transl: do not assume a catch's mode based on the last body form

[maxima.git] / archive / books / schelter / 408c-1.bk

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((face (maxima-eval 2416 2436 3485 3523 3902 4053 4258 4408 4509 4510 4964 5068 5090 5318 5332 5460 5504 5564 6064 6167 6735 7191 7604 7760 8363 8414) (book-shell-eval 2107 2126 2170 2185) (dfplot-eval 1761 1789 1959 2005) (book-result 1274 1275 1447 1626 3013 3177 6534 6561 7393 7411 8057 8118 8541 8600) (maxima-eval-insert 1257 1260 1415 1437 2987 2999 6474 6522 7340 7384 7972 8047 8475 8530) (book-elisp-eval 536 563) (bold 26 39 5567 5579)) (book-command-arg ("xv /d19/staff/wfs/izic-maxima/moebius.gif" 2170 2185) ("xv /d19/staff/wfs/izic-maxima/klein.gif" 2107 2126) ((write-file "~/408c-1.bk") 536 563)))\f
                        408c Tutorial

The highlighted regions of this tutorial are sensitive to clicking
with the mouse with the right button.   Use the arrow keys to move
up or down in the tutorial.

For this assignment you should find the following:

1) integral of sec(x)^5
2) upper sum for x^3+2*x on the interval [2,4] using
partition [2,2.3,2.5,2.8,3.2,4].

[Notation: x^2 means "x to the power 2" and a*b means "a times b"]

Please type your name on the next line:

NAME:
DISCUSSION HOURS: 

Once you are finished click here: <write file to ~/408c-1.bk>
to record what you have done.

You should modify the integrate command below, and the upper_sum
command to do what you are asked to do.   Note for the upper_sum
you must click on the definitions preceding it, so the system
will have them available.  Otherwise you will not get an answer.

This mode allows expressions in the editor to be evaluated in a
symbolic algebra system such as Maxima, or to otherwise cause an
action.  You may cause the action by moving to a highlighted region
and either clicking with the right button, or if you prefer
you may type the two character sequence `Ctrl-c e'.   All special
key sequences in this mode begin with Control C.

For example if you do this for 2+3 then the box 5 will have
its value set to 5.   If you edit the 3 to become a 4, and reclick,
then the value will change to 6.


A more complex example is integrate(1/(1+x^3),x) yields 

                                         2 X - 1
                       2            ATAN(-------)
                  LOG(X  - X + 1)        SQRT(3)    LOG(X + 1)
                - --------------- + ------------- + ----------
                         6             SQRT(3)          3

Try changing the input from 1/(1+..) to be x/(1+..) and click again!

An example of plotting trajectories and direction fields is

   ode{d[x,t]=y,d[y,t]=-sin(x)}

where mouse-2 on the plot window computes an orbit.   Typing any character
does a quit, and records the plot in ~/dfplot.ps.

An xrange or yrange may be specified:

   ode{d[x,t]=y,d[y,t]=-sin(x)};set xrange [-4,4]


It is also possible to view pictures.  To see a bottle with no
interior or exterior by clicking on <view klein bottle> or to see a
surface with only one side see <moebeius band>.

These are images produced by MAXIMA, and then saved to a gif file.
The mathematical formula for this image is a few paragraphs down.

An example of a simple 3d plot using the system defaults
is found by doing

                   plot3d(2^(x^2-y^2))

This uses the default plotter izic.  When the graph comes up you may
use the mouse to scale the image, and also to rotate it.

 Scale objects:  mouse left,  move up to grow image, down to shrink.
 Move object:    mouse middle, move up to grow image, down to shrink.
 Rotate objects: mouse right and move to rotate.

Most things can be done with menus in izic.  Look at the help, but if
you have a colour display, then do ctrl-z after you have found a nice
orientation of your object.  This will bring up a nicely shaded image.

For Maxima the value plot_options evaluates to 
       [[X, - 3, 3], [Y, - 3, 3], [GRID, 30, 30], [VIEW_DIRECTION, 1, 1, 1], 

[COLOUR_Z, FALSE], [TRANSFORM_XY, FALSE], [RUN_VIEWER, TRUE], 

[PLOT_FORMAT, ZIC]]
  
affects the default behaviour.  One can also use GEOMVIEW and PS as
output formats instead of zic.  In every case a file
maxout.{plot_format} is created in your current directory.

To view the plot with other options, these are supplied after the
expression to be plotted.  For example

                   plot3d(2^(x^2-y^2),[x,-1,1],[y,-2,2])

will plot the expression 2^(x^2-y^2) for x in the interval [-1,1] and
for y in [-2,2].  The second and third arguments to plot3d must always be
the first variable range and the second variable range.  These ranges
determine the names of the variables which are used for the first and
second coordinates respectively.  
The other options may occur in any order.  For example:

 plot3d(r^.33*cos(th/3),[r,0,1],[th,0,6*%pi],['grid,12,80],['plot_format,zic],
  ['transform_xy,polar_to_xy],['view_direction,1,1,1.4],['colour_z,true])
  
This transforms to polar coordinates the first 2 variables.  It uses
a grid of 12 x 80.   The other options dont affect the ZIC plot, but
would be significant if we changed the format to postscript.

 plot3d(r^.33*cos(th/3),[r,0,1],[th,0,6*%pi],['grid,12,80],['plot_format,ps],
  ['transform_xy,polar_to_xy],['view_direction,1,1,1.4],['colour_z,true])

To get ghostscript to go away you need to do M-x book-maxima-interrupt
or click on the control-c:  \x03.  If you are viewing with ghostview just
use the menu provided.

While we are doing the plots, we can still use the editor.  It is not
waiting for the plot to complete.  This is because we are not wanting
to insert a result.  The face for the above command is 'maxima-eval'
which does not wait, or try to insert a result.  'maxima-eval-insert'
evaluates the expression and inserts the result in the next result
box.

Another example is a moebius band:

plot3d([cos(x)*(3+y*cos(x/2)),sin(x)*(3+y*cos(x/2)),y*sin(x/2)],
   [x,-%pi,%pi],[y,-1,1],['grid,50,15])

or a klein bottle:

plot3d([5*cos(x)*(cos(x/2)*cos(y)+sin(x/2)*sin(2*y)+3.0) - 10.0,
          -5*sin(x)*(cos(x/2)*cos(y)+sin(x/2)*sin(2*y)+3.0),
           5*(-sin(x/2)*cos(y)+cos(x/2)*sin(2*y))],[x,-%pi,%pi],[y,-%pi,%pi],
          ['grid,40,40])

or a torus

plot3d([cos(y)*(10.0+6*cos(x)),
           sin(y)*(10.0+6*cos(x)),
           -6*sin(x)], [x,0,2*%pi],[y,0,2*%pi],['grid,40,40])

We can output to gnplot too:

             plot3d(2^(x^2-y^2),[x,-1,1],[y,-2,2],[plot_format,gnuplot])


Riemann sums

Let f(x) be a continous function of one variable, then we wish to
study the upper and lower riemann sums for a partition.  A partition
is a subdivision of an interval into a number of subintervals, and can
be specified by the list of end points of the subintervals.

To calculate an upper sum we must find the maximum of a function
on a closed interval.  We know how to do this if the function is
differentiable.  [Click on the following definitions so that they
will take effect].

critical_points(expr,var):= block([numer:true,f:diff(expr,var)],
    realroots(num(f)*denom(f),10^-7));


where realroots finds the real roots of a single variable polynomial
to a given precision, and diff takes the derivative.   The above will work
for rational functions, since in general we need to include as critical points
roots of the numerator and denominator.

An example of the realroots function is
block([numer:true],realroots(x^5+3*x^1+1,10^-7)) returns
   [X = - 0.33198902010917664]  

The numer flag says give a numerical (floating point) answer.  Now the
maximum must occur either at a critical point or at an end point of
the interval, so we can write

function_max(expr,var,a,b,critical_points):=
   block([y:max(at(expr,var=a),at(expr,var=b)) ],
          for v in critical_points
             do (v:at(x,v),
             if a<v and v<b and at(expr,var=v) > y then y:at(expr,var=v)),
          y);
upper_sum(expr,var,partition):=
  block([crit:critical_points(expr,var)],
     sum(function_max(expr,var,partition[i],partition[i+1],crit)*
             (partition[i+1]-partition[i]),i,1,length(partition)-1));

In the above at(x^2+3,x=5) gives 28 .  

We can now compute the upper riemann sum for the expression x^2
for a partition of the interval [0,2]

   upper_sum(x^2,x,[0,.3,.5,.8,.9,1.8,1.9,2.0]) returns 4.0270000000000001 


We wish to form the lower riemann sum as well and to compare it to
the upper riemman_sum.   The next function returns a list containing
the upper sum, the lower sum, and their difference.

upper_and_lower_sums(expr,var,partition):=
   block([up:upper_sum(expr,var,partition),
          low:-upper_sum(-expr,var,partition)],
    [up,low,up-low]);

You can try refining the partition by adding more items to it.  Each
time you add an item the two sums get closer, with the upper one
getting smaller and the lower one getting bigger.   Can you explain
this?
 
upper_and_lower_sums(x^2,x,[0,.3,.35,.4,.7,.8,1.0,1.2,1.3,1.5,1.7,1.9,2.0]) 

yields [3.0591249999999999, 2.3066249999999999, 0.75249999999999995] 

They are still quite far apart.  It is tedious to add the partition
elements by hand, and it also will give an incorrect answer if we
accidentally misorder them!.  The following function makes a partition
into n evenly spaced sub intervals.

make_partition(a,b,n):=makelist(a+(b-a)*i/n,i,0,n);

Trying with a partition with a larger number of elements,

upper_and_lower_sums(x^2,x,make_partition(0.0,2.0,60)); 

returns [2.733703703703704, 2.6003703703703707, 0.1333333333333333]