Mario,

I'm not going to touch Alec's comments, but I do want to add my support to ALEX's comments. It's not realistic to give a "rigorous but simple foundation of the derivative of functions". Even powers, x^n, can be problematic. Integer powers, including negative integers, are not difficult. Rational powers can be handled without too much trouble. But, irrational powers are a different story. Think about it, what do you mean when you write x^(sqrt(2)).

How do you compute this for a specific value of x, say x=2, other than using a computational device?

The standard answer is to write it using exponentials and logarithms: x^sqrt(2) = exp(sqrt(2)*ln(x)). So, even the derivative of powers needs to use exponentials and logarithms.

It's not necessary to have a complete mastery of all of the details of these arguments to appreciate the complexity of the issues. These difficulties do not need to be an obstacle to the learning of the main concepts of calculus. You do not have to be dishonest with the students; I just tell them that if they are intrigued by these issues they should consider taking higher-level MATH courses.

Many people underestimate the complexity of this matter. It does not disqualify anyone from being an effective educator. I think you'll find people on MaplePrimes willing to help you better understand the issues at hand, if you are interested.

Doug

---------------------------------------------------------------------
Douglas B. Meade  <><
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu       
Phone:  (803) 777-6183         URL:    http://www.math.sc.ed

Mario,

I'm not going to touch Alec's comments, but I do want to add my support to ALEX's comments. It's not realistic to give a "rigorous but simple foundation of the derivative of functions". Even powers, x^n, can be problematic. Integer powers, including negative integers, are not difficult. Rational powers can be handled without too much trouble. But, irrational powers are a different story. Think about it, what do you mean when you write x^(sqrt(2)).

How do you compute this for a specific value of x, say x=2, other than using a computational device?

The standard answer is to write it using exponentials and logarithms: x^sqrt(2) = exp(sqrt(2)*ln(x)). So, even the derivative of powers needs to use exponentials and logarithms.

It's not necessary to have a complete mastery of all of the details of these arguments to appreciate the complexity of the issues. These difficulties do not need to be an obstacle to the learning of the main concepts of calculus. You do not have to be dishonest with the students; I just tell them that if they are intrigued by these issues they should consider taking higher-level MATH courses.

Many people underestimate the complexity of this matter. It does not disqualify anyone from being an effective educator. I think you'll find people on MaplePrimes willing to help you better understand the issues at hand, if you are interested.

Doug

---------------------------------------------------------------------
Douglas B. Meade  <><
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu       
Phone:  (803) 777-6183         URL:    http://www.math.sc.ed

Using printlevel gives you all of the output. If you want to see only selected results you should insert explicit print statements.

Another way to control output is to use the userinfo and infolevel commands. The userinfo command specifies the "level" required to have the output displayed and what to print. The infolevel command controls the setting of the "level" which is used to decide when to print the information in the userinfo. Unfortunately, to the best of my knowledge, userinfo only displays messages, not 2D output. But, you can mimic this functionality with flags that you set and check on  your own.

For example,

MyTest := (a,b) -> isprime(a) and isprime(b):
for i from 1 to 10 do
  for j from 1 to 10 do
    a[i,j] := i+j;
    if MyTest(i,j) then print( i, j, a[i,j] ) end if;
  od;
od;
                                   2, 2, 4
                                   2, 3, 5
                                   2, 5, 7
                                   2, 7, 9
                                   3, 2, 5
                                   3, 3, 6
                                   3, 5, 8
                                  3, 7, 10
                                   5, 2, 7
                                   5, 3, 8
                                  5, 5, 10
                                  5, 7, 12
                                   7, 2, 9
                                  7, 3, 10
                                  7, 5, 12
                                  7, 7, 14

I hope this is helpful.

Doug

---------------------------------------------------------------------
Douglas B. Meade  <><
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu       
Phone:  (803) 777-6183         URL:    http://www.math.sc.ed

This reminds me that:

sin(x)
------  = sin
   x

Doug

---------------------------------------------------------------------
Douglas B. Meade  <><
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu       
Phone:  (803) 777-6183         URL:    http://www.math.sc.ed

This reminds me that:

sin(x)
------  = sin
   x

Doug

---------------------------------------------------------------------
Douglas B. Meade  <><
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu       
Phone:  (803) 777-6183         URL:    http://www.math.sc.ed

If you want to utilize the commands from the student or Student'Calculus1] packages, you are going to have to type them out - or rely upon command completion. If you want to use the basic commands (diff, eval, and solve) then you have some options. For diff and eval, you can use the palettes (clickable). You can probably use the context menus to access solve, but only after you explicitly construct the equation that you want to solve. You have some flexibility in how you actually enter the commands. Personally, I find it fastest to simply type the commands (and use command completion for the commands with longer names).

Doug

---------------------------------------------------------------------
Douglas B. Meade  <><
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu       
Phone:  (803) 777-6183         URL:    http://www.math.sc.ed

If you want to utilize the commands from the student or Student'Calculus1] packages, you are going to have to type them out - or rely upon command completion. If you want to use the basic commands (diff, eval, and solve) then you have some options. For diff and eval, you can use the palettes (clickable). You can probably use the context menus to access solve, but only after you explicitly construct the equation that you want to solve. You have some flexibility in how you actually enter the commands. Personally, I find it fastest to simply type the commands (and use command completion for the commands with longer names).

Doug

---------------------------------------------------------------------
Douglas B. Meade  <><
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu       
Phone:  (803) 777-6183         URL:    http://www.math.sc.ed

To remove the minus signs, use simplify:

isolate( eq, P^2 );
                                          3  
                                 2       a   
                                P  = - ------
                                       -m - M
simplify( % );
                                        3  
                                  2    a   
                                 P  = -----
                                      m + M

(normal and factor also work, but not expand)

Doug

---------------------------------------------------------------------
Douglas B. Meade  <><
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu       
Phone:  (803) 777-6183         URL:    http://www.math.sc.ed

To remove the minus signs, use simplify:

isolate( eq, P^2 );
                                          3  
                                 2       a   
                                P  = - ------
                                       -m - M
simplify( % );
                                        3  
                                  2    a   
                                 P  = -----
                                      m + M

(normal and factor also work, but not expand)

Doug

---------------------------------------------------------------------
Douglas B. Meade  <><
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu       
Phone:  (803) 777-6183         URL:    http://www.math.sc.ed

You are a student learning calculus. Let's keep the focus on calculus and make use of basic Maple commands (not specialized commands that completely avoid the relevant mathematics).

Here is how I would use Maple to answer each of your three questions:

1.)  Find the slope of the tangent line to the curve y=x^3+3x-8 at (2,6)

F1 :=x^3+3*x-8;
                                 3          
                                x  + 3 x - 8
dF1 := diff( F1, x );
                                     2    
                                  3 x  + 3
m1 := eval( dF1, x=2 );
                                     15

2.) The tangent line to the circle y=x^3 -6x^2 -34x -9 has slope 2 at two points on the curve. Find the two points.

F2 := x^3-6*x^2-34*x-9;
                             3      2           
                            x  - 6 x  - 34 x - 9
dF2 := diff( F2, x );
                                 2            
                              3 x  - 12 x - 34
solve( dF2=2, x );
                                    6, -2
p1 := eval( [x,F2], x=6 );
                                  [6, -213]
p2 := eval( [x,F2], x=-2 );
                                  [-2, 27]

3.)Find the slope of the curve y=x^5 at x= -2

F3 := x^5;
                                      5
                                     x 
dF3 := diff( F3, x );
                                       4
                                    5 x 
m3 := eval( dF3, x=-2 );
                                     80

Notice that I answered these questions using only three basic Maple commands: diff, eval, and solve. No extra packages and no fancy optional arguments.

I hope this is helpful,

Doug

---------------------------------------------------------------------
Douglas B. Meade  <><
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu       
Phone:  (803) 777-6183         URL:    http://www.math.sc.ed

You are a student learning calculus. Let's keep the focus on calculus and make use of basic Maple commands (not specialized commands that completely avoid the relevant mathematics).

Here is how I would use Maple to answer each of your three questions:

1.)  Find the slope of the tangent line to the curve y=x^3+3x-8 at (2,6)

F1 :=x^3+3*x-8;
                                 3          
                                x  + 3 x - 8
dF1 := diff( F1, x );
                                     2    
                                  3 x  + 3
m1 := eval( dF1, x=2 );
                                     15

2.) The tangent line to the circle y=x^3 -6x^2 -34x -9 has slope 2 at two points on the curve. Find the two points.

F2 := x^3-6*x^2-34*x-9;
                             3      2           
                            x  - 6 x  - 34 x - 9
dF2 := diff( F2, x );
                                 2            
                              3 x  - 12 x - 34
solve( dF2=2, x );
                                    6, -2
p1 := eval( [x,F2], x=6 );
                                  [6, -213]
p2 := eval( [x,F2], x=-2 );
                                  [-2, 27]

3.)Find the slope of the curve y=x^5 at x= -2

F3 := x^5;
                                      5
                                     x 
dF3 := diff( F3, x );
                                       4
                                    5 x 
m3 := eval( dF3, x=-2 );
                                     80

Notice that I answered these questions using only three basic Maple commands: diff, eval, and solve. No extra packages and no fancy optional arguments.

I hope this is helpful,

Doug

---------------------------------------------------------------------
Douglas B. Meade  <><
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu       
Phone:  (803) 777-6183         URL:    http://www.math.sc.ed

piecewise evaluates arguments only as needed.

The prototypical example is

F := x -> piecewise( x=0, 1, sin(x)/x ):

If all arguments were evaluated when piecewise is called with x=0, this would produce an error. But, it doesn't.

F(0);
                                      1

Doug

---------------------------------------------------------------------
Douglas B. Meade  <><
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu       
Phone:  (803) 777-6183         URL:    http://www.math.sc.ed

piecewise evaluates arguments only as needed.

The prototypical example is

F := x -> piecewise( x=0, 1, sin(x)/x ):

If all arguments were evaluated when piecewise is called with x=0, this would produce an error. But, it doesn't.

F(0);
                                      1

Doug

---------------------------------------------------------------------
Douglas B. Meade  <><
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu       
Phone:  (803) 777-6183         URL:    http://www.math.sc.ed

Peter,

I do not believe odeplot has an explicit log-plot option. But, you can create this by putting the log of whichever component you want to have plotted. One of the nice things about odeplot is that you can specify any combination of dependent and independent variables in the ODE system. I have used this to study conserved quantities, phase plots, .... With a little practice, I'm confident you will be able to produce a plot that suits your needs. If not, post what you have and see if someone on Mapleprimes can help you out.

Good luck,

Doug

---------------------------------------------------------------------
Douglas B. Meade  <><
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu       
Phone:  (803) 777-6183         URL:    http://www.math.sc.ed

Peter,

I do not believe odeplot has an explicit log-plot option. But, you can create this by putting the log of whichever component you want to have plotted. One of the nice things about odeplot is that you can specify any combination of dependent and independent variables in the ODE system. I have used this to study conserved quantities, phase plots, .... With a little practice, I'm confident you will be able to produce a plot that suits your needs. If not, post what you have and see if someone on Mapleprimes can help you out.

Good luck,

Doug

---------------------------------------------------------------------
Douglas B. Meade  <><
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu       
Phone:  (803) 777-6183         URL:    http://www.math.sc.ed