LAB #1: INTRODUCTION TO MAPLE V

Using the UP/DOWN - ARROW keys, you can scroll through the worksheet.Areas with BLACK test (such as this) are TEXT REGIONS.int(sin(x),x);Areas with RED text are INPUT REGIONS.Areas with BLUE text are OUTPUT REGIONS.Note that all input statements must be followed by semicolon (or colon if you don't want the output to be displayed). Try deleting the semicolon from the previous MAPLE statement.

Most users of modern software packages never will buy reference books but rather entirely (or primarily) rely on the on-screen-help. Try clicking on the on-screen help on the right upper corner of the MAPLE window. The alternative is type ? keyword (in an input region), and execute by hitting the ENTER key, see the example below.?integrateCan you copy the examples from the HELP pages into INPUT regions of the worksheet for editing and executing?

MAPLE is a package for SYMBOLIC ALGEBRA, that ALSO can do the usual numerical calculations, and produce first rate graphical output.

Execute the following examples (with or without modification), and memorize the effects. Move the cursor to the next input area, and repeatedly hit the ENTER KEY until you reach the end of this section. Each statement needs to be closed with a semi-colon (or a colon) before MAPLE will execute it. Omitting this semi-colon is the most frequent cause for trouble.restart;2/7-1/9;3+4;Frequently there is a need to go from exact results to approximations by decimal numbers.soln:=solve(x^2+x=3);To list each solution separately trysoln[1];soln[2];Now get a decimal answerevalf(soln);The default number of digits after the dot is 10. To change this number for example to 20 we writeevalf(soln,20);Notice the difference between the := (assignment) and the = (logical equality) in the above statement. Do not confuse these!A handy short-cut is the ditto-symbol:solve(x^3-5*x=2);To obtain a numerical approximation with 12 decimals:evalf(%,12);and with 4 decimalsevalf(%%,4);?"solve(a+b*c=d,b);Notice in the above equation we have specified the variable with respect to which we want to solve the equation.MAPLE is case-sensitive:evalf(Pi);evalf(pi);Multiplication has to be denoted explicitly by * , i.e. juxtaposition does not work:x:=3;y:=2;x * y - xy;Indeed in this case xy is a new variable, just like "x1" or "vol".

Let's look at some basic plotting. To begin enter an expression and tag it with the name f. The assignment operator is the := operator. Note that you have not created a function in the mathematical sense. The notation f(2) will not be recognized by Maple with the assignment made here.restart;f:=x^2;plot(f);No domain was given. Hence, no graph. The correct syntax for the plot command:plot(f,x=-3..3);To gain control over the vertical scale, use a second range in the plot command. Maple interprets the first range as the horizontal scale and the second range as the vertical.plot(f,x=-3..3,y=-1..15);Now suppose we want to examine the intersection of some simple curves and calculate the points of intersection. Enter a second "function" g.g:=7+3*x-5*x^2;Plot both functions on one set of axes to see what to expect for intersections.plot({f,g},x=-3..3);There are two intersections points to compute, and the x-coordinates of each point has magnitude of about one. To compute the points exactly, we can set f and g equal and solve for x.q:=solve(f=g,x);The solve command has returned an expression sequence containing the two roots, listed in sequence. The ease with which teh solutions can be referenced and used is an essential feature of a computer algebra system.q[1];q[2];Now let's see how to obtain the y-coordinates corresponding to the two x-coordinatesjust calculated. Note that mathematically you would "plug in" or substitute the x-value into the appropriate expression.y1:=subs(x=q[1],f);y2:=subs(x=q[2],f);Obviously, there is a need for some simplification. Two options come to mind:we can try to expand the expression for y1 and y2,or we can try to simplify these expressions.simplify(y1);expand(y1);The more natural action is to expand the term being squared. Hence, the expand command produced the more appropriate result.Sometimes it is necessary to work numerically. xx1:=evalf(q[1],30);subs(x=xx1,f);Note that Maple did not transfer to the calculation of y the need for more digits. One way to dictate that outcome is via the evalf command.evalf(subs(x=xx1,f),30);Now let's look at another example where the choice of the correct domain is important.plot({sqrt(9-x^2),-sqrt(9-x^2)},x=-3.5..3.5,y=-3.5..3.5);plot({sqrt(9-x^2),-sqrt(9-x^2)},x=-3..3,y=-3..3);Can you explain the difference between the two plots above?With Maple we can also do implicit plots but in order to do so we need to load the package with(plots);with(plots);implicitplot(x^2+y^2=9,x=-3.5..3.5,y=-3.5..3.5);We can also do 3-dimensional plotsplot3d({sqrt(9-x^2-y^2),-sqrt(9-x^2-y^2)},x=-4..4,y=-4..4);What kind of solid do these equations correspond to? How would you go about filling the "holes" in the plot?plot3d(-x/(1+x^2+y^2),x=-5..5,y=-3..3); plot3d(x*exp(-x^2-y^2),x=-2..2,y=-2..2);The plots package also has a display command that can be used to display multiple plots on the same graph.curve1:=plot(x^2,x=-2..2,color=blue): curve2:=plot(x^3,x=-2..2,color=green):Notice that the output was suppressed when you executed these statements. What has happened is that the actual plotting commands have been stored in the variables curve1 and curve2.curve1;curve2;If you want do display these both in the same picture use the display command:display({curve1,curve2});To get a more accurate picture, use constrained scaling:display({curve1,curve2},scaling=constrained);NOTE: When you label plots as above with curve1 and curve2, it is ESSENTIAL that you suppress the output with a colon. If you put a semicolon, MAPLE will output the list of plotting commands and it will look like what happens here, only perhaps lots longer when you are plotting complex objects.:nocoloncurve:=plot(x^2,x=-2..2);If this happens to you it is a sure sign that you have labelled a plot but didn't suppress the printout. Remember though, if you don't label a plot and use a semi colon it will draw the graph.For extra practice plotting in 3D consider the following examples: implicitplot3d(z=3, x=0..5, y=0..5, z=0..5, axes=boxed);implicitplot3d(x+y=1, x=-3..3, y=-3..3, z=-3..3, axes=boxed);implicitplot3d(x^2+y^2+z^2=1, x=-1..2, y=-2..2, z=-2..2, axes=boxed);

The derivative of a function can be found by using the command diff. The process of computing a derivative is called differentiation, and that iswhere the notation diff comes from.restart;f:=3*x^2-5*x+2;diff(f,x);Let's construct, at the point (2,4), the line tangent to the graph of f(x).d:=diff(f,x);The slope of this tangent line is the value of the derivative at x=2.m:=subs(x=2,d);The equation of the tangent line is thereforey:=m*(x-2)+4;A graph of the tangent line and of the function f is created via plot({f,y},x=0..3);We see that the line through (2,4) with slope 7, as calculated from the derivative, is tangent to the graph of f(x).Let's explore the calculation of a derivative for another function.f:=sin(x);The difference quotient (which represents the slopes of the secant lines) ism:=(subs(x=x+h,f)-f)/h;The limit of the difference quotient is found by the computationlimit(m,h=0);Finally, we compare this result with a direct call to the differentiation command.diff(f,x);REMARK: A common mistake results when a name has been assigned a value earlier. See the example below:x:=2;diff(x^2,x);Consequently requests like differentiating x^2 with respect to x don't make any sense at all (you can't differentiate 4 with respect to 2): everyone runs into this trouble -- there is simply no way a person can keep in mind all the previous assignments. There are two ways: First we may selectively reset x to simply mean x (the letter), and nothing else:x:='x';diff(x^2,x);More radically, we may restart MAPLE, thus wiping out all prior assignments, loaded packages, calculations etc. (but the worksheet is safe!).x:=3;y:=x^2;diff(y,x);restart;y:=x^2;diff(y,x);

There are different ways to define a function in MAPLE:The "assignment" definitionf:=x^2;The value of the function at a point, say x = 2, is then obtained by evaluating the expression f for x = 2 (i.e., we substitute x = 2 in the expression f):subs(x=2,f);The "arrow" definition:f:=x->x^2; Then we can find f(2), f(0), etc.f(2);The "procedure" definition:g:=proc(x) x^2; end;g(2);The procedure definition is useful when we need to define very complicated functions whose definition requires more than one line commands.

Work through the following problems.

AA1: subtract 1/9 from 2/7, and obtain a decimal approximation. AA2: "Factor" the polynomial x^5-1.AA3: "Solve" the equation x^2=x+5/(x-1). AA4: Expand the expression (x^2-3*x+2)*(x^2-5*x+3). Then solve exactly for its roots. If stuck, try help for simplify, expand, collect, combine, .....

C1: Plot the graph of y=sin(5*arcsin(x)) from -1 to 1.C2: Find the fourth derivative of arctan(x) with respect to x. Write the result as a single fraction (combine,simplify,....) (do a help on diff to find out how to compute second, third, fourth,... derivatives)C3: Write the difference quotient for the function sin(x). Compute the limit for h->0 and compare with the value of the derivative computed by using diff.C4: Find an antiderivative of sin(x).

P1: On the same graph, plot the functions NiMqJiIiJCIiIi0lJHNpbkc2IyUieEdGJQ==, NiMtJSRjb3NHNiMqJiIiIyIiIiUieEdGKA==, and NiMqJiklImVHJSJ4RyIiIiIjNSEiIg== in three different colors. Let x go from NiMsJCUjUGlHISIi to NiMlI1BpRw==.Find Maple commands which reproduce each of the following plots as closely as possible. (Most of your effort should go into finding the right functions to plot.) Some combination of pencil-and-paper with trial-and-error will probably work best. P2: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P3: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P4:-%%PLOTG6,-%'CURVESG6cu7$7$$!1+++++++%)!#;$!1++++++!y#!#:7$$!10+++++]')F,$!1************>FF/7$F07$$!13+++++v()F,$!1)*********\#o#F/7$7$$!1RLLLLL$=*F,$!1++++++gDF/F67$7$F=$!1************fDF/7$$!1\WWWWWW%*F,$!1abbbbbb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