Math 32A Graphs and Contour Worksheet

Transcription

1 Math 32 Graphs and ontour Worksheet Sections 5., 5.4 November 8, 208 This worksheet helps you practice matching graphs with its equations. Here are some general techniques and tricks that may help: (i) Looking for the range of the function. For example, graphs of z = x 2 + y 2 or z = xy must be above the xy-plane because these functions are non-negative. In some cases, we can even say more. For example, z = x 2 + y 2 + is not only non-negative, it is also. In a similar spirit, z = is always. x 2 +y 2 + (ii) Looking for the domain of the function. This method is usually harder to apply since most functions given in quizzes/exams are continuous. However, if you detect some discontinuities of the function, then the graph should look weird at those discontinuities. For example, z = is not continuous at the origin (in fact at any xy point where x = 0 or y = 0), so its graph consists of separate pieces rather than one smooth piece. (iii) Functions involving sin or cos. Expect these graphs have some kind of oscillations. One useful bound to remember is that cos(...) and sin(...). That is, the graph only fluctuates between and. (iv) Looking for symmetries. Graphs of functions that have some sort of symmetry between x and y should look symmetric with respect to the z-axis. For example, z = x 2 + y 2, z = sin(x 3 + y 3 ). (v) Looking for behaviors near the origin and near. For example, z = x 2 +y 2 + is very small when x, y (but always stay non-negative). So you should expect its graph decays eventually (that is, getting closer to the xy-plane from above). (vi) Try plugging in some specific values. If you encounter a problem that two functions have quite similar graphs, plug in some specific values to distinguish them. (vii) Finding traces or slicing the graph. For example, z = cos(x y). Then with special values x = y, we always have z = cos 0 =. So the plane x = y must intersect the graph at the points whose height z =. (viii) Use online graphing tools to test your answers. Use wolframalpha.com.

2 Question. z = sin(x)e x2 y a) 2 z = b) c) z = cos(xy) d) z = x 2 + y x 2 + y 2 nswer: We see that b) and d) define negative functions, so their graphs should be and. Since b) is not defined at x = y = 0, its graph should be. So d) matches with. Since c) is symmetric with respect to x and y, its graph should be symmetric as well. So c) matches with and a) matches with. 2

3 Question 2. (i) z = (ii) z = (x y) 2 z = e x2 +y (iii) 2 (iv) z = sin(x y) x + y nswer: (iv) must match with because of the oscillation. (i) matches with because of the discontinuity at (0, 0). Since (iii) defines a non-negative function, it must match with. So (ii) matches with. 3

4 Question 3. (i) z = (ii) x2 + y 2 + z = sin(2xy) z = e x2 y 2 (iii) (iv) z = cos(x + y) nswer: We can see immediately that (ii) and (iv) must have oscillatory graphs. For (iv), the plane x + y = 0 cuts the graph at z =, so its graph must be. So (ii) matches with. To distinguish the other two, look at the functions at x = y = 0. For (i), we get z = so it matches with. For (iii), we get z = e < so it matches with. 4

5 Question 4. Match the equations with their contours. (i) z = x 2 + y 2 + (ii) z = 2x 2 + y (iii) z = y (iv) z = x + x 2 2y 2 nswer: For (i), set z = c we can solve x 2 + y 2 =, which are circles (). For (ii), set c z = c we get y = 2x 2 + c, which are parabolas (). For (iii), set z = c we get y = c(x + ), which are straight lines (). For (iv), we get x 2 2y 2 =, which are hyperbolas (). c 5