Mathematica quick reference

• To execute Mathematica code, press Shift-Enter on the keyboard or just Enter on the number pad.
• Also, Mathematica is case-sensitive. That means capital and lower-case letters matter.
• You may edit previously executed code by moving the mouse to it and making changes as you would in a word processor. However, the new output will erase the original output.

You may copy-and-paste boldface code on this page directly into Mathematica.
If you have trouble running some samples, select the text here, copy, go to Mathematica, and choose "Paste As...Plain Text" from the "Edit" menu or right-click menu.
Use Alt-Tab to quickly flip between Mathematica and your Internet browser.

Contents:
Basic Calculations
Simplifying and Solving Equations
Defining a Function

Graphing Curves and Surfaces
Basic Animation
Graphing Coordinate Points
Bar Graph
Pie Chart

Calculus

Need More Help?

Basic Calculations
Mathematica does all of the basic functions of a calculator.

• Addition: 3 + 4
  Subtraction: 3 - 4
  Multiplication: 3 * 4 or 3(4) or 3 space 4
  Division: 3 / 4
• Square Root: Sqrt[34]
• Exponent: 3^4
• Parentheses: Use ( and ) only
• Trigonometry: Cos[Pi/3], Sin[Pi/3], Tan[Pi/3], ArcCos[1/2], ArcSin[1/2], ArcTan[Sqrt[3]], Sec[Pi/3], Csc[Pi/3], Cot[Pi/3], ArcSec[2/Sqrt[3]], etc.
  Default is radians. To use degrees: Cos[60 Degree]
• Factorial: 4!
• Absolute Value: Abs[expression]
• Exponential Function (e): Exp[expression] or E^(expression)
  Natural Logarithm (ln): Log[expression]

Note: Mathematica gives you exact answers—fractions and reduced roots.

To get a numerical approximation use the function: N[expression] or expression //N
Example: N[Pi] or N[Pi, 50] to approximate pi to 50 decimal places.

Contents

Simplifying and Solving Equations
Mathematica will expand and factor expressions, reduce trigonometric expressions, and solve equations.

Simplifying

Expand[(x+3)(x-4)]
Factor[x^2 - x - 12]
Cos[x] / Tan[x] - Sin[ArcCos[x]]

Solving — Be sure to use two equals signs.

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 an error message is produced with your result: Solve[3^x == 5, x]
NSolve[3^x == 5, x] gives you a numerical approximation to the result.

Contents

Defining a Function

• To define a function f₁(x), follow the example. Be sure to use an underscore "_" after the variable.
  f1[x_] = x^2 + 1

• You can now use this to calculate several values.
  f1[3] produces the output "10" as expected.

• As a sample mini-program:
  f1[x_]=x^2 + 1;
  f2[x_]=x^3 - 1;
  f1[3]+f2[2]
  The semi-colon suppresses the intermediate output.

Contents

Graphing Curves and Surfaces
Below is some sample code you can follow to create some basic graphs.

Plot[Cos[x], {x, -Pi, Pi}]

Plot[x^2 - 3, {x, -4, 4}, PlotRange -> {-10, 10}]

Plot[{x, x^2, x^3}, {x, -1, 1}, PlotStyle -> {Thickness[.005], {RGBColor[1,0,0], Thickness[.01]}, Thickness[.02]}]

Graph with a Legend
To use this next sample, select the text, copy, go to Mathematica, and choose
"Paste As...Plain Text" from the "Edit" menu or right-click menu. You may also need to clear the memory for this one: Go to the "Kernel" menu, choose "Quit Kernel...Local", and click "Ok."

<<Graphics`Legend`
f1[x_] = 2^x;
f2[x_] = 4^x;
f3[x_] = 7^x;
Plot[{f1[x], f2[x], f3[x]}, {x, -4, 0},
PlotStyle -> {Dashing[{.01}], Dashing[{.03}], Dashing[{.05}]},
PlotLegend -> {"f1", "f2", "f3"}, LegendPosition -> {1, -0.2}]

In three dimensions:
Plot3D[y*Cos[x], {x, 0, 2Pi}, {y, -3, 3}]

To create an animation, be sure to use "Paste As...Plain Text"

<<Graphics`Animation`
SpinShow[Plot3D[y*Cos[x], {x, 0, 2Pi}, {y, -3, 3}, Boxed->False, Axes->False], Frames -> 20, SpinRange -> {0 Degree, 360 Degree}]

Then double-click on any of the images. You may want to delete the first image for a smoother animation. It is just cycling through the list of images, like the frames of a movie.

You can show two graphs at the same time using "Show" in 2D or 3D.
graph1 = Plot3D[10 - x^2 - y^2, {x, -3, 3}, {y, -3, 3}];
graph2 = Plot3D[-3x + 2y + 6, {x, -3, 3}, {y, -3, 3}];
Show[graph1, graph2, PlotRange -> {0, 10}]

Contents

Graphing Coordinate Points

• To plot a list of coordinates, type it as below. Use "PointSize" to control the size of the points.
  ListPlot[{{3, 4}, {-2, 4}, {4, 8}, {1.5, 0}, {-2.5, -4}}, PlotStyle -> PointSize[0.05]]

• To plot the points of a function, define it and use "Table" as below.
  f[x_] = Exp[x]Sin[x];
  ListPlot[Table[{x, f[x]}, {x, 0, 3, 0.2}], PlotStyle -> PointSize[0.03]]
  The "{x, 0, 3, 0.2}" tells it to make a point for x every 0.2 units between 0 and 3.

• If you use
  PlotStyle -> {PointSize[0.02], RGBColor[0.5, 0, 1]}
  the points will be purple.
  R=amount of red, G=amount of green, B=amount of blue (#s between 0 & 1)

Contents

Bar Graph
The following code produces a bar graph with a legend. You will probably need to enlarge the second graph to see it clearly. Remember, you can cut-and-paste the graph into another program, such as Microsoft Word.

To use this sample, select the text, copy, go to Mathematica, and choose
"Paste As...Plain Text" from the "Edit" menu or right-click menu.

<<Graphics`;
BarChart[{10, 15, 13, 17}, {12, 14, 10, 14},
BarLabels->{"Qtr. 1", "Qtr. 2", "Qtr. 3", "Qtr. 4"},
BarStyle->{Hue[0], Hue[0.6]}, PlotLabel->"Homework Average",
DefaultFont->12];
ShowLegend[%, {{{Hue[0], " Per. 1"},{Hue[0.6], " Per. 2"}}, DefaultFont->12, LegendPosition->{1, -0.5}}]

Contents

Pie Chart
The following code produces a pie chart.

To use this sample, select the text, copy, go to Mathematica, and choose
"Paste As...Plain Text" from the "Edit" menu or right-click menu.

<< Graphics`;
PieChart[{15, 10, 25, 50},
PieLabels -> {"Class Work", "Projects", "Homework", "Tests and Quizzes"},
PieStyle -> {Hue[0.2], Hue[0.4], Hue[0.55], Hue[0.82]},
PlotLabel -> "Grading Categories", DefaultFont -> 14]

Contents

Calculus

Derivatives
Taking a Derivative:           D[3x^2 + Sin[x] , x]
Don't forget to specify the variable of differentiation!
Second Derivatives:           D[D[5x E^(x^3/2), x], x]

Integrals
Taking an Antiderivative:    Integrate[3x^2 + Sin[x] , x]
Finding a Definite Integral: Integrate[3x^2 + Sin[x] , {x, -1, Pi/2}]
Approximate this result:      N[%] or use NIntegrate[expression]

Limits
Finite Limit:                        Limit[(x^2-9)/(x-3), x -> 3]
Limit at Infinity:                   Limit[(4x^3 - 2x)/(5x^3 + 3x^2), x -> Infinity]
Limit at Negative Infinity:    Limit[Sqrt[4x^2]/x, x -> -Infinity]
Limit from the Left:             Limit[1/x, x -> 0, Direction -> 1]
Limit from the Right:           Limit[1/x, x -> 0, Direction -> -1]    (Default)

Infinite Series
Convergent:                        Sum[1/n^2, {n, 1, Infinity}]
Numerical Approx.:           NSum[1/n^2, {n, 1, Infinity}]
Divergent:                          Sum[1/n, {n, 1, Infinity}]

Contents

Need More Help?

Try www.Wolfram.com, the publishers of Mathematica.

Extensive demos and ideas may be found at: http://library.wolfram.com.

Technical Support: http://www.wolfram.com/services/techsupport

E-mail Abby: abby.brown@sduhsd.net.

Abby Brown ~ Spring 2002

Return to:

• TOP
• Mathematica Files
• Getting Started With Mathematica
• Helpful Files Ready to Use
• Teaching and Learning

home