1. ConcepTest Section 12.2 Question 1 Let be the distance above the ground (in feet) of a jump rope x feet from one end after t seconds. The two people turning the rope stand 10 feet apart. Then h(x, ¼) is (a) Concave up (b) Concave down (c) Flat (d) Changes concavity in the middle

2. ConcepTest Section 12.2 Answer 1 (c)At t = ¼ the cosine is zero. Therefore, h (x, ¼) = 3 is constant, so its graph is a straight line. ANSWER COMMENT: Follow-up Question. Find a value for t at which the cross section is concave down.

3. ConcepTest Section 12.2 Question 2 The object in 3-space described by x = 2 is (a) A point (b) A line (c) A plane (d) Undefined

4. ConcepTest Section 12.2 Answer 2 (c), a plane. While x is fixed at 2, y and z can vary freely. ANSWER COMMENT: If all three variables do not appear in a problem, students sometimes forget that the missing variables are important.

5. ConcepTest Section 12.2 Question 3 The set of points whose distance from the z-axis equals the distance from the xy-plane describes a (a) Plane (b) Cylinder (c) Sphere (d) Cone (e) Double cone (two cones joined at their vertices)

6. ConcepTest Section 12.2 Answer 3 (e), double cone. Fix a value for z, for example, z = a, and draw the set of points in the plane z = a that are equidistant from the z-axis and the xy-plane. It is the set of points at distance |a| from the z-axis, namely a circle of radius |a| in the plane z = a. Putting all these circles together, we get two cones, with vertex at the origin, one above the xy-plane and one below it. ANSWER COMMENT: Note that you also have to consider negative values of z. Students have a hard time visualizing surfaces in 3 dimensions.

7. ConcepTest Section 12.2 Question 4 Which of the following objects cannot be obtained as the graph of a function of two variables with a single formula? (a) Paraboloid (b) Plane (c) Cylinder (d) Sphere (e) Line (f) Parabolic cylinder

8. ConcepTest Section 12.2 Answer 4 (c), cylinder; (d), sphere; (e), line. ANSWER COMMENT: Follow-up Question. Give examples of functions whose graphs are the other surfaces.