Question

1 a) Find the domain of r(t) = (2-Int ) and the value of r(to) for to = 1. b) Sketch (neatly) the line segment represented by the vector equation: r=2 i+tj; -1 <t<l. c) Show that the graph of r(t) = tsin(t) i + tcos(t) j + t?k, t> 0 lies on the paraboloid: z= x2 + y². 2. a) Find r'(t) where r(t) = eti - 2cos(31) j b) Find the parametric equation of the line tangent to the graph of r(t) = e?ti - 2cos(3) j, to = 0. c) Evaluate S. [tº i + tại]dt. 3. a) Find the arc length of the parametric V U OSIUT. v - 3sin(t). 2-4t: 0< <T

Answer 1

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Scabion 192: Given: Fly) (H 2 - ME) obiously x-component is well defined for all reapt except 04:2, Domain of x-componenl: te (-00,2), (2,00). . & y-component is well defined for t so ». ... Domain of y, component= 10%). So: Domain of r(t)= common Domain to both components 2. (0, 2)u(2,00). And i Alle Now: rito) - r()= (1+2, -110y = 43,0). Am. A(6), vector Eqn-g the line signent. rict)Žintit;^= 92,61 . ; -15ts 1 =) kritial point: N(+1)=(2,-1) & final (Terminal) Point: vºl1 = (2,). I love the -3 -2 1 01 X \ (c): Giren r()= (tsent, teast, ty N Y X(t)= tSeit, ylt)= t cost, H16)=42 .; x+y=+4lin't+ cos 7) = = = Thus, graph of F(t) lies on the parasolos of