HW3 Solutions13.3 Arc Length and the Unit Tangent Vector T8. T=v|v|= (cost)i-(sint)jand Length =R2√2tdt=ht22i2√210.Let P(t0) denote the point. Thenv= (12 cost)i+ (12 sint)j+ 5kand-13π=Rt00p144 cos2t+ 144 sin2t+ 25t dt=Rt0013dt= 13t0⇒t0=-π, and the point isP(-π) = (12 sin(-π),-12 cos(-π),-5π) = (0,12,-5π)14.r(1 + 2t)i+ (1 + 3t)j+ (6-6t)k⇒v= 2i+ 3j-6k⇒ |v|=p22+ 33+ (-6)2= 7⇒s(t) =R607dτ= 7t⇒Length =s(0)-s(-1) = 0-(-7) = 714.1 Functions of Several Variables8.(a)Domain: all (x, y) satisfyingx2+y2≤9(b)Range: 0≤z≤3(c)level curves are circles centered at the origin with radiir≤3(d)boundary is the cirlex2+y2= 9(e)closed(f)bounded13.f14.e15.a16.c17.d18.b