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UCSD MATH 10C - Graphs of Functions of Two Variables and Contour Diagrams

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Graphs of Functions of Two Variables and Contour Diagrams 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. 3210123396303692642024613 2 10 1 200000000132101226420243 9630369yx336 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. 1Using 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) = x2 + y2. Solution: Again, we can start by creating a table of values. 3210123 18 13 10 9 10 13 18213 8 5 4 5 813110 5 2 1 2 510094101491105 212512138 545813 18 13 10 9 10 13 18yx303 Table 2: Values of g(x, y) = x2 + y2 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. 2Figure 2: Graph of g(x, y) = x2 + y2 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 – x2 – y2 (ii) g(x, y) = (x – 1)2 + (y + 1)2 (iii) h(x, y) = 4x2 + y2 Solution: (i) f(x, y) looks like x2 + y2, except it has been shifted up by 3 units and opens downwards instead of upwards. It has a maximum value at (0, 0, 3). Figure 3: Plot of 3 – x2 – y2 (ii) g(x, y) looks like x2 + y2, except it has been translated by 1 unit to the right in the x-direction and 1 unit to the left in the y-direction. It has its minimum value at the point (1, -1, 0). 3Figure 4: Plot of (x – 1)2 + (y + 1)2 It is a little difficult to see that this is the same shape as x2 + y2. The reason for the slight difference is the clipping introduced with the window size. Compared the Figure 2, the minimum point does appear to be at the point (0, -1, 0). (iii) h(x, y) looks like x2 + y2, except in the x-direction, the values have been compressed (which has the effect of making the y-values appeared to be stretched). As compared to the circular bowl of the graph of x2 + y2, h(x, y) will look like an elliptical bowl. Figure 5: Plot of 4x2 + y2 A good way to visualize what a graph looks like in 3-space is to consider cross-sections of the function. That is, we fix either x or y and see what the resulting graph looks like. This is like standing on one of the axes and describing what we see. We can also fix the z-value. The resulting curve is called a level curve. The collection of such curves is called a contour diagram. The more closely spaced the level curves, the more the function is changing. The more widely spaced the level curves, the function is more constant. Taking the cross-sections and contour diagram into account, we can reconstruct what the shape looks like. 4Cross-Sections and Contour Diagram of a Graph For a function z = f(x, y), the function we get by holding x fixed and letting y vary is called a cross-section of f(x, y), with x fixed. The graph of the cross-section of f(x, y) with x = c is the curve we get by intersecting the graph of f(x, y) with the plane x = c. Similarly, if we fix y and let x vary, we get the cross-section of f(x, y) with y fixed. If we fix the z-value in the graph, the resulting curve is called a level curve and the collection of these curves is called the contour diagram of the function f(x, y). Example 4: What shapes are the cross-sections of the function z = f(x, y) = x2 + y2? What about the contour diagram of f(x, y)? Solution: The cross-sections of x2 + y2 look like upwards pointing parabolas since setting y = c, results in a curve which looks like z = x2 + c2, and if we fix x = c, then the resulting curve is z = c2 + y2. Both of these are upwards pointing parabolas with an intercept of c2. Figure 6: Cross-sections of x2 + y2 The level curves of the function are circles, since k = x2 + y2 denotes a circle with radius k . Collectively, we have the following contour diagram of the function. 0.71.42.12.83.53.53.53.54.24.24.24.24.94.94.94.9-1 1x-11y Figure 7: Contour Diagram of x2 + y2 5Example 5: By first considering cross-sections and the contour diagram, sketch the graph of the function g(x, y) = x2 – y2. Solution: Similar to Example 4, we begin with cross-sections. Notice that if we set y = c, we end up with z = x2 – c2, which is an upwards pointing parabola. But if we x = c, we end up with z = c2 – y2, which is a downwards pointing parabola. Figure 8: Cross-sections of x2 – y2 A contour diagram shows hyperbolas (as compared to circles). Again, this makes sense because if we set z = k, we have k = x2 – y2, which is the equation of a hyperbola. -3-3-2-2-1-1000112233-1 1x-11y Figure 9: Contour Diagram of x2 – y2 Putting this all together, we get an idea of what the resulting shape looks like. From the y-axis, it looks like an upwards pointing parabola, but from …


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UCSD MATH 10C - Graphs of Functions of Two Variables and Contour Diagrams

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