## 2448 Reputation

14 years, 230 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.

## There is no finite max...

In the objective function, replace either x or y with its value from the constraint. In either case an upward-oening parabola results. Each such parabola has a minimum, but no finite maximum.

Using Maple's built-in LagrangeMultiplier command, the single extrema is at (4/3, 2/3).

## evaluation rules are complicated...

Maple's evaluation rules are complicated. In addition to the issues raised, note that Maple will immediately apply the distributive law to 2*(x+1)  but not to a*(x+1). Learn to live with these design decisions.

For your second question, the evalf command converts an exact expression to a numeric (floating-point, or deciman) one. Hence, with evalf, you get the equivalent of 1.0, a floating point number. Without evalf, you get 1, the exact value.

And by the way, there's one other way to get the exponential e: If you are entering in typeset (math) mode, type e and press the Esc key to bring up a list of all things beginning with the letter e. The very first one is the exponential e. Select that and you have the exponential e. (The Escape key performs command completion, which can also be implemented from the Tools menu.

## Arc Length tutor...

In the Tools menu, select Tutors/Calculus-Single Variable/Arc Lengths. In this pop-up tool (housed in the Student Calculus1 package) simply enter the expression for the curve and the x-coordinates of the endpoints between which the arc length is to be calculated. If Maple can't produce a closed-form solution, it will return a numeric value of the arc-length integral.

At the bottom of the tutor you can see the actual command that produces the graph shown. To use this command, copy it and paste it into your worksheet. Be sure to install the Student[Calculus1] package. If the output option in the command is changed to "integral" the arc-length integral is returned. If this option is not there, the value of the integral is returned. (Or use output=value.)

## General case vs specific instance...

If a combine/trig is applied to the integrand in the general case, four cosine terms appear. The arguments in each case are x times one of (K-L-M), (K-L+m), (K+L-M), (K+L+M). When K=14, L-2, M=12, that first term is cos(0)/4, and hence the integral is Pi/4. If none of these factors is zero, and each integer is even, the integral will be zero.

In the general case, add the option AllSolutions=true to the int command and this anomaly will be completely clarified by the piecewise result that is returned.

## It works for me...

I wrote this Task Template, so I'm instantly on alert if someone claims it doesn't work. I just tried it in Maple 2017.3 and it works.

According to the second bullet point on the left, press the Initialize button to make this template work.

To see the code, right-click on the Initialize button and select the option "Edit Click Action."

If there's a problem with this task template, please let me know.

## Use Drawing Tools...

Faced with this task, I'd probably use the Drawing Tools accessed by selecting Drawing in the plotting toolbar after clicking on the graph. It's tedious, but one could enter into a Drawing-Tools text box one letter at a time, hitting the return key after each letter. Clearly, one has to be desperate to go this route, but it would work in a pinch.

The unhappy thing about using the interactive drawing tools is impermanence. If the code that produced the graph is re-executed, the drawings on it are not preserved. I get around this by exporting the annotated graph as a PNG file, then importing that back into Maple. Of course, this gives me a graph that can't be re-executed, and could be lost if accidentally deleted.

## Defaults to the complex field...

In the LinearAlgebra package, the DotProduct command defaults to the complex number field. Maple chooses to conjugate the second vector. Include the option conjugate=false to have the DotProduct calculate over the reals.

There is no DotProduct command in the Student LinearAlgebra package; use the period as an infix operator. This operation defaults to the reals.

There is a DotProduct command in the Student MultivariateCalculus package that defaults to the reals.

Finally, since you ask why conjugation for the dot product when the vectors can be complex: without conjugation, the dot product of a complex vector with itself can be zero. This would then violate the definition of that operator (zero only if the vector is the zero vector). It requires conjugation of one of the vectors to prevent this. Some books will define the dot product with the first vector conjugated, some with the second. Maple conjugates the second.

For the sake of completeness: the norm of <1,I> without conjugation would be zero. With conjugation, it is sqrt(2), as expected.

## testplot...

Maple's algorithm for axis-label placement is a percentage of the axis length. Changing the graphing ranges shifts the position of the label. I tried it with your graph. It works, but the distortion might not be acceptable.

When faced with similar challenges where it really mattered, I suppressed the labels from the plot command and placed my own labels with the textplot command. As far as I know, Maple has never addressed the issue of user-control of axis-label placement.

## Control+Shift+R...

This works in every version of Maple, Maple 15 included.

The first "How do I" question answered in the Maple Portal's Student version is "How do you set a piecewise function in Maple?" The detailed example that gives the answer shows how to add an additional "row" or rule to the piecewise template inserted from the Expression palette.

## Typesetting Rules Assistant...

Select "Typesetting Rules" in the View menu to launch the Typesetting Rules Assistant. In the lower left corner make the two changes shown in the following figure.

The exact syntax needed for this is somewhere in my Little Red Book of Maple Magic, but that's no longer handy; so I tend to use the Assistant rather than go look up the syntax.

## NewtonsMethod command...

If all you want  is to see the iterates, use the NewtonsMethod comand in the Student Calculus1 package. It has an option to return a graph (or an iteration) of the approximation process. Better yet, use the Newton's Method Tutor available from the Tools/Tutors menu. This tutor implements the NewtonsMethod command in a syntax-free way.

## list or set...

I'm going to guess that instead of supplying to dsolve a single list or set, inside the set braces you have grouped the equations and/or initial conditions as a list or set.

## By parts, stepwise...

The following Maple code will apply integration by parts repeatedly, generating the asymptotic expansion "stepwise." I did not see the asymptotic expansion in Maple's FunctionAdvisor.

with(IntegrationTools):
n:=5:
q := Int(1/ln(t), t = 0 .. x);
for k to n do
q := simplify(Parts(q, GetIntegrand(q)));
end do;

The Parts command applies integration by parts to an inert integral (set with Int) or to an expression containing such. The second argument is the factor that is to be differentiated. That factor is the integrand of each integral appearing, so the GetIntegrand command simplifies the coding. Of course, n can be set to any desired positive integer.

## Calculus Study Guide...

Section 1.2 of the Calculus Study Guide deals with the epsilon-delta definition of a limit. The approach taken involves solving equalities rather than inequalities. Use equations such as f(a+delta_right)=L+epsilon and f(a-delta_left)=L-epsilon to obtain delta(epsilon) for an increasing function whose limit at x=a is L. Maple's ability to solve equations for the bounds on delta is greater than its ability to solve the inequalities inherent in the epsilon-delta definition of a limit. For a nonlinear function f, delta_right and delta_left will in general be different; the trick is to pick the smaller of the two bounds to obtain a single delta(epsilon) needed in the definition.

RHIT where I developed most of my Maple materials is primarily an engineering school. Interpret that as it is intended; purists might feel that no calculus course can ever be complete without mastery of the epsilon-delta definition of a limit. The Calculus Study Guide devotes a single section to this definition on the grounds that those in need of a study guide probably do not need to master this definition in the first calculus course. But please, no flaming here. I'm just expressing my opinion in the freedom of retirement.

## Built-in tools...

The built-in command

Student:-VectorCalculus:-TNBFrame(<cos(t),sin(t),t/10>*3/2, 'output'=animation, 'axes'=frame, 'range'=0 .. 4*Pi, 'frames'=30,caption="");

will produce an animation of the TNB frame moving along the helix in Rouben's worksheet. In fact, the animation could be created with the SpaceCurves Tutor, and the command that generates the animation copied and pasted from the bottom of the tutor. The tutor (and the underlying command) will work for other than Cartesian coordinates.

This is not to take away from Rouben's exemplary coding skills, but for many novices, learning where the built-in tools lie can be useful.

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