Final test review
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Part I (Multiple Choice)
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1. Which of the following is an equation for the plane
that contains the point P and is orthogonal
to the vector V (or a line x = Q + tV)

WW. 10.5, 10, 11, 12


2. Which of the following is an equation for the tangent line of r(t) when t = a?

WW. 10.7, 9,10


3. Find the arc length of the curve

WW. 10.8 1-4

4. The Chain Rule

WW. 11.5, 2 (only find z_s)

5. Find the directional derivative of the function f(x,y)
at the point (a,b) in the direction v.

WW. 11.6 3, 4 , 8

6. Find critical points for f(x,y) and classify them.
  (See stingray example in lectures)

  WW. 11.7 , 6 (stingray)

7. Switch the order of integration for the double integral ...

  WW. 12.2, 6, 7, 8, 9 

8. Evaluate a triple integral ... over the region bounded by z = f(x,y) and the plane z = c.

WW. 12.5, 4

9. Use Green’s Theorem to evaluate the line integral ...

WW. 13.4, 1, 2, 3, 4   


10. Calculate curl of a vector field

 WW. 13.5, 1, 2, 3

11. Parametrization of a surface

 WW. 13.6, 1  (pay attention to a parametrization of a cone)


12. The surface area of the parametric surface r(u,v), a < u < b, c < v < d is ...

WW. 13.6, 6, 7


PART II (Free Response)
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1. Find the equation of the tangent plane for the function at a given point

WW. 11.4, 1, 2 


2. Convert the following triple integral to spherical coordinates and evaluate it.

WW. 12.7, 12


3. Find the flux of the vector field "F" uptward through the surface "S" 

WW. 13.7 2,3




4. Evaluate the surface integral of the first type.

WW. 13.7, 1, 8