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Leader: Course: Instructor: Date: Worksheet #8 12.1-12.4 Supplemental Instruction Iowa State University Jon Math 265 Poon & Weber 09/22/11 Section 12.1 1) Sketch the level curve z k for z x2 and k 4, 1, 0,1, 4 [#19] y Section 12.2 2) Find all first partial derivatives of the following functions. A) g ( x, y) e xy [#9] B) f ( x, y ) x2 y 2 [#3] xy C) f ( x, y) y cos( x 2 y 2 ) [#13] 3) The volume V of a cylinder is given by V hr 2 . If h is held constant where h 10'' , find the rate of change of V with respect to r when r 6'' [#29]. 4) Show that the function f ( x, y) x3 y xy 3 is harmonic [ 2 f 2 f 0 ] [#33]. x 2 y 2 Supplemental Instruction 1060 Hixson-Lied Student Success Center 294-6624 www.si.iastate.edu Section 12.3 5) Find the indicated limit or state that is does not exist. sin( x 2 y 2 ) A) lim ( x , y )(0,0) [#7] 2 2 x y B) x2 y 2 lim( x, y )(0,0) x4 y 4 [#9] 1 x2 y 6) If f ( x, y ) 4 , show that f as ( x, y ) (0, 0) along the parabola y x 2 [#37]. 2 2 x y Section 12.4 7) Find the gradient of f ( x, y) x 2 y xy 2 and the tangent plane @ the point p (2,3) [#11]. 8) Find the equation of the tangent hyper-plane to the surface f x, y, z 3x2 2 y 2 xz 2 at the point p (1, 2, 1) [#15].