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1 Answers o Homework. x + and y x 5 y To eliminae he parameer, solve for x. Subsiue ino y s equaion o ge y x.. x and y, x y

2 x To eliminae he parameer, solve for. Subsiue ino y s equaion o ge x y, x. (Noe: The resricion on x is needed for he graph of y o mach he parameric graph.) x. x and y x y To eliminae he parameer, noice ha y. Subsiue ino x s equaion o ge x y.. x and y 9 x y 6 To eliminae he parameer, solve for x. Subsiue ino y s equaion o ge y x,. (Noe: The resricion on x is needed for he graph of y x o mach he parameric graph.)

3 5. x and y 9 x 7 y To eliminae he parameer, solve for x +, x (since ). Subsiue ino y s equaion o ge y x x and y x 6 y To eliminae he parameer, solve for x x y or y. x. Subsiue ino y s equaion o ge

4 7. x and y x y und. To eliminae he parameer, noice ha x. Subsiue ino y s equaion o ge y. x 8. x cos and y sin + π π π π x y - 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 ) +. 9

5 9. x sin and y cos + π π π 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 + ) + y.. x sec and y an π π π π 5π π 7π π x und. und. y und. und. To eliminae he parameer, subsiue ino he rigonomeric ideniy + an sec o ge + y x or x y.

6 Answers o 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 + d y

7 d d.. + ln + and + + d d d y d d cos sin sin and + cos sin an cos d d d y an sec sec sec cos 9 6. (a). + (b) When, 8, x 5 and y, so he angen line equaion is y ( x 5 ). 5

8 7. (a) cos co. sin (b) When π, equaion is y x π co, x and y., so he angen line 8. (a) wih and. (b) A horizonal angen occurs when and, so a horizonal angen occurs when, or a. When, x 7 and y, so a horizonal angen occurs a he poin (7, ). A verical angen occurs when and. Since, here is no poin of verical angency on his curve. 9. (a). (b) A horizonal angen occurs when and, so a horizonal angen occurs when, 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 and so a verical angen occurs when, or. When, x y and, so a verical 8 angen occurs a he poin, 8.

9 . (a) cos co wih sin (b) A horizonal angen occurs when cos and sin. and, so a horizonal angen π π occurs when cos, 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 and, so a verical angen occurs when sin, or when and π. When, 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 e + 9.

10 Answers o Homework. Since x'() ( 7 + ) 6( )( ) when and when, he answer is E.. Noe ha a() 6 +, so ha a'() 6 6 when. Compuing he acceleraion a he criical number and a he endpoins of he inerval, we have a(), a() 9, and a(). The maximum acceleraion is, so he answer is D.. Noe ha v() ( )( 5) and a() 8 ( ). The speed is increasing on < <, 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.. Since d 9 6. cos.., he answer is C Since v ln. 6, he answer is E.

11 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 [, ] and a y- window of [, ]. The wo graphs inersec a hree poins, so he answer is D. 7. Disance v e sin. 6, so he answer is D. 8. (a) a() v'(). or.. (b) v().6. Since a() <, and v() <, he speed is increasing. (c) Noe ha v() when an (e ). The only criical number for y is ln(an).. Since v() > for < ln(an ) and v() < for > ln(an), y() has an absolue maximum a.. (d) y + v. 6 or. 6. Since v() < and y() <, he paricle is moving away from he origin. 9. (a) Average acceleraion of rocke A is v 8 v f / sec. 8 (b) Since he velociy is posiive, 7 v represens he disance, in fee, raveled by rocke A from seconds o 7 seconds. A midpoin Riemann sum is + + f. v v v (c) Le v v B be he velociy of rocke B a ime. Then C B Since v 6 + C B + B 6 +. Hence, v ( 8) 5 > 9 v( 8) and Rocke B is v B raveling faser a ime 8 seconds., hen C and

12 Answers o Homework. d e e e. d. v. v + 5,, v, so., 5, 6 a. v. d x d y,, 6, so a,. π π, cos v, 6 so cos π, π, π.

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

14 A ( ), 8, when 5 Tangen line equaion: y + x sin 5 when v, 5 + cos, + cos + ( 8 ) sin v , (a) Magniude when 5 is 6 or 6 (b) 5 5 Disance ( + ) (c) 6 x + 5

15 . (a) x + x ( + ) y + D. When, y so D. y ( x, y) ln ( + ), ln + C. When, x ln so C. + 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 y( ) x b x a 5 ( ) x x 5 6 ln ln ln5 (d) Insananeous rae of change. (a) cos co sin (b) y + x + (c) x when , ,,, lengh sin cos. 756 or

16 Answers o Day 5 Homework. cos( ) e π. Lengh 9 cos sin + sin cos. + 8 is undefined when. So he curve given by he parameric equaions x and y + 8 has a verical angen when 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 x Speed ( 5sin( 5 )) + ( ).

17 + ( ) 8. (a) Magniude + ( ) (b) Disance 5.86 ( ) (c) The paricle is a res when v,,, so is a res when. Posiion (, ).,., speed + 9. a sin 5 when v ,. 9 ( a) cos( ) sin. + Tangen line equaion: y. x (b) Speed sin an e ( ) + cos ( ).8 (c) Disance sin cos ( d) x sin. 6, y cos (.6,.557) so posiion

18 Answers o 6 Homework +. (a) Magniude (b) Disance + (c) x + (d) Paricle is on he y-axis when + 5, and. (a) x ln + C. Since x ln, C. + y, D. Since y, D. Posiion vecor ln( +), + (b) When, so he angen line equaion is y ( x ln ). (c) Magniude (d) ( + ) when + so +. + cos 7 when v( ). 968, a ( + ) + sin so. 76, (a) When, sin( e ) cos e. 5 so he angen line equaion is y. 5 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)

19 6. (a) a. 9,. 55 (b) cos( ) sin. 68 so he angen line equaion is y. 68 x ( ) + ( ( )) ( ) 9 ( ) (c) Magniude sin cos ( d) x sin., y cos. so he posiion (.9,.) + 7. (a) y 5 sin e.69 (b) (c) Speed e e + sin + sin. 8 so. 8 + sin e. 8 + ( + sin( e ))

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