1) Draw any four level curves of the surface given by 2z + 3x 2 − 2y = 0. Describe the shape of the surface in words.

2) The table below represents the wind chill w(z, y) where x represents the air temperature in Fahrenheit, and y represents the wind speed in mph.

a. Estimate Wx(25, 25). What does it mean in this context?

b. Estimate Wy(25, 25). What does it mean in this context?

3) On a continuous surface f(x, y), assume the tangent slope at point P, in the direction u, has value c. Prove that in the direction opposite u, the tangent slope at point P must have the value −c.

Why is the surface being continuous necessary for this seemingly obvious result?

4) Let h(x,y) = 2x 2 − 2x 2y + y 2 . Find the absolute minimum and maximum values of the surface on the region −2 ≤ x ≤ 2 and −2 ≤ y ≤ 2

5) For the following situations, explain which concept, method, and/or formula from chapter 14 you would need to use to solve the problem, and why:

a. I need to measure a container full of radioactive material in order to calculate the volume of material. But because of the danger, I must measure it from a distance and my measurements may be incorrect.

b. I’m diving to do emergency repair on an underwater component of an oil rig. I sent a drone first and calculated the pressure at various locations and depths in the water, since the machinery can affect the normal water pressure. I need to make sure that I don’t follow a dive path where the pressure increases too fast, or else I might get injured.

c. I have a raw spherical diamond that I am cutting using computer-aided tools, in order to put into jewelry. I need to program the computer to make sure that the facets that it cuts always face directly away from the center.

d. I run a business, and the business makes money from selling two types of products. I want to make the most profit I can, but my factory and the global market have limitations and restrictions that I must consider.