Some Handy Computing and Graphing Tools I. Google s Calculator Google has a graphing calculator application that is integrated into its search bar. For example, you can simply type 3/sin(2) into a search bar and you will get: Note that the default is radians. If you meant 2 degrees, click the radians/degrees switch, retype the expression directly into the calculator, and click =.
2 You must use appropriate syntax. For example, depending on how you type your expression, Google may not recognize it as a formula to feed to the calculator. In that case, it will simply spit out basic search results: On the other hand:
3 To graph two or more functions simultaneously, just separate the formulas with a comma. For example, try graphing: f(x) = cos 7x x 2 along with the envelopes y = 1/x 2 and y = 1/x 2. You should see... Notice the + and zoom buttons. If you click on the tab next to the cross, you also have the option to zoom only vertically, or only horizontally. Click and drag to move around the graph. (You can also try zooming with a scroll wheel or trackpad.)
4 This can also graph functions of two variables:... what is happening near the origin? Does the limit below exist? Why/why not? lim (x,y) (0,0) xy x 2 + y 2 Note the coordinate window on the bottom right. You can click on each to adjust the window. (Firefox doesn t seem to work here.) You can spin this around in 3-d. Play with the different buttons, for zooming, etc. If you click the arrow by the coordinate window, this removes the coordinate axes and grid...
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6 One of the unfortunate limitations of this application is that it can not graph more general equations in two or three variables. For example, it won t graph the circle x 2 + y 2 = 1, or the sphere x 2 + y 2 + z 2 = 1. As far as I know, it can only graph functions, either y = f(x) or z = f(x, y). If you want to graph a sphere using Google s Calculator you have to settle for splitting it up into pieces which are graphs of functions. For example, the top half of the unit sphere from the origin is the graph of the function f(x, y) = 1 x 2 y 2... and Google still struggles a bit with this. (I also changed the viewing window.)
7 II. Graphing with Grapher in OSX If you have access to a Mac, you have another option, which is the Grapher utility. This is a neat program and a proper graphing utility, so you have a bit more control than with Google. It does have its own quirks however. (As I understand it, Apple did not develop this program, but rather bought it from another software company, a long time ago. Because of this, it appears that the program has not been updated in many years, and probably never will.) Grapher is in the Utilities folder, inside the Applications folder. So: Applications > Utilities > Grapher If you have a Mac, I recommend adding this to your dock. When you open the program, you will see: You can play around with the other options later, but just choose default to start. Doing so, you will be prompted to enter a formula. For example, trying y = x 2, (by typing in: x 2), looks like...
8 You can graph several functions at once by clicking on the big plus in the bottom left corner, and selecting new equation. For example, trying y = x 2 sin 50x... You can also save these files for later use.
9 And with Grapher, you are not limited to functions. For example, you can graph: x 3 + y 3 = 3xy Grapher will also graph basic equations in three variables...
Here s a saddle surface, z = y 2 x 2, with a few y-slices shown: 10
To get the pictures above, I graphed the equation z = y 2 x 2, and also the equations y = 0, y =.5. I changed the colors, style, and transparencies using the Inspector. Look for this in a menu, or right-click on the surface you want to edit. Clicking on the color(s) brings up the palette. Turning down the opacity of a color makes it more transparent; 11
12 Remember the ellipsoid from class? (14.5) x 2 + 2y 2 + 3z 2 = 15 We found the equation of the plane tangent to the ellipsoid at (1, 1, 2). (Recall that we did this by viewing the ellipsoid as the 15-level set of the function f(x, y, z) = x 2 + 2y 2 + 3z 2, and using the fact the gradient f is perpendicular to the level sets of f, thus giving a normal vector for the tangent plane.) We can graph both the ellipsoid and its tangent plane together in Grapher:
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14 Of course, Grapher has its own limitations too. For example, for the function f(x, y) = 2y x 2 + y On the other hand, Google spits out a beautiful graph for the same function: