graphs_of_functions_of_two_variables (1)

graphs_of_functions_of_two_variables (1) - Graphs of...

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G raphs of F unctions of T wo V ariables and C ontour D iagrams In the previous section we introduced functions of two variables. We presented those functions primarily as tables. We examined the differences between an equation graphed in 2-space and 3-space. The goal of this section is to introduce a variety of graphs of functions of two variables. The Graph of a Function of Two Variables The graph of a function of two variables, f ( x , y ), is the set of all points ( x , y , z ) such that z = f ( x , y ). In general, the graph of a function of two variables is a surface in 3-space. Example 1: Describe and graph the function f ( x , y ) = xy . Solution: We begin by making a table of values. Then we attempt to plot those points in 3-space and connect the points. As we include more and more points, our graph becomes the desired surface. 321 0 123 3 963 0369 2 642 0246 13 2 10 1 2 0000 0 000 2 1 0 1 2 26 4 2 0 2 4 3 0 369 y x   3 3 6 Table 1: Values of f ( x , y ) = xy We see that if x or y is 0, the function is 0. As x and y both increase, the values of the function get larger. For a fixed value of either x or y , the function looks like a line. 1
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Using this information, we can attempt to make a sketch of the function. Figure 1: Graph of f ( x , y ) = xy We remark that as x and y are both positive or both negative, the function values are getting larger and positive. When one is positive and the other is negative, the function values are getting larger and negative. This is illustrated by the slants of the surface. Example 2: Graph the function g ( x , y ) = x 2 + y 2 . Solution: Again, we can start by creating a table of values. 321 0 12 3 18 13 10 9 10 13 18 21 3 8 5 4 5 81 3 11 0 5 2 1 2 5 1 0 0941 0 14 9 05 21251 38 54581 3 18 13 10 9 10 13 18 y x  3 0 3 Table 2: Values of g ( x , y ) = x 2 + y 2 Notice that there is a lot of symmetry in the table. For any fixed value of x , or any fixed value of y , the values increase as we move further away from 0. It has a minimum value of 0 which occurs at x = 0, y = 0. Also, swapping x and y results in the same value. A graph of the function appears below. 2
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Figure 2: Graph of g ( x , y ) = x 2 + y 2 When we considered functions and graphs of one variable, one of the first things we did was to transform those graphs through shifts and stretches. We can do the same thing with functions of two variables. Example 3: Using the function from Example 2, describe and graph the following functions: (i) f ( x , y ) = 3 – x 2 y 2 (ii) g ( x , y ) = ( x – 1) 2 + ( y + 1) 2 (iii) h ( x , y ) = 4 x 2 + y 2 Solution: (i) f ( x , y ) looks like x 2 + y 2 , except it has been shifted up by 3 units and opens downwards instead of upwards. It has a maximum value at (0, 0, 3).
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This note was uploaded on 09/07/2011 for the course MATH 10C taught by Professor Hohnhold during the Spring '07 term at UCSD.

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graphs_of_functions_of_two_variables (1) - Graphs of...

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