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Comprehensive Practice Final for Advanced Mathematics: Multivariable Calculus and Vectors

This practice final covers a variety of topics in advanced mathematics, focusing on multivariable calculus. Problems include determining the octant of a point, finding angles between vectors, calculating areas of triangles, and deriving equations for planes and lines. Additionally, the final explores concepts such as tangent planes, directional derivatives, and surface areas, along with transformations and vector fields. This resource aims to reinforce understanding and application of mathematical theories and principles.

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Comprehensive Practice Final for Advanced Mathematics: Multivariable Calculus and Vectors

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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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