advertisement

MVC SHOW ALL WORK !! Sample Quiz NAME: #1 Find the directional derivative of F ( x, y) sin( x 2 y 2 ) in the direction of the vector v = i + j, at the point ( , ). #2 Write a formula for w if w f ( x, y ) and x x(r , s, t ), y y (r , s, t ) . s #3 Prove that if z f ( x, y ), with x r cos and y r sin , then 2 1 z z z z 2 r x y r 2 2 2 xe y and P = (0,1,1). Find u such that Du [f(0,1,1)] is a maximum. Evaluate 2z 2 3 Du [f(0,1,1)] for this vector u . #4 Let f(x,y,z) = #5 Find a tangent plane to the surface which is defined implicitly by the equation xy 2 yz 2 zx 2 3 at the point (1, 2,1) . #6. The figure below shows the level curves of a function z k x, y . In this plot, the z-values increase as you move up and to the right. a. b. On this contour plot, sketch vector v with initial point (2, 2) in the same direction as k 2, 2 . On this contour plot, sketch a vector w with initial point (2, 2) so that Dw k 2, 2 0 . That is, the directional derivative at (2, 2) in the direction of w is 0. k k c. For this contour plot, which has the greater value at the point (3, 1), or ? x y Briefly explain your choice. #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 ) (3sin 2t , 4 cos 2t ), 0 t . a. Sketch the path using arrows to indicate direction of travel. b. Calculate the velocity, speed, and acceleration at t . 4 c. Sketch the velocity and acceleration vectors from b. with initial point on the path. #9. Find the curvature of the path x (t ) (t 2 , 2t 2 ,3t 2 ),0 t 5 at t 2