MULTIDIMENSIONAL MATHEMATICS
MATH171
Practice for Final Exam
1. Find and sketch the domain of the function
2. Eliminate the parameter to find a Cartesian equation of the curve. Sketch the curve and
indicate the direction in which the curve is traced as the parameter increases
3. Sketch the curve in polar coordinates
4. Evaluate the expression and write it in the form a+bi.
5. Find a unit vector that has the same direction as
6. Find
if
and
7. Find the scalar and vector projections of
8. Find
if
and
.
onto
.
9. Find parametric equation of the line segment through
and
and perpendicular to both
10. Find an equation of the plane that passes through the point (6,0,-2) and contains the
line
.
11. Sketch the solid of revolution obtained by rotating the region bounded by the given
curves about the specified axis. Sketch its section by a plane perpendicular to the axis of
rotation and indicate how the size of the section depends on the position of the plane.
, 1 ≤ x ≤ 2, x = 2, y = 0 about the y-axis.
12. Sketch the curve with the given vector equation r(t) = t i − cos t j + sin t k
13. Use traces to sketch and identify the surface
.
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14. Reduce the equation to one of the standard forms, classify the surface and sketch it
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15. Draw a contour map of the function
.
16. Find a parametric representation for the part of the surface
cylinder
17. For the matrices
and
inside the
compute BT and all
possible products of A, B and BT
18. Find all solutions to the following system of equations by row operations. What is the
geometric meaning of the solution(s)?
19. Compute the determinant of the following matrix
20. Compute the inverse of the following matrix
⎛ 1 2⎞
⎟
21. Find eigenvalues and eigenvectors of the matrix ⎜
⎝ −1 3⎠
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