ConcepTest • Section 14.5 • Question 1

1. Let f (x, y, z) represent the temperature in °C at the point (x, y, z) with x, y, z in meters. Let be your velocity in meters per second. Give units and an interpretation of each of the following quantities. ConcepTest• Section 14.5•Question 1

2. ANSWER ConcepTest• Section 14.5 •Answer 1

3. For f (x, y, z) suppose The tangent plant to the surface f (x, y, z) = 0 through the point (a, b, c) is given by z = p + mx + ny. Which of the following is correct? ConcepTest• Section 14.5•Question 2 • m > n > 0 • n > m > 0 • m < n < 0 • n < m < 0 • None of the above

4. ANSWER ConcepTest• Section 14.5 •Answer 2

5. Let Which of the following is/are not a possible equation(s) for the tangent plane to the surface f (x, y, z) = c at (x0, y0, z0)? Why not? ConcepTest• Section 14.5•Question 3

6. ANSWER ConcepTest• Section 14.5 •Answer 3 Only (b) and (c) are incorrect. The equation for the tangent plane is a scalar equation; (b) is a vector equation. In (c), the left side should be –fz (x0, y0, z0)(z – z0). COMMENT: Have the students justify and explain the other equations.

7. The vector grad f is perpendicular to the level curve f (x, y) = f (a, b). • The vector grad f is perpendicular to the surface z = f (x, y) at the point (a, b, f (a, b)). • The vector is perpendicular to the surface z = f (x, y). • If the vector is any vector which is perpendicular to the surface at the point where x = a and y = b, then is a scalar multiple of (grad f - ). The function f (x, y, z) has gradient f at the point (a, b). Which of the following statements is/are true? ConcepTest• Section 14.5•Question 4

8. ANSWER ConcepTest• Section 14.5 •Answer 4 • True • False. The vector grad f is a 2-vector; the vector perpendicular to the surface has a z-component. • False. The normal to the surface z = f (x, y) is obtained by writing it in in the form • f (x, y) – z = 0, • giving the normal as • True. One normal is • so any other normal is a multiple of this one.