Question 1:

(a) Suppose that u, v, w and x are coplanar unit vectors as in the diagram below.

List the quantities u u, u x, u w and u v in increasing order. You cannot assume anything about actual values of the angles between vectors except that the angle (measured counterclockwise) from u to w is less than radians. Explain your answer.

(b) If three of the vertices of a parallelogram are (0; 0; 1), (0; 2; 0) and (3; 0; 0), what is the area of the parallelogram?

Question 2. Suppose that f(x; y) is a function of two variables with continuous first and second partial derivatives whose quadratic approximation about the point (2; 1) is Q(x; y) = 4 8(y  1)  (x  2)2 + 8(x  2)(y  1):

(a) Complete the following table
(b) Write an equation for the tangent plane to the the graph of f(x; y) at the point (2; 1).
(c) Use a linear approximation to estimate the value of f(2:2; 0:95).

Question 3. Let f(x; y) = p 81  x2  y2.

(a) Sketch the level curve f(x; y) = 0 on the axes below
(b) Find an equation for the graph of f(x; y) in terms of cylindrical coordinates.

Question 4. Evil Person Being is climbing to the top of Mount Pineapple in an e ort to spy on the triangle beings during their vacation in Triangle Archipelago. The height of Mount Pineapple is given by the function h(x; y) = 200  3x2  3y2 + 6xy.

(a) While resting at the point (1; 2), Evil Person Being sees the triangle beings having a picnic at
the point (3; 2). What is the initial rate of change of the altitude if Evil Person Being walks directly towards the triangle beings?

(b) While Evil Person Being is at the point (1; 0), volcanic activity causes lava to ow from the side of Mount Pineapple. To escape the lava, Evil Person Being will need to follow a path of fastest decrease. Unfortunately, Evil Person Being has panicked and cannot determine which direction to move. The benevolent triangle beings have noticed the plight of Evil Person Being and wish to save their fellow being. In which direction should the triangle beings tell Evil Person Being to begin descending?

Question 5. Find all values of d which make the following function continuous everywhere or explain why no such value for d exists

Question 6. In an ideal gas, the temperature, T, of the gas is a function of the pressure, P, and volume, V , of the gas given by where n and R are (positive) constants. Suppose that the pressure and volume of the gas are varying with time. Suppose that, at the time t0 when the pressure is 5 and the volume is 200, the pressure is increasing at a rate of 0:05 and the volume is decreasing at a rate of 2:6. Is the temperature increasing or decreasing at t0?

Question 7. Let C be the curve parametrized by r(t) = D
t ; 4 ; t2 2 + 4 E. Let L be the
line parametrized by x = s + 4; y = 4; z = 4s + 12:

(a) Find all points where the curve C intersects with the line L
(b) Find the curvature of C at each point of intersection from (a).

Question 8. Let S1 be the surface ezy+xz2 = 2, and let S2 be the surface z = ax2+by2+c. Suppose that S1 and S2 intersect at the point P = (1; 1; 1).

(a) Find an equation for the tangent plane to S1 at P.
(b) If S2 has the same tangent plane as S1 at P, nd a, b and c.

Question 9. Find and classify all critical points of the function
g(x; y) = xy  x3  y2 + 2x + y:

Question 10. Wile E. Coyote has placed a boulder at the top of a ramp, which is at an angle of 45 relative to the horizontal, as a trap for the Road Runner. The force of gravity acting on the boulder, in Newtons, is F = h0;100i. If the work done by gravity to roll the boulder down the length of the ramp is 1000 p 2 J, how long is the ramp?
Hint: Recall that 1 J = 1 N m.

Question 11. A new prototype projectile has an interior motor that causes an acceleration of h4et; 2i at the time t after it is launched. The projectile is launched from a platform of height 60 with an initial speed of 8 p 2 at an angle of 4 radians relative to the horizontal. If the acceleration due to gravity is h0;10i, what is the horizontal distance traveled by the projectile from the time it is launched until it hits the ground

Question 12. Triangle Being has a plan to deal with Evil Person Being once and for all. Triangle Being has been granted a warrant that allows surveillance cameras to be placed on the border of Evil Person Being's palatial estate to collect evidence of Evil Person Being's dastardly deeds. However, EPB has a bug detecting eld covering the estate. At the point (x; y), the strength of the eld is given by the function

f(x; y) = x2 + y2 + 2xy + 4:

In order to avoid detection, the cameras must be placed at the points where the strength of the eld is minimized. If the border of Evil Person Being's estate is the circle x2 +y2 = 8, where should Triangle Being place the cameras?