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...Calculus iii practice final exam solutions spring let c be the curve described by vector function r t hsin cos i a find and hcos sin h b tangent to at point corresponds value is corresponding sor parametric equations of line parallel passes through therefore x y z orx l given show that lies on both we must there for which clearly satises all three so equation plane containing nd need v perpendicular as shown in part can take cross product direction vectors coecients are components contains or dz suppose f g if dt zdx zdy chain rule says xdt ycos p denote directional derivative hf fi ysin unit u duf increases most rapidly zz evaluate da wheredisthetriangularregionwithvertices d left boundary right express double integral an iterated inte gral integrating rst with respect dx dy yx xy yz potential toshowthatf isapotentialforf wemustshowthat sof dr where helix starting terminal fundamental theorem integrals zzz zdv e region between spheres use spherical coordinates dv forming boundaries sp...