408DS09assign8 - ( x,y ) (0 , 0) xy x 2 + y 2 as ( x,y )...

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M408D Spring 2009 Assignment 8 Due Thursday, April 9 Be sure that you have read and understood sections 15.1, 15.2, and 15.3 and worked the assigned text exercises before you complete this assignment. You must show sufficient work in order to receive full credit for a problem. Please write legibly and label the problems clearly. Circle your answers when appropriate. Multiple papers must be stapled together. Write your name and the time of your discussion section on each page. Homework is to be turned in at the beginning of the discussion section. 1. Let f ( x,y ) = ln(1 - xy ). (a)Find and sketch the domain of f . (b) Sketch 3 level curves for the surface z = f ( x,y ) corresponding to z = - 1, z = 0, and z = 1. 2. Evaluate lim
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Unformatted text preview: ( x,y ) (0 , 0) xy x 2 + y 2 as ( x,y ) approaches the origin along (a) the x-axis (b) the line y = 2 x (c) the spiral r = , > (d) along the path ~ r ( t ) = 1 t ~ i + sin t t ~ j , t > 0. Does lim ( x,y ) (0 , 0) xy x 2 + y 2 exist? Explain. 3. A petrochemical company is designing a cylindrical tank with hemispher-ical ends to be used in transporting its product. (a) Find the volume of the tank as a function of the radius r and height h of the cylindrical portion of the tank. (b) If h held xed, nd the rate of change of the volume with respect to r . 4. Calculate the four second order partial derivatives of f ( x,y ) = e xy 2 . 1...
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This note was uploaded on 10/14/2009 for the course M 408d taught by Professor Sadler during the Spring '07 term at University of Texas at Austin.

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