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Math 281 Practice Final

Math 281 Practice Final. 1. If we are looking at the axes in this direction, which octant will the point (3, -2, -4) belong to?. z. y. x. Answer:. Front, left and below. 2. What is the angle (in degrees) between the two vectors 2,-1,-2 and -3,1,-1 ?. Answer:. 120.16679 °.

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Math 281 Practice Final

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  1. Math 281Practice Final

  2. 1.If we are looking at the axes in this direction, which octant will the point (3, -2, -4) belong to? z y x Answer: Front, left and below.

  3. 2. What is the angle (in degrees) between the two vectors 2,-1,-2 and -3,1,-1 ? Answer: 120.16679°

  4. 3. What is the area of the triangle with vertices (2, 0, 0), (0, 4, 0) and (0,0,1) ? Answer:

  5. 4. Find an equation for the plane with intercepts (-2, 0 ,0), (0, 5, 0) and (0, 0, -3). Answer:

  6. 5. Find an equation for the line passing through the points (3, -7, 6) and (-1, -5, 2). Answer:

  7. 6. Find an equation for the plane perpendicular to the line and containing the point (2, 1, 1). Answer:

  8. 7. Find the following limit Answer: 1

  9. 8. Compute fy if Answer:

  10. 9. Find the only point at which the tangent plane to the surface z = x2 + 2xy +2y2 – 6x + 8y is horizontal. Answer: (10, -7)

  11. 10. Find an equation of the tangent plane to the surface z = x2 – 4y2 at the point (5, 2, 9). Answer:

  12. in terms of u and v if 11. Find and x = cos 2u, y = sin2v Answer:

  13. 12. True or false. Let f(x,y) be a function of two variables. If all directional derivatives of f exists at (0,0), then f is differentiable at (0,0). False Answer:

  14. 13. If f (x, y) = 3x2 – 5y2 , find the direction (in terms of degrees from the positive x-axis) where the slope is greatest at the point P(2, -3). 68.1986º Answer:

  15. and 14. If Can f be differentiable at (0,0)? Why? No, because otherwise Answer:

  16. 15. Reverse the order of integration in the following integral (but you don’t have to integrate it.) Answer:

  17. Write down a double integral for surface area of the saddle-shaped surfacez = xybounded inside the cylinder x2 + y2 = 1. Answer:

  18. Convert the following triple integral from rectangular to spherical coordinates. (Do not evaluate.) Answer:

  19. 18. Find the Jacobian for the transformation u = xy, v = xy1.4 Express your answer in terms of u and v. Answer:

  20. 19. Find the curl of the vector field defined by F(x, y, z) = y, sinx, 0 At which points will the field be irrotational? Answer: 0, 0, cosx−1, this becomes 0 if x = 2nπ

  21. 4 y 2 0 -4 -2 0 2 4 x -2 -4 20. Is the following line integral independent of path? Why? where C is the green curve below from left to right. Answer: No, because you gain energy by going up first then right, but you lose energy going right first then up.

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