/* [wxMaxima batch file version 1] [ DO NOT EDIT BY HAND! ]*/
/* [ Created with wxMaxima version 11.08.0 ] */

/* [wxMaxima: title   start ]
Derivatives, Slopes, Tangent Lines, and
Making Movies 
   [wxMaxima: title   end   ] */

/* [wxMaxima: comment start ]
Chapter 3 Technology Application Project
   [wxMaxima: comment end   ] */

/* [wxMaxima: section start ]
Introduction
   [wxMaxima: section end   ] */

/* [wxMaxima: comment start ]
OBJECTIVE: To visualize the derivative and the linearization 
of a function at a point.

In this module, we explore the derivative as the slope of a 
nonlinear function and find the equation of the line tangent
to a curve at a point. You will learn how to plot the curve 
and selected tangents on the same graph. In addition, you will 
see how to use Maxima to make a movie animation by 
generating a sequence of plots, each showing a different 
tangent to the curve. When the sequence of graphs is animated, 
the tangent lines appear to roll along the graph of the 
function.
   [wxMaxima: comment end   ] */

/* [wxMaxima: subsect start ]
Technology Guidelines
   [wxMaxima: subsect end   ] */

/* [wxMaxima: comment start ]
NOTE:  If you have just finished a document, restart Maxima 
before executing a new document.  This can be done by choosing
"Restart" from the Maxima menu.

TO OPEN OR CLOSE CELLS
Click on the arrow at the top of the cell bracket. 

TO STOP AN EXECUTION
Click on STOP button from the toolbar. 

ORDER OF EXECUTION
Execute commands in the order given. Do not skip any Maxima
Input lines within a given document. 

Alternatively, you can execute the entire worksheet by 
selecting the "Evaluate All Cells" command from the "Cell" drop
down menu or simply press Ctrl-r. 

SAVING WORKSHEETS
You can save anytime to any directory you choose, and it is 
wise to save often.

EXPERIENCING MAJOR PROBLEMS
Save if appropriate, and then shut down Maxima and start it up 
again. 
   [wxMaxima: comment end   ] */

/* [wxMaxima: subsect start ]
You Try It
   [wxMaxima: subsect end   ] */

/* [wxMaxima: comment start ]
First, work through Parts I - V with the example function  
f(x) = x^2, and then repeat the steps in Parts I - V for some 
functions that you select. Here are some suggestions.

1. x^3 for -2<=x<=2
2. sin(x) for 0<=x<=2*%pi
3. %e^(-x^2) for -2<=x<=2
4. sqrt(x) for 0<x<=4  
    (Note that we do not include x=0.  Do you know why?)
   [wxMaxima: comment end   ] */

/* [wxMaxima: section start ]
Part I:  The Derivative at a Point
   [wxMaxima: section end   ] */

/* [wxMaxima: comment start ]
Section 3.1
   [wxMaxima: comment end   ] */

/* [wxMaxima: comment start ]
Define a nonlinear function of your choice and call it f(x) 
and graph it. For an example, we choose the function 
f(x) = x^2 for -2<=x<=2 (To put in a different function and 
domain, change the entries in the following input cell.)
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
numer:true$
kill(all)$
load(draw)$
ratprint:false$
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
x0: -2;
xf: 2;
f(x) := x^2;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
wxdraw2d(
    nticks=200,
    color=blue,
    explicit(f(x),x,x0,xf),
    xlabel="x",
    ylabel="y");
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
Now pick a point on the function and use the definition of the 
derivative to find the slope of the function's graph at the 
point you pick, calling it mtangent. We choose x = 1 for our 
example.
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
mtangent: limit((f(1+h)-f(1))/h,h,0);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: section start ]
Part II: The Linearization of a Function
   [wxMaxima: section end   ] */

/* [wxMaxima: comment start ]
Sections 3.1 and 3.8 (ET Sections 3.1 and 3.10)

Form a new function for the line that is tangent to the 
function that you chose in the "You Try It" section at that 
point you picked in Part I, and call it ytangent = L(x). This 
function is called the linearization of y = f(x) at the 
point (1, f(1)).
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
L(x) := f(1)+mtangent*(x-1);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
wxdraw2d(
    nticks=200,
    color=red,
    explicit(f(x),x,x0,xf),
    color=blue,
    explicit(L(x),x,x0,xf),
    xlabel="x",
    ylabel="y, ytangent");
/* [wxMaxima: input   end   ] */

/* [wxMaxima: section start ]
Part III: The Derivative Function
   [wxMaxima: section end   ] */

/* [wxMaxima: comment start ]
Section 3.1 

Form the derivative function that will give the slope of the 
tangent to your chosen function at any point with coordinates 
(x, f(x)). Call the derivative function yprime, and graph 
it.
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
yprime: limit((f(x+h)-f(x))/h,h,0);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
wxdraw2d(
    nticks=200,
    color=green,
    explicit(yprime,x,x0,xf),
    xlabel="x",
    ylabel="yprime");
/* [wxMaxima: input   end   ] */

/* [wxMaxima: section start ]
Part IV: A Whole Bunch of Tangents
   [wxMaxima: section end   ] */

/* [wxMaxima: comment start ]
Section 3.1 

Form a new Maxima function that gives the equation of the line 
tangent to y = f(x) at the point (a, f(a)). Call the new 
function tanline(x,a).
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
tanline(x,a) := f(a)+subst(a,x,yprime)*(x-a);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
Test your tanline(x,a) for several values of a by plotting the 
tangent lines and y = f(x) together on the same graph. First,
use the tanline(x,a) function and a short loop to generate a 
list of equations for the tangents to the curve at points 
(a, f(a)), for values of a varying from -2 to 2 in increments 
of 0.1. Then graph the tangent lines and y = f(x) together on 
the same graph.
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
numer:true$
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
g: []$
listoftangents: makelist(tanline(x,x0+(xf-x0)*a/20), a, 0, 20);
for i: 1 thru 21 do g: append(g,[explicit(listoftangents[i],x,x0,xf)]);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
p1: explicit(listoftangents, x, x0, xf)$
p2: explicit(f(x), x, x0, xf)$
wxdraw2d(
    nticks=200,
    color=blue,
    g,
    color=red,
    line_width=5,
    explicit(f(x),x,x0,xf),
    xlabel="x",
    ylabel="y");
/* [wxMaxima: input   end   ] */

/* [wxMaxima: section start ]
Part V: Making Movies
   [wxMaxima: section end   ] */

/* [wxMaxima: comment start ]
Section 3.1

The following command generates a sequence of graphs and then 
saves them as an animated GIF file named "tanplot_anim.gif". 
This resulting file can typically be found in the following 
locations:

mac:  in your applications folder
windows: C:\Program Files\Maxima-5.24.0\wxMaxima 

Note that here we are using the draw( ) function instead of the 
draw2d( ) function.  This is because of our need to create a 
graphic which is composed of several scenes (thus creating our
animation).  The draw2d( ) function is designed to create a 
graphic using only one scene.  Thus draw2d( ... ) and 
draw( gr2d(...) ) do the same thing.
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
b: 1$
tanplot: []$
for a: 1 thru length(listoftangents) do block(
    tanplot: append(tanplot, [gr2d(
        nticks=200,
        color=blue,
        explicit(listoftangents[a],x,x0,xf),
        color=red,
        line_width=5,
        explicit(f(x),x,x0,xf),
        xlabel="x",
        ylabel="y",
        yrange=[-12,3])]))$
draw(
    terminal = animated_gif,
    delay = 40,
    file_name  = "tanplot_anim",
    tanplot)$
/* [wxMaxima: input   end   ] */

/* Maxima can't load/batch files which end with a comment! */
"Created with wxMaxima"$