162 MATH Calculus II Computer Laboratory
Topic: Intro to Mathematica, Part II, & Intersections of Polar Functions
Goals of the lab: To re-familiarize students with some basic operations in Mathematica, such as how to define a function, and the basic structure to plotting commands. To understand polar coordinates better and how polar functions work.
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1.1
Review and Extensions
A Summary of Previous Experience
During our last lab, we learned the basic structure of commands in Mathematica. The built-in commands are always capitalized, and their arguments go inside square brackets [ ]. used a few of these basic commands, including several that produce plots: in the plane: Plot, ContourPlot, ParametricPlot in 3D-space: Plot3D, ParametricPlot3D The basic inputs to these commands included the thing (function or vector function) to be plotted as well as a range, enclosed in curly brackets { }, of values for the input variable(s). Two examples are Plot[Sin[x], {x, 0, 2*Pi}] ParametricPlot3D[{Sin[t], Cos[t], t/5}, {t, 0, 20}] Start Mathematica and try out each of the above commands. Notice that the purpose of Plot is to graph functions of one variable (here sin x), while ParametricPlot3D is used, in this example, to plot the vector function r(t) = (sin t)i + (cos t)j + (t/5)k.
Mathematica Lab, MATH 162
Polar Graphs
Recall, also, that these represent minimal inputs to the two commands. Switches, like ones affecting how axes are labeled, the "viewing window", etc., may be added. several that carry out calculations and/or algebraic operations. These include: Together (used to combine fractions) and D (used to find derivatives or partial derivatives). How to define a function of your own. The syntax involves naming the function, indicating (in square brackets) what its inputs will be (each followed by an underscore character), using "colon equals" (`:=') instead of "equals", and then giving how the function should behave. For a function of two variables like f (x, y) = sin(xy - y 2 ), this is accomplished by f[x_, y_] := Sin[x*y - y^2] 1.2 Plotting More than One Thing
Often we wish to look at several graphs together. Let's say we wish to have the sine and cosine functions appear together. A graphing calculator can do it, so it would be a severe deficiency not to be able to do it in Mathematica. The key is to place the things you want plotted, separated by commas, into curly brackets. Try out the following modifications to the two plotting commands you entered earlier. Plot[{Sin[x], Cos[x]}, {x, 0, 2*Pi}] ParametricPlot3D[{ {Sin[t], Cos[t], t/5}, {1-t/10, Cos[t], t/5} }, {t, 0, 20}] Of course, when plotting two vector functions in Mathematica with ParametricPlot3D, the syntax for each vector function itself requires a pair of curly brackets, which explains why the second command above has as many such brackets as it has.
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Mathematica Lab, MATH 162
Polar Graphs
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2.1
Several New Commands
PolarPlot
Two commands we will use to accomplish further tasks in this lab are PolarPlot and Solve. PolarPlot is not among the core list of commands available to Mathematica at startup, and must be imported via the command (execute it yourself) <<Graphics`Graphics` From this point on, we will be able to use the command as if it had been available from the start. This command does not do a whole lot more than what your graphing calculator can do. In particular, you can plot polar functions like r = 5 - 3 cos(2) in your graphing calculator by placing your calculator in polar mode, or in Mathematica with the command PolarPlot[5 3*Cos[2*t], - {t, 0, Pi}] Of course, there are options which may be tacked onto the base command above. For a list of the options PolarPlot will accept, type Options[PolarPlot] In particular, this modification of the above command PolarPlot[5 - 3*Cos[2*t], {t, 0, Pi}, AspectRatio -> 1] skews the tick-marks on the axes so that the figure has an exaggerated width in comparison to its height. Exercise 1: The figure at right was produced in Mathematica. Write a single Mathematica command that reproduces it. (Hint: we converted the equation for a similar circle to polar form yesterday.)
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Mathematica Lab, MATH 162
Polar Graphs
2.2
Solve and NSolve
By itself, PolarPlot does not offer much that most graphing calculators cannot do. Perhaps, at least for some calculators, the same may be said of the Solve command. Its basic syntax is Solve[equation, variableName], where the equation should be written with a double-equals "==" instead of a single one. Building on this, the commands Solve[x^2 - 5*x + 3 == 0, x] Solve[x^3 - 5*x^2 + x == 1, x] Solve[x^4 - 5*x^2 + x == 1, x] Solve[x^5 - 5*x^2 + x == 1, x] Solve[{3*x - 5*y == 1, 2*x + y == Solve[Sin[x] == 1/2, x]
11}, {x, y}]
all attempt to solve equations. Enter them one by one to see the results. (Note that the 2nd-to-last one puts several equations in curly brackets, as well as several variables to solve for. What kinds of problems have we done recently in which that kind of functionality would be useful?) Which would you say result in no answer at all? Which would you say give "usable" answers? Now, put a capital `N' in front of each of the commands, as in NSolve[x^2 NSolve[x^3 NSolve[x^4 NSolve[x^5 5*x + 5*x^2 5*x^2 5*x^2 3 + + + == 0, x] x == 1, x] x == 1, x] x == 1, x]
etc. The upshot is that Solve is uncompromising about seeking only exact answers, whereas NSolve, whose inputs are identical, looks for decimal approximations (approximations even to the non-real answers).
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Intersections of Polar Functions
A familiar idea is that, when we want to know where two functions of x intersect, we may determine this by setting them equal and solving. For
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Mathema...