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Math 206A Final Exam page 1 Thursday 12/16/2010 Name. ——_——_> 1. An object moving around an ellipse has position at time t € [0, 27] given f(t) = (6sin(t), —4cos(t)). (1A.) In terms of x and y, what is the equation of that ellipse parameterized by f? AL t <0, h, 1,3, m Aiuhi at vel, hi by shag ah gota fe ban (6.0 b, “Teas t) lo (0) 64 AF 60), Sophy ‘ Satoh lui. C67 = Bes me A. cMpei e+e 2/ {Yo an (1B.) In vector form, what is the equation of the line tangent to this ellipse at t = 1/4? He poirt on he ele 6 tI) >(0%%, ~1%) 3 : Ae —s To = ae = th ae “gene $8) eres “eR bs (FEB) re sh be ie)~ (6%, -1%) + (6%, "% ) (netsh) ¢ (sealer) (velred, exh) (1C.) Find and solve the equation that gives all t € [0, 27] for which the velocity vector is parallel to the vector v = (1,1). So Wwe jot he velochy vechr fo have He frm K(i)) (we Aad MOF Gaug,* ant fF =(y)" fo we ste Ffe)= Ku) co 10, ache Gye LE iesroentee (seat, Ymre ) 2 (®K) co £= 0.9828 4 7 (i th xpd tr shh j er t €[0,%) pot £62 coh > bast = tort pau 2S, ant _] DE Oe, 9 => ¢ ae qe L= 0,962 BA OZ 5 41 F 41299 (1D.) Find and simplify the function that gives the speed of the object at time ¢. (Recall, this is just the length of the velocity vector). Hh te Viberd £ 4 , Sia Greet (6) = || elect (Ol 5 ed 4 / h [o,") guild)= (Fa fae ~ [Sofa tile & rumba of Vays i ih ts nae 4. 2/5cs't +4 / 1E.) Use (1D) and your math 105 skills to find all t € [0, 27] at which the speed it maximized. Then identify the positions (1E.) Use ( y 5 P Ps on the ellipse where this maximum speed is obtained. : , ws 10 fphadua P cally crbh ph, the ide. Lhe fy * max mh Ha ped Be “ay : . lett = + (2ebaa euler YR (r2fadend +82 (inh) Ueaeaa ar of” (splat ~'ofsitlixe) ho, (pall =O DE = G4, GH (lnc H mb OO ) Oy hf por & dent £70, 7, or te Mane; careypond, Tukes om te Myje me (0-4) OY € (2-4) 
