Getting Started With MapleMaple is what is known as a "computer algebra system" often abbreviated "CAS". The difference between a CAS and a typical calulator (even graphing calculators) will be expanded upon later. Simply put however, a computer algebra system does "symbolic" manipulations. Ie. it can perform operations on expressions containing variables that have not been given a numerical value. In addition, Maple has a built in word processor that allows you to write text, including mathematical notation, in the same document containing mathematical input and output. The Maple Worksheet combines these two features. Click on the "+" sign to open the section.<Text-field layout="Heading 1" style="Heading 1">The Maple Worksheet - Text and Executable Groups:</Text-field>The file you are reading is called a Maple Worksheet. It is seperated into text and executable "groups". Each group is delineated by the square bracket to the left of this paragraph. You can choose between text groups and executable groups by clicking on the "T" or the "[>" located on the tool bar (the yellow boxes). The paragraph you are reading is a "text" group. If you want Maple to perform some type of operation, you can switch to an executable group by clicking on the "[>" from the tool bar. Then you can type in a command for Maple to execute. For example:3 * 5;After you execute this command, the result is displayed centered in blue (in this example, "15" is the Maple output) and Maple wil give you an executable" [>" prompt for the next command. If you wish instead to put in some text, just click on the "T" from the tool bar and the executable prompt goes away.. then you are, by default, in a text group. Note, the semicolon after each command is necessary. This lets Maple know you are done entering a command. This may be replaced by a colon and the output is supressed. For example: 3 * 5:If you do not end the statement with a colon or semicolon, you will get an error message:3 * 5Get used to this ... you will probably see it often. If this happens, or some other error message, you can edit the line by left clicking the mouse on that line, or moving up to it by using the arrows on your keyboard. Make the necessary changes and then re-execute the command by hitting "enter" while the cursor is anywhere on the command line before the colon or semicolon. # end of section<Text-field layout="Heading 1" style="Heading 1">Basic Commands, Defining Variables, Defining Functions </Text-field>If you are reading this from a downloaded Maple worksheet you must hit "enter" somewhere on each command line (in red) before the colon or semicolon. This is called "executing" the command. If you don't execute each command, the values (or expressions) assigned to variables will not be made and this will lead to problems. I generally start each session with the command "restart". This clears the assignments made to any variables and allows you to "start from scratch" without opening a "new" worksheet. restart;Notice, there is no output from this command. Assigning a value to a variable. You must use "colon equals (:=)" to assign a value to a variable. The "=" sign alone is a test character and returns a value of true or false. Below, "a" is assigned the value 10.a := 10;and the equal sign results in the rather strange statement:a = 3;This can be tested as a "true" or "false" by the command "evalb"evalb(a = 3);Despite the "assignment" a = 3, the value is still the one assigned with the ":=". Notice:a;Calculator Operations: Create a new variable "b" and assign it the value 132 by:b := 132;Multiplication must use the " * " symbol:a * b;You will probably forget this. Below is what happens if you do. a b;Division is as you would expect:a/b;Notice, Maple simplifies but does give a decimal expansion. In order to do this, type evalf(a/b);The above command evaluates 5/66 as a floating point number with 10 significant digits. evalf(5/66,n) gives the result with n significant digits. evalf(5/66,3);Notice: Preceding zeros do not count as significant digits and evalf rounds the result. Stored Functions: Maple has many functions stored in its memory. The list is long and we will investigate them as needed. Here we only look at the simple trig and exponential function. Trig functions are as you would expect and NiMlI1BpRw== is denoted Pi. sin(Pi/3);Maple will give you an exact value for trig expressions. If it cannot, it will reiterate your input: cos(Pi/13);If you want a decimal approximation to this number, use evalf:evalf(cos(Pi/13));The exponential function NiMtJSRleHBHNiMlInhHis executed by exp(x). Therefore, you can get Maple to evaluate the number e by:evalf(exp(1));Defining a function: This is rather odd notation but you can think of it as "f takes x and assigns it to". f := x -> (3*x^2 + 2*x + 5);Do not forget the * for denoting multiplication. 3 x does not mean "3 times x". You must use 3 * x. Also, do not forget the ":" in front of the "=" and of course "->" created by typing in the "-" followed by the " >" symbol . You can now evaluate f(x) at x = 3 with the command: f(3);Clearing Variables and comments. To clear the value, or expression, assigned to a variable, you must use one of the following.a := 'a';unassign('f');The latter does not confirm your command but "f" is cleared"f(3);A # sign may be used for comments and everything after it is ignored by Maple:3*a + b; # Hi, How are you?Notice that variable a has been cleared but b still has a value of 132. # end of section<Text-field layout="Heading 1" style="Heading 1">Plotting Functions</Text-field>You can plot a function with the "plot" command that has the form: plot(f(x), domain, etc.). Where the "etc." may be a range, a color designation or many other options. For example we plot the following function: f := x -> 3*x^2 + 2*x + 5;plot(f(x),x=-5..5,color=green,linestyle=SOLID,thickness=2,title="A Sample Plot", labels=["x","f(x)"]);You should label the axes in all graphs. If you're only plotting one function, the "color" and "linestyle" are rather irrelevant. However, if you are printing up a graph, the thickness command is useful as the default thickness may be hard to see on a printed document.Plotting more than one curve at once. More than one function may be plotted on the same graph by listing them, and their features, in square brackets. For example, we introduce another function g(x) = NiMtJSRleHBHNiMlInhH by g := x -> exp(x);plot([f(x),g(x)],x=-5..5,color=[red,blue],linestyle=[DOT,SOLID],thickness=[2,2]);Here, the color and linestyle are important. If you are printing on a color printer the color can differentiate which curve is which. However, if you are printing in black and white, you must differentiate the curves by their style. Inserting a legend is done by right-clicking somewhere in the graph window and choosing "legend". You can left-click anywhere on the graph and in the upper left you will see the coordinates of the cursor. Combining graphs with the "display" command. We illustrate this with an example regarding inverse functions. We know that arccosine is the inverse of cosine over an appropriate domain. Here we do this by combining graphs with the display command. Notes: End the first two lines with colons. Load the "plots" library with the command "with(plots)". Add legends. graph1 := plot([x,cos(x)],x=0..Pi,linestyle=[DOT,DASH],color=[black,red],thickness=1):graph2 := plot(arccos(x),x=-1..1,linestyle=SOLID,color=green,thickness=1):with(plots):display({graph1, graph2});From this figure it certainly appears as though arccosine is a reflection of cosine about the line y=x, as it should. Plotting Functions of more than one variable. You may define a function of two variables, such as area, by:A := (l,w) -> l*w;A(4,12);We will not be doing much of this but its cool to see how Maple plots a function of two variablesplot3d(A(x,y),x=0..3,y=0..3,axes = boxed, labels = ["length","width","area"]);Place your cursor in the box containing the graph. left click and drag. This rotates the image. I think this is cool. <Text-field layout="Heading 1" style="Heading 1">The "Algebra" in Computer Algebra Systems</Text-field>Here you will see what makes a computer algebra system so special. There are many features other than those demonstrated here but you should get a good idea what Maple can, and cannot, do. restart; # this clears our previous variables.Solving Equations: A simple linear equation can be solved for x by:solve(3*x + 5 = 7,x);Here you first type in the equation, and then the variable you want to solve for. Hopefully, you could have done this by hand. Furthermore, your calculator probably could do the same thing. So .. this is not so special. However, suppose we have a line described by the equations "a x + b y = c", where no numercal values are yet assigned to a, b or c. We can find the y-intercept by solving the equation for y when x is set equal to zero. yint := solve(a*0 + b*y = c,y);You can also solve the same linear equation for the x intercept by solving for x when y is set equal to zero. xint := solve(a*x + b*0 = c,x);Maple can solve more difficult equations such as solve(cos(x) = sin(x), x);Not bad, but Maple doesn't let you know that you can add any integer multple of NiMlI1BpRw== to this solution to obtain another solution. Still, its a nice starting point. It is not hard to come up with an equation that Maple cannot solve algebraically. Notice from the graph below that NiMvLCYlInhHIiIiKiQtJSRjb3NHNiNGJSIiI0YmLSUkZXhwR0Yq at NiMvJSJ4RyIiIQ== .plot([x + (cos(x))^2, exp(x)],x=-1..1); But Maple has a hard time finding this solution algebraically. solve(x + (cos(x))^2 = exp(x),x);Does this equal zero. Its hard to say. Lets try evaluating it as a floating point number: evalf(%); Notice, the % symbol denotes the last Maple output, and it turns out that the "RootOf ...." is equal to zero. However, what is hidden in this sequence of commands is the fact that Maple did not find this solution algebraically. It actually used a numerical routine to find where these two curves intersect. Simplify. Maple can take the pain out of simplifying some expressions. For example Maple knows the most common identity from trigonometry: NiMvLCYqJiklJGNvc0ciIiMiIiIlInhHRilGKSomKSUkc2luR0YoRilGKkYpRilGKQ==. simplify((cos(x))^2 + (sin(x))^2);But, you don't have to look very far to find an expression that can be simplified easily by hand but stumps Maple. For example, the laws of logarithms shows that the expression given byexpression := ln(exp(x)) - ln(exp(-x));equals 2 x. Confirm this. But when Maple is asked to simplify this expression bysimplify(expression);It just returns the original expression. Lesson: Maple can't solve all of your simplification needs. Expand and Factor: Maple can expand and factor some polynomial expressions. For example, you can have Maple multiply monomials by expand((x-2) * (x-2) * (3 - x));Maple can then factor this cubic polynomial. factor(%);That's pretty nice because factoring anything greater than quadratic polynomials by hand can be very time consuming. # end of section<Text-field layout="Heading 1" style="Heading 1">Assignment </Text-field>First. Open a new worksheet by clicking on "File" from the menu bar and choosing new. Start by clicking on "T" from the tool bar and type your name, class, and section. Then click on the "[>" from the tool bar and generate two the two plots described next. Plot 1. Plot the functions f(x) = NiMsJiokSSJ4RzYiIiIjIiIiIiImRig= and g(x) = NiMtJSRleHBHNiMlInhH on the same graph over a domain that includes where these curves intersect. Make sure that when printed up, these two curves are differentiated by color (if you have a color printer) or by linestyle = "SOLID, DASH, DOT, or DASHDOT" and that the thickness is large enough to be easily read. Use a legend to differentiate the two curves or label them by hand. Label the point (by hand) where the two curves intersect (round to two decimal places). This point of intersection can be found by using the "solve" command and your coordinates must be correct to the first two decimal places. Plot 2. Plot the equations y = sin(x), y = x, and y = arcsin(x) over an appropriate domain that illustrates reflection about the line y = x. Remember arcsine is the inverse of sine only over the domain [-NiNJI1BpR0kqcHJvdGVjdGVkR0Yk/2, NiNJI1BpR0kqcHJvdGVjdGVkR0Yk/2] and range [-1,1]. Finally. Edit out the commands used to generate the graphs. Print it. This must not be more than one page. If your graphs turn out to be too big, you can adjust the size by clicking on the graph and shrinking the borders. I do not want to see the commands you used to generate the graphs.