A Brief Introduction to Mathematica

Transcription

1 A Brief Introduction to Mathematica Objectives: (1) To learn to use Mathematica as a calculator. (2) To learn to write expressions in Mathematica, and to evaluate them at given point. (3) To learn to plot one-variable functions. (4) To learn how to use a differential equations solver. Mathematica as a Calculator Mathematica can be used as a calculator. We will start by trying to do with Mathematica the basic operations that a calculator does Basic rules of engagement: In order to execute any command in mathematica, you have to hold down shift and press return. Try now 145 * ˆ The basic operations are + for sum, the asterisk for multiplication, / for division and the caret ˆ to indicate powers. It is important to observe the following for rational numbers: 45/9 5 1

2 45/7 45/7 Mathematica displays the exact answer. If you want an approximation by a decimal number, you can use the Mathematica function N[ ]. N[45/7] You can use round parenthesis as usual in algebra. Do NOT use square brackets, [ ] or curly braces { } as algebraic delimiters, since they have special meaning in Mathematica. Mathematica includes all of the basic functions, e.g. sin, cos, exp. The functions are case sensitive and their arguments go inside braces. For example: Cos[0] 1 The constant π = is denoted by Pi and the symbol E stands for the base of natural logarithms ( = ). Cos[Pi] -1 Mathematica as a symbolic manipulator Mathematica can also perform many symbolic operations between expressions. For example try: f[x ] = 3*xˆ2-12*x+Pi 3x 2 12x + π 2

3 Entering f[x ]=3*xˆ2-12*x+Pi assigns to the function f the expression 3x 2 12x+ π. Mathematica can differentiate the expression f with respect to x : f [x] 6x 12 Mathematica can compute the values of the expression f for a given value of x: f[0] f[1] π 9 + π So the command f[a] substitutes x = a in y. Notice two things: (i) the first substitution did not make Mathematica forget the relationship between x and f, and (ii) Mathematica s answers are not always decimal. In fact Mathematica tries not to evaluate an expression until it is absolutely necessary. If you want to obtain a decimal answer, you should use N[ ] as above: N[f[0]] Hence the command N[f[value]] where value is some number, returns the numerical value of f for a given x. 3

4 Plotting one-variable functions. Mathematica can plot one variable functions very easily: Plot[ 3*xˆ2-12*x+Pi, {x,0,10}] Mathematica will return a window with the graph of the function y = 3x 2 12x+ π for x [0,10]. In general, to plot a function of one variable, the command is Plot[ expression, {variable,minvalue,maxvalue}] Here expression can be something like 3*xˆ2-12*x+Pi, or if you have assigned this expression to a function, say f, then expression can be just f[x]. Also, the variable is the independent variable of f and minvalue and maxvalue are the extreme values of the graph. It s entered in the following way. Suppose the independent variable is x, and you want a plot with x [0,10]. Then, to indicate that x should range from zero to 10 you type {x,0,10}. Notice that these three things must be entered in curly braces separated by commas. As another example, to indicate the range x [0.1, π], type {x,0.1,pi} Example. Plot the function g(x) = sin(2x) with x [ π,π]. SOLUTION 1: SOLUTION 2: g[x ] = Sin[2*x] Plot[g[x],{x,-Pi,Pi}] Plot[Sin[2*x],{x,-Pi,Pi}] Exercise. Plot the function y = x 4 5x 2 + 4, first for x [ 3,3] and then for x [ 10, 10]. Notice how the change in scale changes the overall picture. This is a first example of the limitations of the computer: the pictures provide only approximate information about the real functions. It is sometimes helpful to plot two or more functions simultaneously. Mathematica can do this, provided the range of the independent variable is the same for all functions. Here s how it is done. Suppose you have defined two functions of x, say f and g. To plot them both simultaneously on the interval [a,b], enter Plot[{f[x],g[x]},{x,a,b}] 4

5 Note that the two functions go in the expression field inside curly braces and separated by a comma. You can do this with more than two functions. Example. Plot the function y = x 4 5x and its derivative on the same graph, with x [ 2.1, 2.1]. (Notice that this is the same function as in the previous example the range contains all the local max and min of the function.) SOLUTION: f[x ]= xˆ4-5*xˆ2+4 x 4 5x Plot[{f[x],f [x]},{x,-2.1,2.1}] (Mathematica returns a window with the graphs of f and f.) Notice how the local max, min of f correspond with the zeroes of f, f is increasing where f is positive, etc. Exercises: a) Plot the function y = x 3 2x + 1 for x [ 2, 2]. b) Find a function z = mx + b whose graph is the tangent to the graph of y = y(x) at x = 0. c) Plot both functions simultaneously for x [ 2, 2] and then for x [ 1, 1]. 3d Graphics. Another useful feature of Mathematica is its ability to produce realistic graphics of three-dimensional objects. Try the following exercise: Enter exactly the following Plot3D[Sin[x*y],{x,-Pi,Pi},{y,-Pi,Pi}] You should get a graphic in your output window. You can now rotate it in space, as follows: 1. With the pointer in the 3D screen, click the left mouse button. 2. Holding the left button down, drag the pointer. This has the effect of rotating the graph. 5

6 The display style of the graph can be varied as well by the following. Plot3D[Sin[x*y],{x,-Pi,Pi},{y,-Pi,Pi},Axes->False, Mesh->None] Experiment with the other manipulations of the graph by looking up Plot3D in the help menu. Exercises: a) Plot the function f (x,y) = cos(xy + y), x [ 3,3], y [ 3,3]. b) Find the equation of the plane tangent to f at ( 0, π 2). Plot f and its tangent plan for x [ 3,3], y [ 3,3]. Differential Equations Solver Mathematica has a build in differential equations solver. Follow the steps to solve the example differential equation. The command DSolve[ equation, function,variable] attempts to solve a differential equation in terms of the indicated variable. Try the following examples: DSolve[y [x]==x+y[x],y[x],x] DSolve[{y [x]+y[x] == 0,y[0]==0,y [0]==1},y[x],x] Mathematica also has a numerical differential equation solver. Try the following S=NDSolve[{y [x]==xˆ2-y[x],y[0]==0},y[x],{x,0,10}] Plot[Evaluate[y[x]/.S],{x,0,10}] S=NDSolve[{y1 [x]==-y2[x],y2 [x]==-y3[x],y3 [x]==-y1[x], y1[0]==1,y2[0]==0,y3[0]==1},{y1,y2,y3},{x,0,10}] ParametricPlot3D[Evaluate[{y1[x],y2[x],y3[x]}/.S],{x,0,10}] This gives a numerical solution to the system of differential equations with initial conditions y 1 = y 2 y 1 (0) = 1 y 2 = y 3 y 2 (0) = 0 y 3 = y 1 y 3 (0) = 1 6

7 on the interval [0,10]. Exercises: a) Use the solver to find the solution to the differential equation y = xy(x 1). b) Use the solver to find the solution to solve the differential equation y 2y = 0 with initial condition y(0) = 0, y (0) = 1. Plot your solution on the interval [0,4π]. More useful commands. Use the help menu of the internet to learn how to use each of the following functions. Make some examples that demonstrate how each of the following functions works. a) Solve b) Expand c) Simplify d) Rand 7