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MVC SHOW ALL WORK !! Quiz #4 NAME: #1 Find the directional derivative of F ( x, y) xye y v = i + j , at the point (1,0). #2 Write a formula for 2 in the direction of the vector f if w f ( x) and x x(r , s, t ). t yx zx yx zx , Suppose that w f and v . is a differentiable function of u xz xy xz xy Prove that: w w 2 w x2 y2 z 0 x y z #3 #4. For the contour map for z f ( x, y ) shown below, estimate each of the following quantities. Explain briefly how you are getting your answer. (a) f x (1, 2) and f y (1, 2) (b) f (1, 2) (c) Du f (1, 2) , where u is a unit vector in the direction of f (1, 2) (d) Sketch the vector f (1, 2) on the contour map for f using (1, 2) as the initial point. (e) Sketch a unit vector v with initial point (1, 2) such that Dv f (1, 2) 0 . 6 4 4 2 0 2 4 6 5. Find an equation for the tangent plane to the surface xy z 1 z 5 at the point (4, 1, 3). #6. Parameterize the straight line segment from (0,0) to (1,2) in terms of the arc-length parameter s. #7. Look at the path given by x (t ) (t cos t , t sin t ), 0 t 6 . a. Sketch the path using arrows to indicate direction of travel. Label scales on the x- and yaxes. b. Calculate the velocity and speed at t 2 . c. Sketch the velocity vector from b. with initial point on the path. d. Which will be larger: ( ) or (5 ) . Explain your reasoning.