve C wih x is he rae a which he x-coordinae is changing wih respec o or he velociy of he paricle in he horizonal direcion. y is he rae a which he y-coordinae is changing wih respec o or he velociy of he paricle in he verical direcion The College Board.

28 Vecors x, y is he posiion vecor a any ime. x, y is he velociy vecor a any ime. x, y is he acceleraion vecor a any ime. velociy vecor. b a + + is he speed of he paricle or he magniude (lengh) of he is he lengh of he arc (or arc lengh) of he curve from a o b or he disance raveled by he paricle from a o b. Mos exbooks do no conain he ypes of problems on vecors ha are found on he AP Exam, so I supplemen wih he examples and workshees below. Example (no calculaor): A paricle moves in he xy-plane so ha a any ime, he posiion of he paricle is given 4 by x 4, y. + (a) Find he velociy vecor when. Soluion: d d v, ( + ) ( ) +, 4 4 8, 4 v, (b) Find he acceleraion vecor when. Soluion: a a d 0, 6 d d d, + 8, , 6 Example (no calculaor): A paricle moves in he xy-plane so ha a any ime, 0, he posiion of he paricle is given by x, y. Find he magniude of he velociy vecor when The College Board.

29 Vecors Soluion: The magniude or lengh of he velociy vecor can be found by using he Pyhagorean Theorem, since he horizonal and verical componens make a righ riangle, wih he vecor iself as he hypoenuse. Therefore is lengh is given by: Magniude of velociy vecor + Figure 5 For our problem, d + + and d 6. + ( ) Magniude of velociy vecor Noice ha he formula for he magniude of he velociy vecor is he same as he formula for he speed of he vecor, which makes sense since speed is he magniude of velociy. Example (no calculaor): A paricle moves in he xy-plane so ha x 4cos and y sin, where 0 π. The pah of he paricle inersecs he x-axis wice. Wrie an expression ha represens he disance raveled by he paricle beween he wo x-inerceps. Do no evaluae. Soluion: The pah of he paricle inersecs he x-axis a he poins where he y-componen is equal o zero. Noe ha sin 0 when sin. For 0 π, his will occur when π and 6 5π 6. Since d 4cos 4 d sin and sin cos, 5π he disance raveled by he paricle is Disance 6 4sin cos. π 6 + ( ) 9 00 The College Board.

30 Vecors Day 4 Homework Use your calculaor on problems 0 and c only.. If x and y e, find.. If a paricle moves in he xy-plane so ha a any ime > 0, is posiion vecor is ( ), find is velociy vecor a ime. ln + 5,. A paricle moves in he xy-plane so ha a any ime, is coordinaes are given by 5 4 x and y. Find is acceleraion vecor a. 4. If a paricle moves in he xy-plane so ha a ime is posiion vecor is π sin,, find he velociy vecor a ime π. 5. A paricle moves on he curve y ln x so ha is x-componen has derivaive + for x 0.A ime 0, he paricle is a he poin (, 0). Find he posiion of he paricle a ime. 6. A paricle moves in he xy-plane in such a way ha is velociy vecor is +,. If he posiion vecor a 0 is 5, 0, find he posiion of he paricle a. 7. A paricle moves along he curve xy 0. If x and, wha is he value of 0. If x and, wha is he value of? 8. The posiion of a paricle moving in he xy-plane is given by he parameric equaions x and y For wha value(s) of is he paricle a res? 9. A curve C is defined by he parameric equaions x and y 5 +.Wrie he equaion of he line angen o he graph of C a he poin (8, 4). 0. A paricle moves in he xy-plane so ha he posiion of he paricle is given by + ( )( ) x 5 sin and y 8 cos. Find he velociy vecor a he ime when he paricle s horizonal posiion is x 5.. The posiion of a paricle a any ime 0 is given by x and y. (a) Find he magniude of he velociy vecor a ime The College Board.

31 Vecors (b) Find he oal disance raveled by he paricle from 0 o 5. (c) Find. Poin P x, y and for 0. + as a funcion of x. moves in he xy-plane in such a way ha and for 0. + (a) Find he coordinaes of P in erms of given ha, when, x ln and y 0. (b) Wrie an equaion expressing y in erms of x. (c) Find he average rae of change of y wih respec o x as varies from 0 o 4. (d) Find he insananeous rae of change of y wih respec o x when.. Consider he curve C given by he parameric equaions x cos and π π y + sin, for. (a) Find as a funcion of. (b) Find he equaion of he angen line a he poin where π 4. (c) The curve C inersecs he y-axis wice. Approximae he lengh of he curve beween he wo y-inerceps. Answers o Day 4 Homework. d e e e. d. v. v + 5,, v, so 4., 5, 6 a 4. v 4. d x d y, 0, 6, so a 0, 4. π π, cos v, 6 so cos π, π, π. 00 The College Board.

32 Vecors 5. x ( + ) + + C. x( 0) C so x Since x and y ln, Posiion, ln. 5 5 Or, since x + ( + ) and y 0 ln, Posi ion 5 5, ln. 6. x ( + ) + + C. x( 0) 5 so C 5 and x y + D. y( 0) 0 so D 0 and y Posiion vecor + + 5,. A, Posiion 9, ( + ) ( 9, 4). Or, since x and y 4, Posiion 0 7. When x, y 5. x + y so + Or find ha Then subsiuing ino x gives 0 4 so ha 8. x ( ) 8 ( ) ( + ) 0 y ( ) + 9 ( ) ( ) 0 when and. The paricle is a res when v 0, 0 so a res when. when and. 00 The College Board.

33 Vecors A ( ) 0, 8, 4 when 5 Tangen line equaion: y + 4 x sin 5 when v v, 5 + cos, + cos + 8 sin , (a) Magniude when 5 is 600 or 0 6 (b) 5 5 Disance ( + ) (c) x The College Board.

34 Vecors. (a) x + x ( + ) y + D. When, y 0 so D. y ( x, y) ln ( + ), ln + C. When, x ln so C 0. + ln (b) e x e x y e x + so and e x e x. y ( b ) y( a) (c) Average rae of change y 4 y( 0) x b x a 5 ( ) x 4 x ln ln ln5 (d) Insananeous rae of change. (a) cos co sin (b) y + x 4 + (c) x 0 when , ,,, lengh sin cos. 756 or The College Board.

35 Vecors Day 5: Moion Along a Curve Vecors (coninued) Example (calculaor): ( ) An objec moving along a curve in he xy-plane has posiion x, y a ime wih sin ( ), cos ( ). A ime, he objec is a he posiion (, 4). (a) Find he acceleraion vecor for he paricle a. (b) Wrie he equaion of he angen line o he curve a he poin where. (c) Find he speed of he vecor a. (d) Find he posiion of he paricle a ime. Soluion: (a) Sudens should use heir calculaors o numerically differeniae boh and when o ge a. 476,. 07. (b) When, cos4 or sin 0. 66, so he angen line equaion is 8 cos4 y 4 ( x ) or y ( x ). sin8 Noice ha i is fine o leave he slope as he exac value or o wrie i as a decimal correc o hree decimal places. (c) Speed + + or sin8 cos Noice ha i is fine o leave he speed as he exac value or o wrie i as a decimal correc o hree decimal places. (d) Sudens should apply he Fundamenal Theorem of Calculus o find he x and y componens of he posiion. sin 4 cos x x x y y y Therefore he posiion a ime is (0.78, 4.44) The College Board.

36 Vecors Day 5 Homework Use your calculaor on problems 7 only.. If x e and y sin ( ), find in erms of.. Wrie an inegral expression o represen he lengh of he pah described by he parameric equaions x cos y sin π and for 0.. For wha value(s) of does he curve given by he parameric equaions x and y have a verical angen? 4. For any ime 0, if he posiion of a paricle in he xy-plane is given by x + and y ln +, find he acceleraion vecor. 5. Find he equaion of he angen line o he curve given by he parameric equaions x 4 + and y 4 a he poin on he curve where If x e and y e are he equaions of he pah of a paricle moving in he xy-plane, wrie an equaion for he pah of he paricle in erms of x and y. 7. A paricle moves in he xy-plane so ha is posiion a any ime is given by x cos ( 5 ) and y. Wha is he speed of he paricle when? 8. The posiion of a paricle a ime 0 is given by he parameric equaions x ( ) + y ( ) 4 and (a) Find he magniude of he velociy vecor a. (b) Find he oal disance raveled by he paricle from 0 o. (c) When is he paricle a res? Wha is is posiion a ha ime? ( ) 9. An objec moving along a curve in he xy-plane has posiion x, y a ime 0 wih + an ( ) and e. Find he acceleraion vecor and he speed of he objec when A paricle moves in he xy-plane so ha he posiion of he paricle is given by + + x cos and y sin, 0 π. Find he velociy vecor when he paricle s verical posiion is y The College Board.

37 Vecors ( ). An objec moving along a curve in he xy-plane has posiion x, y a ime wih sin( ) and cos ( ) for 0 4. A ime, he objec is a he posiion (, 4). (a) Wrie an equaion for he line angen o he curve a (, 4). (b) Find he speed of he objec a ime. (c) Find he oal disance raveled by he objec over he ime inerval 0. (d) Find he posiion of he objec a ime. Answers o Day 5 Homework. cos( ) e π 4. Lengh 9 cos sin + 4 sin cos is undefined when 0. So he curve given by he parameric equaions x and y has a verical angen when 0 and v,, a, When, x, y. 6 Tangen line equaion: y + x 6. e x so e x x +. Then y e so y x 4x Speed ( 5sin( 5 )) + ( ) The College Board.

38 Vecors + ( ) 4 8. (a) Magniude ( ) (b) Disance ( ) (c) The paricle is a res when v, 4 0, 0, so is a res when. Posiion (4, 0).,., speed + 9. a sin 5 when v , ( a) cos( ) sin 0. + Tangen line equaion: y 4 0. x (b) Speed 4sin an e ( ) + cos ( ).084 (c) Disance 4sin cos ( d) x sin. 46, y 4 cos (.46,.557) so posiion 8 00 The College Board.

39 Vecors Day 6: Moion Along a Curve Vecors (coninued) I don work any examples on Day 6. The sudens usually need a lile more pracice on vecors, bu no new maerial is covered in he Day 6 homework. Day 6 Homework Use your calculaor only on problems 7.. The posiion of a paricle a any ime 0 is given by x, y. (a) Find he magniude of he velociy vecor a. (b) Se up an inegral expression o find he oal disance raveled by he paricle from 0 o 4. (c) Find as a funcion of x. (d) A wha ime is he paricle on he y-axis? Find he acceleraion vecor a his ime. ( ). An objec moving along a curve in he xy-plane has posiion x, y a ime wih he velociy vecor v +,. A ime, he objec is a (ln, 4). (a) Find he posiion vecor. (b) Wrie an equaion for he line angen o he curve when. (c) Find he magniude of he velociy vecor when. (d) A wha ime > 0 does he line angen o he paricle a x, y have a slope of? ( ). A paricle moving along a curve in he xy-plane has posiion x, y, wih x + sin and y + cos, where 0 0. Find he velociy vecor a he ime when he paricle s verical posiion is y 7. ( ) 4. A paricle moving along a curve in he xy-plane has posiion x, y a ime wih + sin ( ). The derivaive is no explicily given. For any ime, 0, he line angen o he curve a x, y has a slope of +. Find he acceleraion vecor of he objec a ime The College Board.

40 Vecors ( ) 5. An objec moving along a curve in he xy-plane has posiion x, y a ime wih cos( e ) and sin ( e ) for 0. A ime, he objec is a he poin (, ). (a) Find he equaion of he angen line o he curve a he poin where. (b) Find he speed of he objec a. (c) Find he oal disance raveled by he objec over he ime inerval 0. (d) Find he posiion of he objec a ime. ( ) 6. A paricle moving along a curve in he xy-plane has posiion x, y a ime wih sin ( ) ( ) cos and. A ime, he paricle is a he poin (, 4). (a) Find he acceleraion vecor for he paricle a. (b) Find he equaion of he angen line o he curve a he poin where. (c) Find he magniude of he velociy vecor a. (d) Find he posiion of he paricle a ime. ( ) 7. An objec moving along a curve in he xy-plane has posiion x, y a ime wih + sin ( e ). The derivaive is no explicily given. A, he objec is a he poin (4, 5). (a) Find he y-coordinae of he posiion a ime. (b) A ime, he value of is. 8. Find he value of when. (c) Find he speed of he objec a ime The College Board.

41 Vecors Answers o Day 6 Homework (a) Magniude (b) Disance (c) x + (d) Paricle is on he y-axis when + 4 5, and. (a) x ln + C. Since x ln, C 0. + y 4, D. Since y 4, D. Posiion vecor ln( +), + (b) When, 4 so he angen line equaion is y 4 4( x ln ). (c) Magniude (d) ( + ) when + 0 so +. + cos 7 when v( ) , a ( + ) + sin so. 746, (a) When, sin( e ) cos e so he angen line equaion is y x ( ( e )) + ( ( e )) (b) Speed cos sin ( ) (c) Disance cos e sin e ( d) x cos e., y sin e d. 676 so posiion (.896,.676) 4 00 The College Board.

42 Vecors 6. (a) a. 09,. 545 (b) cos( ) sin so he angen line equaion is y x ( ) + ( ( )) ( ) ( ) (c) Magniude sin cos ( d) x sin., y cos so he posiion (0.9, 4.00) + 7. (a) y 5 sin e.69 (b) (c) Speed e e + sin + sin. 8 so. 8 + sin( e ). 8 + ( + sin( e )) The College Board.

43 Vecors Abou he Auhor Nancy Sephenson eaches a Clemens High School in Sugar Land, Texas. She was a member of he AP Calculus Developmen Commiee from 999 o 00 and is a College Board consulan The College Board.

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