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Calculus 3: midterm

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  1. Find all points on the curve \[x=\cos3t,y=2\sin t\] where the tangent is either horizontal or vertical.
  2. Determine whether these points lie on a straight line: \[A(0,-5,5),B(1,-2,4),C(3,4,2).\]
  3. Sketch the parametric curve \[x=\frac{1}{t}\cos t,y=\frac{1}{t}\sin t,t>0.\]
  4. Find the unit tangent vector of the curve \[r(t)=t\mathbf{i}+t\mathbf{j}+(1+t^{2})\mathbf{k}\] at the point $<0,0,1>.$
  5. Find the plane through the point $(-1,6,-5)$ and parallel to the vectors $a=<1,1,0>$ and $b=<0,1,1>.$
  6. Show that the limit doesn't exist: \[\lim_{(x,y)\rightarrow(0,0)}\frac{xy}{x^{2}+y^{2}}.\]
  7. Find the linearization of the function \[f(x,y)=e^{xy}\] at $(0,1).$