Solutions1

Solutions1 - 1 Mathematics 1c: Solutions to homework Set 1...

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Unformatted text preview: 1 Mathematics 1c: Solutions to homework Set 1 1. (10 Points) Using the computing site or otherwise, draw the graphs of the following functions: (a) f ( x,y ) = 3( x 2 + 2 y 2 ) e- x 2- y 2 ; Tip: On the computing site use E to take the exponent and there is no need to type a * for multiplication; we suggest taking x and y between- 2 and 2. (b) f ( x,y ) = ( x 3- 3 x ) / (1 + y 2 ) Indicate some key features of these graphs, such as the location of the maxima and minima, important sections, etc Solutions. (a) The graph is shown in the accompanying figure: As we see this function has one minimum at the center, two maxima and two saddle points. Interesting sections would be obtained by slicing the graph with planes parallel to the two axes. Interesting level sets are obtained by cutting the graph with horizontal planes at various heights. The level sets, computed using the computing site are shown in the next figure. 2 (b) The graph is shown in the accompanying figure: Perhaps the most interesting section is obtained by cutting the graph using the vertical plane y = 0. Level curves are obtained using horizon- tal planes and these are distorted circles surrounding the maximum and minimum. 2. (10 Points) Section 2.1, parts of Exercises 15, 18. Sketch the zero level set 3 of the function f ( x,y,z ) = xy + yz and the level set for c = 1...
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Solutions1 - 1 Mathematics 1c: Solutions to homework Set 1...

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