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>  restart; 
>  2^500; 
The := assigns a variable name to an expression.
>  p:=4*x^24*x3; 
This will factor the expression.
>  factor(p);

First we give a new polynomial the name p2:
>  p2:=(x+3)*(x2)*(x4)^2; 
Now we multiply it out.
>  expand(p2); 
The size of our expressions is only limited by the computers memory (arbitrary precision).
>  expand((x+y)^20); 
The following is an example of simplifying expressions.
>  p3:=(x^24)^2+(x3)^2;
simplify(p3); 
The following example illustrates the syntax for creating a function. Note the format: name:= variable>expression.
>  f1:=x>x^3;
f1(2);f1(3);f1(4); 
The following is an example of shows how to plot functions.
>  f2:=x>x^2*(x+2)*(x1);
plot(f2(x),x=3..3,y=10..10); 
Here is an example of how to solve an equation:
>  expr1:=x^2+x+1;
solve(expr1=0,x); 
Solving nonlinear inequalities:
>  solve(3*x^24*x4>=0,x); 
Now we will take a first derivative (diff) with respect to x.
>  eq:=3*x^22*x3; 
>  diff(eq,x); 
The second derivative. Note the use of the dollar sign ($).
>  diff(eq,x$2); 
This is the command to access the extensive online help that is available. The help usually includes nice examples. When you have finished looking over the information just close the window to return to this Maple session.
>  help(differentiation); 
Now we will take some limits.
>  f4:=x>x^3;
limit((f4(x+h)f4(x))/h,h=0); 
Here are examples of taking righthand and lefthand derivatives.
>  f5:=x>abs(x)/x; 
>  plot(f5(x),x=5..5,y=1..1); 
>  limit(f5(x),x=0,right);
limit(f5(x),x=0,left); 
We input a rational function.
>  rat:=(3*x^22*x+4)/(6*x^23*x+5); 
We can even take limits as x goes to infinity.
>  limit(rat,x=infinity); 
When written with a capital L, the limit is not evaluated. The output is a nice looking limit expression.
>  Limit((1+1/n)^n,n=infinity); 
Now we will evaluate the
>  limit((1+1/n)^n,n=infinity); 
The evalf command computes the decimal value of its argument. The % refers to the last output. It is like the ANS key on a graphing calculator.
>  evalf(%); 
More limits
>  plot(1/x,x=5..5,y=5..5); 
>  limit(1/x,x=0, right); 
>  limit(1/x,x=0,left); 
Now you solve some problems on your own. Include a function, a plot, a derivative, and a limit. Use ctrl j to enter a few lines so you can do your work.
II. The Rules of Differentiation
We will create three functions ProductRule, Quotient Rule and ReciprocalRule. Each rule will take two functions as input and generate the derivatives associated with each of the rules. For example, ProductRule(x^2,sin(x)) should give 2xsin(x)+x^2cos(x). The syntax for a function with two arguments is f:=(x,y)>expression. Use the diff command for the derivative. Check your results with a few examples. I have done the product rule as an example. You should enter a few lines and do the Quotient Rule and the Reciprocal rule.
>  ProductRule:=(f,g)>diff(f,x)*g+diff(g,x)*f; 
>  ProductRule(x^2,sin(x)); 
>  simplify(%); 
III. Homegrown Derivatives
Create a function using the definition of the derivative to take derivatives.
The definition: f '(x) =
Look back to Part I at the example involving f4 for a hint on how to do this. Try out your function on a few examples including a trigonometric function. Use it twice to take a second derivative. Remember % recalls the last output.
IV. What about Implicit Differentiation?
Read the examples in the help associated with the implicitdiff command. Find some problems in your homework that involved implicit differentiation. Do at least two problems using this command. Check the results.
>  help(implicitdiff); 
V. Creating Pretty Pictures: Plotting Several Function at the Same Time
To plot several function at the same time. Put them in a list like this [f1, f2, f3, ...]. Each will be plotted in a different color. The seq command creates a sequence and can be used to create a list. See the example below.
>  PlotList:=[seq(x^n,n=1..8)]; 
>  plot(PlotList,x=1.5..1.5,y=1.5..1.5); 
Take a few derivatives of
. Suppose you graphed this function and several derivatives on the same graph. Explain what you would expect it to look like.
Describe what you expect to see:
Create a plot like the one above using sin(0.8*x) and its successive derivatives. Does it looks like you expected it too?
VI. A Table for the Derivatives of the Trigometric Functions
We need to learn the derivatives of the trigonmetric functions. To help us we will make a table of them. First create a spreadsheet by going to the insert menu and selecting spreadsheet. Next, label the columns of your spreadsheet. The first column should be labeled function. The second column should be label 1st derivative. The third column should be 2nd derivative. Input the six basic trig functions under the function heading. Remember to include parenthesis around the x. For example type sin(x). Then use the diff command to take the deriviatives of the functions in the first column. To do this you must refer to the previous cell. This is done using the label ~ColumnLetterRowNumber. For example if cos(x) is in cell A2, then to put the first derivative into B3 you would input diff(~B3, x) into B3. Repeat for all the first and second derivatives. If this spreadsheet was easy for you then add the functions exp(x) and ln(x) and their derivatives at the bottom of the list. Do you notice anything interesting?
VII. The Derivative Tutor
Use the DiffTutor command to launch the derivative tutor. Once you enter the commands below it will take a few seconds for Maple to start a maplet (maple applet). This maplet will allow you to input a function and then go through the steps of taking the derivative by clicking on the appropriate step. If you are stuck ask the maplet for a hint. Try out functions that use the sum and difference rules, the power rule, the product rule, the quotient rule and the chain rule. Try it until you can get through them without asking for any hints. Show me one of your results
>  restart;
with(Student[Calculus1]): 
>  DiffTutor(); 