# The directional derivative

Solve the following problems in detail, showing your work in an uploaded file, either pdf or jpeg (or, if necessary, MS Word). Partial credit is given in proportion to how close your work is to being correct. 1. Let f(x, y) = (sin y)(cos x) + y sin (e 2x). Find f xy [It may help to know that this equals fyx]. [5 points] 2. Let f(x, y, z) = (x + xy)/z. Find the following [5 points each part). a. The gradient of fat (1,1,-1). b. The directional derivative off at(1, 1,-1) in the direction parallel to i – 5 j. c. The directional derivative of fat (1,1, -1) in the direction of most rapid increase. 3. Find equations for the tangent plane and normal line to the level surface x 3y+y?z + 2z2 = 14 at (-1, 2, -2). Put the equation of the tangent plane in the form Ax + By + Cz = D. the line.] [6 points for the plane, 4 for 4. Find an equation for the tangent plane to the graph of z = xe -Y at (2, 1, -3), again putting it into the form above. [It may help to know that this graph is the level suface F(x, y, z) = 0 where F(x, y, z) = xe – – z.] [6 points]

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