Introduction to Maple: Part IIOne of the most powerful features of Maple is its capability to perform symbolic algebraic manipulations and create high quality graphs. The goal of this project is to introduce you to these important features of the Maple software package. Open the Introduction: Part II worksheet (Microsoft Word document) found on my webpage. You will complete Section I of the worksheet as you proceed through the material below. You may complete Section II of Introduction: Part II once you have completed this Maple worksheet.

Maple allows you to algebraically manipulate symbolic expressions. Below are some of the most useful Maple commands that allow us to quickly perform some of the most frequent algebra tasks.

simplify (simplifies an algebraic expression)For example, to simplify NiMqJiwmKiRJInhHNiIiIiMiIiJGKSEiIkYpLCZGJkYpRilGKUYq , we input the command that follows.simplify((x^2-1)/(x+1));The syntax for this command has the same basic format for most Maple algebra commands. That is, the command is followed by parentheses, and the expression upon which the algebra will be performed is placed in the parentheses. Thus the basic syntax for most Maple algebra commands is command(expression);Use simplify to simplify NiMqJiwmKiZJJHNpbkc2IiIiJUkieEdGJyIiIkYqKipGJiIiI0YpRipJJGNvc0dGJ0YsRilGKkYqRiosJkYqRioqJkYtRixGKUYqISIiRjA= , on the command line that follows. Be very careful how you input this expression. Paste your answer into the Introduction: Part II Word document.

expand (expands or multiplies expressions)For example, to multiply NiMqKEkieEc2IiIiIiwmKiRGJCIiI0YmRiZGJkYmLCZGJEYmRiYhIiJGJg== , we input the command that follows.expand(x*(x^2+1)*(x-1));Recall that Maple is case-sensitive; the following command has no meaning in Maple -- the output is basically a repeat of what was input. However, it is sometimes valuable to use the uppercase of a command to see how Maple is interpretting what you entered.Expand(x*(x^2+1)*(x-1));Use expand to expand NiMqJiwmJSJ4RyIiIiIiJkYmIiIkLCZGJUYmRichIiJGJg== on the command line that follows. Paste your result into the Introduction: Part II Word document.

factor (factors an expression)For example, to factor NiMsJiomSSRzaW5HNiIiIiRJInhHRiYiIiJGKSoqRiVGKUYoRilJJGNvc0dGJiIiI0YoRikhIiI= , we input the command that follows.factor((sin(x))^3-sin(x)*(cos(x))^2);Use factor to factor NiMsJiomIiIlIiIiKiQtSSRleHBHNiI2I0kieEdGKiIiI0YmRiYqKEYtRiYtSSNsbkdGKkYrRiZGKEYmISIi on the command line that follows. Paste your answer into the Introduction: Part II Word document.

subs (substitutes a value for a variable into an expression)For example to compute NiMsJi1JJHRhbkc2IjYjSSJ4R0YmIiIiKiZGKEYpLUkkY29zR0YmRidGKSEiIg== where NiMvJSJ4RyUjUGlH, we input the commands as follows.subs(x=Pi,tan(x)-x*cos(x));simplify(%);We see that Maple will not always automatically simplify expressions, so we had to use the simplify command above to get the simplified result.Use subs to compute and simplify NiMtSSNsbkc2IjYjKiYiIiMiIiItSSRzaW5HRiU2I0kieEdGJUYp where NiMvJSJ4RyomJSNQaUciIiIiIichIiI= on the command lines that follow. Paste your answer into the Introduction: Part II Word document.

solve (solves an equation or a system of equations for unknown variable(s))For example to solve the equation NiMvLCgqJEkieEc2IiIiJSIiIiomRihGKSokRiYiIiNGKUYpRihGKSIiIQ== , we would type the command that follows.solve(x^4+4*x^2+4=0,x);Note the output of I which is the Maple expression for NiMtJSVzcXJ0RzYjLCQiIiIhIiI= or i. Use solve to solve the equation NiMvLCgqJEkieEc2IiIiJiIiIiomIiIlRikqJEYmIiIkRikhIiIqJkYtRilGJkYpRikiIiE= on the command line that follows. Paste your answer into the Introduction: Part II Word document.

If we attempt to use solve to find the solutions to the equation NiMvLCgqJEkieEc2IiIiJiIiIiomIiIlRilGJkYpRikiIiRGKSIiIQ== , we input the command as shown below.solve(x^5+4*x+3=0,x);The output indicates that Maple could not find the solutions exactly, and the command that follows will extract approximate solutions using the evalf command discussed in Introduction to Maple I.evalf(%);Use the commands above to solve the equation NiMvLCgqJEkieEc2IiIiKCIiIiomIiIlRikqJEYmIiIkRilGKUYtRikiIiE= . Paste all real solutions into the Introduction: Part II Word document.

fsolve (finds approximate solutions to an equation)For example, if we attempt to approximate a solution the equation NiMsJkkieEc2IiIiIi1JJHNpbkdGJTYjKiYiIiRGJkYkRiYhIiI= = 0 using solve and evalf, we would type the commands that follow.solve(x-sin(3*x)=0,x);evalf(%)We obtain a solution of 0. While this is a solution, there are two other solutions that the steps above did not yield. Using the fsolve command we can approximate the other solutions. fsolve(x-sin(3*x),x);Using fsolve as in the above example only yields 0 as the solution; however, we can give Maple an interval in which to search for the other solutions. The following input instructs Maple to look for a solution to the equation on the interval from -1 to -0.25 (the interval must be entered as x = smallest .. largest).fsolve(x-sin(3*x),x=-1..-0.25);Use the fsolve command to find the last solution to this equation (the solution is positive). You can save having to retype the entire command by using cut-and-paste (i.e., copying the command above, pasting below, and modifying) or by simply modifying the command above and re-executing. Paste the positive solution into the Introduction: Part II Word document.

You may assign a variable name with the colon-equal symbol (:=). You may use any variable name that you want that is not a preprogrammed Maple command. For example, we will assign joe to be NiMsJi0lJHNpbkc2IyUieEciIiItJSVzcXJ0R0YmISIi.joe:=sin(x)-sqrt(x);Once we have defined joe as above, we may use the algebraic commands discussed previously on joe.expand(joe^2);That is, the previous result shows that NiMqJEkkam9lRzYiIiIj = NiMqJCwmLUkkc2luRzYiNiNJInhHRiciIiItSSVzcXJ0R0YnRighIiIiIiM= = NiMsKComSSRzaW5HNiIiIiNJInhHRiYiIiJGKSooRidGKS1GJTYjRihGKS1JJXNxcnRHRiZGLEYpISIiRihGKQ==. Below are other examples of algebraically manipulating joe. Be certain that you understand what algebraic manipulation is being performed in each case.subs(x=Pi^2,joe);simplify(%);fsolve(joe=joe^2-1,x=1..5);Use Maple to find the approximate value of the square root of joe evaluated at NiMvJSJ4RyUjUGlH. Paste your answer into the Introduction: Part II Word document.

You may obtain the graph of Maple expressions using the plot command. For example to plot the graph of NiMqJkkkc2luRzYiIiIjSSJ4R0YlIiIi on the interval from 0 to 2NiMlI1BpRw==, we execute the command that follows. plot((sin(x))^2,x=0..2*Pi);You may also specify the y-interval over which to view the graph; the command follows.plot((sin(x))^2,x=0..2*Pi,y=-1..1);Use plot to obtain the graph of NiMtSSNsbkc2IjYjKiRJInhHRiUiIiM= on the interval from -5 to 5.Execute the following two plot commands.plot(1/(x+1),x=-4..2,y=-4..4);plot(1/(x+1),x=-4..2,y=-4..4,discont=true);The first plot includes a vertical line that is not actually part of the graph. (The line is not exactly a vertical line, but we will call it vertical for this discussion.) This vertical line results from Maple's default method of plotting graphs (i.e., plotting points and connecting the dots). In cases where the function we are plotting has a denominator, we can override Maple's default plotting method by using discont = true as done in the second plot command above. This will eliminate any vertical lines that are not actually part of the graph.

The plot command can be used to plot several graphs on the same set of axes. In the commands below, we define f to be NiMqJi1JJHNpbkc2IjYjKiYiIiMiIiJJInhHRiZGKkYqLUkkYWJzR0kqcHJvdGVjdGVkR0YuNiNGK0Yq, we define g to be NiMqJiwmKiYiIiMiIiIlInhHRidGJ0YnRidGJy0lJGV4cEc2I0YoRic=, and we plot both graphs on the same set of axes.f:=sin(2*x)*abs(x);g:=(2*x+1)*exp(x);plot({f,g},x=-1..1);Suppose we wanted to approximate the x-coordinates where the graphs of f and g intersect. We can use the plot and fsolve commands to obtain the approximation. We plot the two graphs on the interval from -3 to 1 using the command below.plot({f,g},x=-3..1);From this plot, it is clear that the two graphs intersect in the interval from -2 to -1, so we use the fsolve command with this interval to obtain one of the x-values where the two graphs intersect.fsolve(f=g,x=-2..-1);Find another point where the two graphs intersect using the plot and fsolve commands. Paste your answer into the Introduction: Part II Word document.

If you have a question about any Maple command, you can use the ? command. Typing ? followed by any Maple command will open a help-page on that command. The page will contain information on proper syntax for that command, explain what the command does, and provide examples and links to similar topics. To ensure that you are using proper syntax, it is often very useful to copy the examples on the help-page, paste them into the worksheet, and modify them for the problem you are considering. For example, suppose you wanted to plot the graph of NiMtJSRzaW5HNiMlInhH in yellow. The following command will provide us with a help-page on the various options for the plot command which will indicate how to obtain the yellow plot of NiMtJSRzaW5HNiMlInhH. After you have determined how to obtain such a plot, type the proper command on the second line that follows.?plot,options;

You may also use the pull-down Help menu found at the top of the page. This menu has a search feature which is useful when you are unsure whether Maple has a certain command to carry out a specific task that you are performing.

You may right click on any output to view potential ways that Maple can manipulate the output. You may simply select the desired command, and Maple will enter the syntax for that command and execute the command for you. For example, first execute the command that follows. Next right click on the output and choose Factor. Then right click on that output and choose Plots followed by 2-D Plot . Right click on the plot and choose Axes then Range to adjust the axes to the intervals -3 to 3 for both x and y. H:=x^6-3*x^5+x^4+3*x^3-2*x^2;

Complete Section II of the Introduction: Part II worksheet (Microsoft Word document) found on the J drive (ClassroomWork -- Mathematics -- Introduction to Maple folder). You may use Maple to help you complete the assignment.. You may use Maple to help you complete the assignment.