1. (1 point) Find all the first and second order partial derivatives of f (x, y) = 4 sin(2x + y) 10 cos(x y). A. xf = fx = B. yf = fy = 2 f = fxx = x2 2 f D. y2 = fyy = 2 f E. xy = fyx = 2 f F. yx = fxy = C. A B C D E F Answer(s) submitted: (incorrect) 2. (1 point) Find the first partial derivatives of f (x, y, z) = z arctan( xy ) at the point (2, 2, -5). A. xf (2, 2, 5) = B. yf (2, 2, 5) = C. f z (2, 2, 5) = Answer(s) submitted: Answer(s) submitted: (incorrect) (incorrect) 3. (1 point) 4. (1 point) Match the equations below with the pictures of the of the level surfaces they represent. 1. 21 y2 3z2 x2 = 1 2. 2y2 3z2 x2 = 1 3. 2x2 y2 3z2 = 1 4. 2z2 3x2 y2 = 1 (You can drag the images to rotate them.) Match the functions below with their level surfaces at height 3 in the table at the right. 1. f (x, y, z) = 2x2 3z 2. f (x, y, z) = 2z2 3y 3. f (x, y, z) = 2y2 3x 4. f (x, y, z) = 2x2 + 3y (You can drag the images to rotate them.) 1 (incorrect) 6. (1 point) For each surface shown on the left, select the image on the right that represents the level sets for that surface. Drag a surface in order to rotate it. In the level sets, the red areas are the lowest levels while the purple are the highest. A B C 1. 2. 3. 4. D Answer(s) submitted: (incorrect) 5. (1 point) The graphs below show level sets for six different functions, where the red areas represent the the lowest heights and the purple areas are the highest heights. A B C D E F A B C D E F Match the following functions with the proper graphs above. 1. f (x, y) = 3x2 + y2 2. f (x, y) = 3y x2 3. f (x, y) = x2 + 3y2 4. f (x, y) = 3x y2 Answer(s) submitted: Answer(s) submitted: 2 (incorrect) 7. (1 point) If f (x, y) = 2x2 + 4y2 , find the value of the directional derivative at the point (4, 4) in the direction given by the angle = 2 5 . (incorrect) 9. (1 point) Suppose f (x, y, z) = xy + yz , P = (2, 1, 2). A. Find the gradient of f. i+ j+ k f = Note: Your answers should be expressions of x, y and z; e.g. \"3x - 4y\" B. What is the maximum rate of change of f at the point P? Answer(s) submitted: (incorrect) 8. (1 point) Suppose f (x, y) = xy , P = (4, 2) and v = 4i 4j. A. Find the gradient of f. i+ j f = Note: Your answers should be expressions of x and y; e.g. \"3x 4y\" B. Find the gradient of f at the point P. ( f ) (P) = i+ j Note: Your answers should be numbers C. Find the directional derivative of f at P in the direction of v. Du f = Note: Your answer should be a number D. Find the maximum rate of change of f at P. Note: Your answer should be a number Answer(s) submitted: (incorrect) 10. (1 point) Let f (x, y) = 4x3 3xy2 . Then the direction in which f is increasing the fastest at the point (2, 1) is , and the rate of increase in that direction is . The direction of the fastest decrease at , and the rate of decrease in that the point (2, 1) is direction is Note: Your answer should be a number E. Find the (unit) direction vector in which the maximum rate of change occurs at P. u= i+ j Note: Your answers should be numbers Answer(s) submitted: Answer(s) submitted: (incorrect) 3