Mathematica Basics. Exponential Functions Exp[expression] Natural Logarithms (ln) Log[expression]

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1 Mathematica Basics To evaluate a Mathematica command, press [Shift]+[Enter]. Pay attention to spaces! Mathematica interprets some of them as multiplication. Syntax, capitalization and punctuation are meaningful. Mathematica is case-sensitive. Basic Calculations Mathematica as a calculator Addition 3+4 Trigonometry Cos[Pi/3] Subtraction 3-4 Sin[Pi/3] Multiplication 3*4 3(4) 3 space 4 Tan[Pi/3] ArcCos[1/2] ArcSin[1/2] Division 3/4 ArcTan[Sqrt[3]] Square Roots Sqrt[34] Factorial 3! Exponents 3^4 Absolute Value Abs[-3] Parenthesis 3*(3+4) Exponential Functions Exp[expression] Natural Logarithms (ln) Log[expression] E^(expression) Logarithms Log[base,expression] Defining a Variable x=expression Note: Mathematica outputs exact answers as fractions and roots. To output numerical approximations, use: N[expression] or expression//n Example: N[Pi] or N[Pi,50] Simplifying and Solving Equations Mathematica as a Computer Algebra System Simplifying: Expand[(x+3)(x-4)] Factor[x^2-x-12] Cos[x]/Tan[x]-Sin[ArcCos[x]] Solving: *note the double-equal-sign* Solve[x+4==2x-3] Solve[x^2+5x+7==0] Solve[Log[2x]==2Log[3x], x] Solve[x+4-3y == 2x 3+y/3, y] solves for y in terms of x Sometimes a message is produced with your result: Solve[3^x==5, x] For a numerical approximation, use NSolve: NSolve[3^x == 5, x] Defining a Function To define a function f 1 (x), follow the structure of the example. Note the underscore, _, after the variable. f1[x_]=x^2+1 You can now use this to calculate several values f1[3] f1[4] f1[vector] You can use multiple functions together f1[x_]=x^2+1; f2[x_]=x^3-1; f1[3]+f2[2] The semi-colon is used to link steps. Only the last result will show.

2 Graphing The following are some of the different ways to graph ListPlot (can be used to graph data points) ListPlot[{{x1, y1 },{x2, y2 }...{xn, yn }}] ListPlot 0, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 5, 6, 8 Plot (can be used to graph 2-D functions; can show single or multiple functions) Plot[f(x), {x, lower x limit, upper x limit }] Example:: Plot 1 x, x, 0, 2 Plot[{f(x)1, f(x)2,...,f(x)n}, {x, lower x limit, upper x limit }] Plot Sin x, Cos x, Tan x, x, 0, 2 Pi Plot3D (can be used to graph 3-D functions) Plot3D[f(x,y), {x, lower x limit, upper x limit }, {y, lower y limit, upper y limit }] Plot3D x y, x, 0, 2, y, 0, 2 Plot3D Cos x Cos y Cos x Sin x Sin y, x, 0, 2 Pi, y, 0, 2 Pi ParametricPlot (for parametric functions) ParametricPlot[{fx(u), fy(u)}, {u, lower u limit, upper u limit }] ParametricPlot Sin u, Cos u, u, 0, 2 Pi PolarPlot (for polar functions) PolarPlot[r(Θ), {Θ, lower Θ limit, upper Θ limit }] PolarPlot Sin 1 t, t, 0, 2 Pi

3 Table Perhaps the easiest way to explain the command "Table" is to show it first. Table n, n, 1, 10 Watch what happens. Table lists the number "n" from n=1 to n=10. Basically, it listed the numbers 1 to 10. Note that the output is a list. What happens if instead of listing n, we list n^2? Table n^2, n, 1, 10 What if we didn't want integer intervals, but rather intervals of 0.1? Table n^2, n, 1, 10,.1 Let's incorporate some lists in here. What do you think will happen here? Table n, n^2, n, 10, 10, 2 Is it just me or are these lists starting to look like coordinate points? hmm..let's take a look. ListPlot Table n, n^2, n, 10, 10, 2 Hey, this looks almost like a parabola! Wow! What a surprise... What if we wanted to turn this into a teaching tool? First we're going to define the function f as f(x)=x^2. The command is "f[x_]:=x^2;". Then we're going to define our list. f x : x^2; mylist Table x, f x, x, 10, 10, 2 ; Row TableForm mylist, TableHeadings None, x, x^2, ListPlot mylist, ImageSize 200 OK, it looks complicated, but we can take this one part at a time. You're smart. First, we're defining the function f(x) as f(x)=x^2. We do this by typing "f[x_]:=x^2;". Quick exercise, how can you define g(y) as g(y)=y+4? Next, we're defining our list and simply naming it "mylist". So according to this, mylist is the Table of {x,f[x]} from x=-10 to x=10 with an interval of 2. Remember what f[x] was? The next line of code is actually formatting. The command "Row" simply puts the graphics in a row. The command "TableForm" actually puts the table in the form of a table. Creative name for a command eh? The "TableHead - ings" part essentially names the table. So the first part is the actually table listing the points x and x^2. The other part is actually plotting the points on the graph. We do this by "ListPlot" and ListPlot plots the points in the list "mylist". Remember what was in "mylist" again? The "ImageSize" just makes the image larger. Can you do the same thing with g(y)=y+4 listing the table and plotting the points on a graph too? Go ahead, you can do it.

4 Manipulate ANATOMY OF A MANIPULATE What's the most powerful command in Mathematica? The question-mark query. It takes the form of?<name of command>. Try it now.? Manipulate But what does that mean? Let's break down the basic Manipulate command. Manipulate expr, u,u min,u max. Manipulate itself is a wrapper: you wrap it around an expression (expr) to give it interactivity by changing values (u). A basic example: Manipulate x, x, 0, 1 Okay, not terribly interesting. But remember, "expr" doesn't have to be a single variable like we used. Anything can go into a Manipulate: plots, tables, charts, graphics, whatever, and every aspect of what you put in can be interactively manipulated. Got a trig class to teach? RANDOM EXAMPLES ARE EXAMPLES "expr" doesn't have to be the same as u, you know... Manipulate Row "e ", N E, places, places, 1, 100 Want someting more...um, useful? Try this. Manipulate Plot Sin b x, x, 5, 5, b, 0.1, 5 Ah, I hear you. Any trignometric equation requires more than one parameter? Like y =a sin(b(x -c)) + d? Try this. Manipulate Plot a Sin b x c d, x, 0, 2 Pi, PlotRange 6, a, 1, 3, b, 1, 8, c, 0, 2 Pi, d, 2, 2 Suppose you have a function that makes no sense without integer values. No problem. Manipulate Expand x y ^n, n, 0, 10, 1 Or maybe you have, ah, exotic needs. You name it, Mathematica has a control type for it. Manipulate Plot x^2, x, 3, 3, PlotStyle colors, Thickness num, colors, ColorSlider, num, 0, 0.5 Note how Mathematica chooses the correct input: sliders for numbers, buttons for options...2-d sliders!? Manipulate pt, pt, 1, 1, 1, 1 3D graphics are no problem either. As long as you have a computer that can keep up. Manipulate Plot3D Sin n x y ^2, x, 0, 3, y, 0, 3, Mesh None, n, 1, 20 More advanced? This is what an expert Mathematica user can create, using Locators. Hold alt and click to add a point to the plot, and watch interpolation work! Manipulate Plot InterpolatingPolynomial ptlist, x, x, 10, 10, PlotRange 10, ptlist, 0, 0, Locator, LocatorAutoCreate True

5 Target Example Simple example of using Graphics to make a circle. Graphics Red, Circle 0, 0, 1, Axes True This generates a list of circles with radii varying between 1 and 10. Table Graphics Red, Circle 0, 0, n, PlotRange 10, n, 1, 10 This shows them all together in the same image. Show Table Graphics Red, Circle 0, 0, n, n, 1, 10 Change the Circle to Disk and now they are all filled in. Show Table Graphics Red, Disk 0, 0, n, n, 1, 10 Make a list of red disks for odd radii and white disks for even radii. Remove the semi-colon ( ; ) at the end of each line if you want to see the output. reddisks Table Red, Disk 0, 0, 2 n 1, n, 0, 10 ; whitedisks Table White, Disk 0, 0, 2 n, n, 1, 10 ; The command below shows them all together. The trouble is that all of the white disks are stacked on top of all of the red disks. Graphics reddisks, whitedisks To alternate elements from each list, we will use the command Riffle. See the example below. Riffle a, b, c, d, 1, 2, 3, 4 This shows them all together, but the bigger ones are on top of the smaller ones. Graphics Riffle reddisks, whitedisks If we reverse the order of the list we just made with Riffle, the smaller ones will be listed last. Let's make this interactive. Wrap it all in Manipulate and change the "10" at the end of the Table to "nmax." (Note: you can copy-andpaste code from earlier in the notebook.) Add a comma to separate the control {nmax, 1, 20}. Manipulate reddisks Table Red, Disk 0, 0, 2 n 1, n, 0, nmax ; whitedisks Table White, Disk 0, 0, 2 n, n, 1, nmax ;, nmax, 1, 20 For a little more fun, make the colors variable. Change "Red" to "clr1" and "White" to "clr2" and then add ColorSlider controls for each one as shown below. Manipulate reddisks Table clr1, Disk 0, 0, 2 n 1, n, 0, nmax ; whitedisks Table clr2, Disk 0, 0, 2 n, n, 1, nmax ;, nmax, 1, 20, clr1, Red, ColorSlider, clr2, White, ColorSlider