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MATH 237 Online Calculus 3 for Honours Mathematics
Spring 2024
Practice exam
1.(20 points)
(i) (10 points) Determine the limit of the following function at (0, 0):
(ii) (10 points) Determine the differentiability of the following function at p0, 0q:
2.(25 points)
(i) (10 points) If z = f(x, y), y = g(x), find dx/dz.
(ii) (15 points) Find the first- and second-degree Taylor polynomials for the following function at the given point:
f(x, yq = px + y)sin(x - y), at (π, π).
3.(25 points)
(i) (10 points) Find and classify the critical points of the function f(x, y) = xyex+2y.
(ii) (15 points) Find the maximum and minimum of the function f(x, y) = x3 - 3x + y2 + 2y on the region bounded by the lines x = 0, y = 0, x + y = 1.
4.(25 points)
(i) (10 points) Let 
be a region in R3. Give descrip-tions of the region in spherical coordinates and cylindrical coordinates.
(ii) (15 points) Evaluate
where D is the region bounded by x + y + z = 2, z = 2, x = 1 and y = x.
5.(5-10 points) Miscellaneous problem. This problem will be proof-based. Similar to the last problem in Written Assignment and Midterm. There will be bonus points in the final.