Web Works #6 Spring 2009

Web Works #6 Spring 2009 - Assignment 6 The unit normal...

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Unformatted text preview: Assignment 6 The unit normal line at this same point oriented to make an acute angle with the positive z axis is i j k 7.(1 pt) The equation of the tangent plane to the surface ¡ ¡ ¥ § § ¤ ¦¢ ¥ ¡ ¤ £ ¢ ¡ a collection of unequally spaced parallel lines two straight lines and a collection of hyperbolas a collection of concentric ellipses a collection of unequally spaced concentric circles a collection of equally spaced parallel lines a collection of equally spaced concentric circles ¡¤ 3yz ¡ ¥¤ 3xz ¥¤ ¡ ¡ ¡ ¡   ¥  ¤ ¥ ¤ ¡ ¡ § ¤ ¥ ¤ ¡ §¤ ¥ ¤ ¤ ¥ ¤  § ¥   ¤ 9 F 9 10s t  ¥ ¡ ¡ § § § § ¡   ¥ ¡ § § §  § ¥  ¥   ¤ ¥ ¡ ¡  © ¨  © ¨ ¥ ¥ ¥  £ ¥  ¤ ¥  ¤ ¤ ¡ ¡ ¤ ¥ ¤ ¤ ¤ ¥¤ 1 F  § ¥   ¤ ¡  © ¨  © ¨ § ¡ ¡ ¥   §  ¥   ¤ ¤ § 1 2 12 is . ¡¤ ¥ gst xy4 § ¡  © ¨  © ¨ at the point x y z z x y sin 9xy ¥ ft 6.(1 pt) The equation of the tangent plane to the surface 2x2 y 0. 00 Note that the answers are different. The existence and continuity of all second partials in a region around a point guarantees the equality of the two mixed second derivatives at the point. In the above case, continuity fails at 0 0 . z ∂ ∂F ∂x ∂y 2 10.(1 pt) Show that the function f x y ux4 x3 y 2 y2 3 4 is harmonic, i.e. satisfies Laplace’s equa12x vxy wy tion ∂2 f ∂2 f 0 ∂x2 ∂y2 if and only if the constants u v w are given by u v w 11.(1 pt) Let F be a differentiable function of x y z and suppose that ∂F ∂F ∂F 111 3 111 1 111 8 ∂x ∂y ∂z For the functions f t g s t defined by 5.(1 pt) Consider the function defined by 0 0 where F 0 0 f xyz ∂f ∂x ∂2 f ∂x∂y ∂3 f ∂x∂y∂z 4.(1 pt) Find the limit, if it exists, or type N if it does not exist. (Hint: use polar coordinates.) 6x3 9y3 lim y2 xy 0 0 x2 except at x y Then we have ∂ ∂F ∂y ∂x 0 0 2y 9.(1 pt) Find the following partial derivatives of the function 3.(1 pt) Find the limit, if it exists, or type N if it does not exist. x 17y 2 lim xy 0 0 x2 172 y2 xy x2 7y2 x2 y2 ¡ £ ¥  ¤ ¤ ∂f ∂x ∂f ∂y ∂2 f ∂x2 ∂2 f ∂x∂y ∂2 f ∂y2 2.(1 pt) Find the limit, if it exists, or type N if it does not exist. 2x2 lim 2 1y2 xy 0 0 4x Fxy 5x  ¦ ¦¥ 3y y2 ¡ 2x xy x2 3y2 25 5y 2 ¡ 2x2 ¡ x1 5xy at the point 4 3 129 is x y . z The parametric equations of the normal line at this point are x t, y t, z t, 8.(1 pt) Find the first and second partial derivatives of the function 2 2 f xy 7 2xy e 6x 7y y2 x2 ¡¡ ¥  § § ¤ A. B. C. D. E. F. 1 3x2 ¡ z y2 ¡ ¡ x2 z z z z z z z ¡¡¡ 1.(1 pt) Match the surfaces with the verbal description of the level curves by placing the letter of the verbal description to the left of the number of the surface. 1. 2. 3. 4. 5. 6. 7. MATH262, Winter 2009 ¡ Hector Hernandez due 4/1/09 at 11:59 PM. 19t 9t 2 3 st 7 2s 4t compute the following derivatives: f1 ∂g ∂s 1 1 2t 2 4st 3 t 5t 2 13 4s 9t 9st ¥  ¤ ¥  ¤ ¥  ¤ ¥  ¥¥  ¤  ¥  ¤ ¤ ¥  ¤ ¤ §  7 4  ∂s2 ∂2 f ∂s2 ∂2 f ∂s∂t  1 ∂2 u Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR 2 11 11 ∂2 F ∂y2 ∂2 u ∂t 2 7 ∂v ∂s 9 ∂v ∂2 v ∂2 v ∂2 v ∂2 v 6 2 6 2 ∂t ∂s2 ∂s∂t ∂t ∂s ∂t 2 If f s t F u s t v s t compute the following derivatives: ∂f 11 ∂s ∂f ∂t 1 1 4 6 1 § ∂2 F 1 ∂x2 ∂2 u ∂t ∂s  ∂u 7 ∂t § § ¥ ∂u ∂s ∂2 F ∂y∂x  and at t s ∂2 F ∂x∂y 11 ∂F 4 ∂y  ∂F ∂x ∂2 u ∂s∂t 12.(1 pt) Let F x y be a differentiable function of x y, and let u s t v s t be differentiable functions of s t . Suppose that u11 v11 1, that at x y 11 §  § ¥  £ ¥  ¤ ¤    ¤ £ ¥  ¤ ¥  ¤   ¥  ¤ ¥¤¥¤ ¥¤ ¥  ¤ 11 § § ∂g ∂t ...
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This note was uploaded on 11/09/2011 for the course MATH 262 taught by Professor Faber during the Spring '08 term at McGill.

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