MAT235Y1: Multivariable Calculus
Practice Final Exam – Fall 2024/Winter 2025
1. (10 points) (Multiple choice) For each part, write either A, B, C, D or E on the indicated line. Only your final answer will be graded for this question. Each part is worth 2 marks.
(i) Let S be the surface parametrized by Which of the following is a unit normal vector to S at the point (1, 1, 0)?
(ii) Let f(x, y) = x2 - xy2 + y2 . Which of the following points is a saddle point of f?
A (0, 0) B (1, -1) C (-1, 1) D (-1, 0) E None of the above
(iii) Which, if any, of the following limits is wrong?
(iv) Which of the following vector fields is not a curl vector field?
(v) Consider the graph of the given vector field (x, y, z), where only the slice z = 0 is shown. Assuming has no -component and is independent of z, which of the following statements best describes the divergence of the vector field at the points (1 , 0, 0) and (1/2, 1/2, 0)?
A div (1, 0, 0) > 0 and div (1/2, 1/2, 0) = 0
B div (1, 0, 0) > 0 and div (1/2, 1/2, 0) < 0
C div (1, 0, 0) < 0 and div (1/2, 1/2, 0) < 0
D div (1, 0, 0) > 0 and div (1/2, 1/2, 0) = 0
E None of the above statements describe the divergence at the given points
2. (4 points) The following plot shows the level sets of a function
f(x, y) = k for k = -5, -3, -1, 1, 3, 5, 7.
Sketch ▽f evaluated at each of the labelled points A, B, C, D.
3. (4 points) Consider the graph of the following vector fields on R3 labelled (a), (b) and (c), which shows the slice of each vector field where z = 0 and where —1 ≤ x ≤ 1, —1 ≤ y ≤ 1.
(You may assume that each vector field has no -component and is independent of z.)
(a) (2 points) Which vector field(s) above, if any, satisfy curl (1/2, 1/2, 0) · > 0?
(b) (2 points) Which vector field(s) above, if any, could be gradient vector fields? Give a brief justification.
4. (12 points) For each part, write your final answer on the indicated line. Only your final answer will be graded for this question. Each part is worth 3 marks.
(i) Let z = f(x, y) where x = g(t), y = h(t) and f, g, h are differentiable functions. Given the information in the following table, evaluate when t = 1.
(ii) Let f(x, y) = sin(x2y). Find the second-order Taylor approximation of f(x, y) at the point (x, y) = (0, 1).
(iii) Write down (but do not evaluate) an iterated integral which gives the volume of the solid bounded by the planes z = 0, z = 4, the cylinder x2 + y - 1 = 0, and the cylinder x2 - y - 1 = 0.
(iv) Find the area vector of the surface given by y = -10, 0 ≤ x ≤ 5, 0 ≤ z ≤ 3, oriented away from the xz-plane.
5. (7 points) Let
Find the set of all points where f(x, y) is continuous. Justify your answer.
6. (9 points) Use Lagrange multipliers to find the point (x, y, z) on the sphere of radius 2 centered at (0, 0, 0) in the first octant such that the value of x2yz is maximized. What is the value of the maximum? (Note: You may assume that the maximum exists.)
7. (7 points) Evaluate where E is the solid bounded by the cylinder x2 + z2 = 1 and the plane x + y = 1 in the first octant.
8. (8 points) Find the surface area of the part of the paraboloid z = 1-2x2 -3y2 that lies inside the cylinder 16x2 + 36y2 = 1.
9. (9 points) Evaluate the flux integral
where (x, y, z) = ⟨1, 1, 1⟩, S is the part of the sphere x2 + y2 + z2 = 4 in the first octant, and where S is oriented outward.
10. (10 points) Let C be the curve of intersection of the cylinder x2 +y2 = 1 and the plane y+z = 2. Define the vector field by
(x, y, z) = ⟨sin x + yz,—yx, x3 + ey〉
Evaluate the line integral assuming C is oriented counterclockwise when viewed from above.
11. (10 points) Evaluate the flux integral
where S is the part of the sphere x2 + y2 + z2 = 2 inside the cone and where S is oriented outward.
12. (10 points) Let S be the boundary of the solid that is enclosed by the surfaces z = 1 — y2 ,y = √x, z = 0, and x = 0. Use the Divergence Theorem to evaluate , where
(x, y, z) = ⟨z sin(z)ey + 2x,—y + sin(ex), x + y〉
and where S is oriented outward.