2805 Reputation

14 Badges

17 years, 207 days

Dr. Robert J. Lopez, Emeritus Professor of Mathematics at the Rose-Hulman Institute of Technology in Terre Haute, Indiana, USA, is an award winning educator in mathematics and is the author of several books including Advanced Engineering Mathematics (Addison-Wesley 2001). For over two decades, Dr. Lopez has also been a visionary figure in the introduction of Maplesoft technology into undergraduate education. Dr. Lopez earned his Ph.D. in mathematics from Purdue University, his MS from the University of Missouri - Rolla, and his BA from Marist College. He has held academic appointments at Rose-Hulman (1985-2003), Memorial University of Newfoundland (1973-1985), and the University of Nebraska - Lincoln (1970-1973). His publication and research history includes manuscripts and papers in a variety of pure and applied mathematics topics. He has received numerous awards for outstanding scholarship and teaching.

MaplePrimes Activity

These are answers submitted by rlopez

The Tangent command in the Student Calculus1 package will do exactly what you want. It will provide the equation of the tangent line and also draw a graph showing the curve and the tangent line. No need to write any code. Just look at the help page: ?Tangent


First, draw a graph of the left and right sides of the inequality.

plot([abs(sin(x)-x),4/1000],x=-0.3 .. 0.3)

Then, solve explicitly for those x values that make the inequality an equality. Two choices here:



The solve command returns a RootOf structure that does not readily yield exact solutions. Hence, the evalf option.


The odeplot command in the plots package is designed for exactly this task.

Assign a name to the dsolve command, as in


Then, use the odeplot command with, for example, the following syntax.


Access the help page for odeplot with ?odeplot to see the full range of options provided by this graphing tool.

Make use of the Lines&Planes tools in the Student Multivariate Calculus package to find the distance from the unknown line to each of the known lines. Each distance must be zero if the intersections are to occur. Thus, two equations in the three unknown parameters of the direction vector for the unknown line. Solve these two equations for two unknowns in terms of the third. But the length of the direction vector is immaterial. Choose a convenient value for that third parameter.


#Define the three lines in this exercise as "Line Objects" in the Maple package.


#Determine the distance from the unknown line to each of the two given lines.


#Set each distance to zero, and solve for, say, b and c in terms of a.

S:=solve({d1,d2},{b,c}) assuming a<>0:
#Insert the values of b and c into the unknown line, and return its equation in parametric form.

#The choice of a=6 removes fractions and the arbitrary length of the direction vector.

The conversions between input forms are in the Format menu. If you start with ALT R to launch the Format menu, then you will see the hot keys for "Convert To: and then 2-D Math Nonexecutable, 2-D Math Input, and 1-D Math Input. (If you use the mouse to launch the Format menu, the hot keys are not underlined.)

So, for example, if you type an expression in 2-D Math Input, use ALT-R-v-i to convert it to linear (1-D) input form.

Student[Calculus1][Roots]( (f, t = 0 .. 2*Pi, numeric=false );

I think you want an implementation of the Method of Lines (MoL). It's not built into Maple, but a 1-dimensional example can be found in the following video


in which Dr. Sam Dao shows how to use Maple's dsolve command to obviate the need to solve the time-dependent ODEs by RK4. In fact, he shows how to implement MoL using many other of Maple's labor-saving devices.

In a forthcoming webinar (November or December) I will show how to extend Dr. Dao's methods to a PDE over a 2D (spatial) region.


To assign a value to a name, use the syntax name := value, not just name = value.

Maple provides "literal subscripts" that are in essence, Atomic variables, and "indexed" subscripts that are essentially table values. If your Maple is set to form indexed variables when you subscript, then note that you should not assign to a name such as y if you intend assigning to a subscripted version of y. It is because of this "gotcha" that the literal subscript was made the default in Maple. You can see all such atomic variables if you select "Atomic Variables" in the View menu. All atomic variables show up in a maroon color.

The Maple syntax y[1] creates an indexed variable. If you had assigned to y[1], you would then not want to assign to y itself. The name y[1] pretty-prints as a subscripted y, so it can easily be confused visually with the Atomic version.

Assuming that your Maple is set up in its default mode, your subscripted variables will be Atomic, and you can ignore the complications referred to above.

I didn't find where y_sub_one was assigned a value. Hence, in the plot command, that quantity is unknown to Maple.

Finaly, in the plot command, the square brackets around what you want to graph are superfluous. Square brackets represent a list. You can graph what's in a 1-element list, but making a list for the graph of a single expression isn't necessary.


And, by the way, the error message is telling you precisely what the problem is. It's saying that y[1] contains no t, the variable that the plot command is looking for.

For many years I have used "Advanced Find and Replace" by Abacre Software to manage all my Maple files. Although it's a commercial product, not a Maplesoft product, it has served me very well as I rummaged through the thousands of files in the AEM ebook and the three Study Guides I have maintained for Maplesoft.

At the left edge of the lower toolbar for a graph, there's a drop-down box. It shows the word "Plot" by default, but the box also contains the option "Drawing." So, the Drawing toolbar still exists in Maple 2021.

For graphs drawn with the Plot Builder, the Drawing toobar is the default toolbar because all the Plot options are provided by the Plot Builder.

Also, after an initial scare that the animation toolbar no longer was available in Maple 2021, further experiments (Windows 10) verify that this toolbar does appear for animations created with the Plot Builder and with the plots:-animate command.


Under the radicals, the RootOf expressions require an explicit solution of a fifth-degree polynomial. In the 1800s, such formulas were proven not to exist. What you want is a mathematical impossibility. I believe that is the "something small" that has escaped you.

Let's simplify the issue by considering polar coordinates.

Points in polar coordinates are represented in the VC packages as "vectors," sums of components times fake unit vectors in a rectangular version of the polar plane. Thus, the point where r=a, theta=b has a polar representation in terms of unbarred basis vectors e_r and e_theta. This allows the polar point to be represented in this rectangular polar plane as a "position vector" in that plane, a vector from the "origin" of that plane to the "point" (r,t). This is what is meant by the "vector" r*e_r+t*e_t, where these basis vectors are the unbarred basis vectors.

That's why the VC packages also have barred basis vectors as the moving, point-wise determined basis vectors needed for vector fields. When changing coordinates in a vector field, the basis vectors also have to change. But this is not the case for "vectors" that represent points and use the unbarred basis vectors.

MapToBasis(Vector(<r,t>,polar),cartesian[x,y]) produces the Cartesian vector (and hence, the Cartesian point (r cos(t), r sin(t))) expressed with the unbarred basis vectors e_x and e_y that can be interpreted as i and j.

Thus, the help page is correct when it says just components are converted because the argument to PositionVector is never a VectorField, just a Vector that represents the points along a curve or surface. 

I do not see how Equation A becomes Equation B. Eqn A has c^2 on the right, but it becomes c^4 in Equation B. Could that be why the Equation Manipulator gave a messy result and not Equation B?

Here's the device I use. Not super convenient, but it works.

Into a GUI table place the input that creates the equation you want displayed. Execute the input. Then, use the table properties dialog to hide the input. (It's a checkbox near the bottom.) You can also hide the bounding lines of the table if you wish.

Re-execution of the worksheet re-executes code hidden in a table cell, so the equation label is re-constructed.

Execute the command 


The help page for this Task Template will be returned. You can use it to determine the radius of convergence of any power series. Convergence at an endpoint always has to tested separately.

The Task Template itself can be accessed directly from Tools/Tasks/Browse/Calculus-Integral/Series/Radius of Convergence.

1 2 3 4 5 6 7 Last Page 1 of 26