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• Gradient and Directional derivative

True or False

1.The gradient of f (x, y) is a scalar value.

2.Letf (x, y) = x2y. Then∇f(x, y) = x2 i + 2xyj = (x2, 2xy).

3.A directional derivative represents a rate of change of a function in any given direction.

4.A directional derivative is a vector.

5.The direction of the greatest decrease of f(x, y) is -∇f(x, y).

Choose exactly ONE answer for each of the following questions.

6.Let f (x, y) = (2x2+y)6. Find fx(x, y) =?
(a) 6(2x2+y)5
(b) (4x+y)6
(c) 6(2x2+y)5⋅4x
(d) None of the above

7.Let f (x, y) = x2y. Find ∇f(1,2).
(a) i + 4j= (1, 4)
(b) i − 4j= (1, −4)
(c) 4i −j = (4, −1)
(d) 4i+ j = (4, 1)

8.If f(x, y, z) = exyz. Find the direction of the maximum rate of change of f at (0, 1, −1).
(a)(0, −1, 0)
(b) (−1, 0, 0)
(c) (0, 0, −1)
(d) None of the above

9.What is the directional derivative of f (x, y) in the direction u that is not a unit vector?
(a) ∇f(x,y) · (u / ||u||)
(b) ||u||∇f(x,y)
(c) ∇f(x,y) ·u
(4)None of the above

10.Letf (x, y) = x2y. Find the derivative off at the point (1, 2) in the direction of u = (3, 4).
(a)1*(3/5) + 4*(4/5) = 19/5 = 3.8
(b) 13 + 44 = 19
(c)4*(3/5) + 1*(4/5) = 16/5 = 3.2
(d) 43 + 14 = 16

Answers: 1. F 2. F 3. T 4. F 5. T 6. c 7. d 8. b 9. A 10. c