Animating Taylor polynomialsMapleNet - Demonstration Demonstration Worksheet by Mike May, S.J.- maymk@slu.eduThis worksheet is an adaptation of an older worksheet on animating Taylor series. It is intended as a demonstration of what can be done with MapleNet. For that reason it includes more text on the construction of the worksheet.First some technical details. We restart Maple to clear memory and load some packages we need. The button below executes the commands:restart:with(plots):with(Student[MultivariateCalculus]):with(DocumentTools):restart:
with(plots):
with(Student[MultivariateCalculus]):
with(DocumentTools):
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use:
# GetProperty( component_name, attribute_name )
# and
# SetProperty( component_name, attribute_name, value )
# to access any of the attributes of the component.
# See ?CustomizingComponents for more information.
When working with Taylor polynomials there are two features that are worth demonstrating with animations that compare higher and higher degree Taylor polynomials with the original function. The first feature is that, as we would hope with something that is called an approximation, the Taylor polynomials morph into the function. The second feature is the radius of convergence. If the Taylor series has a finite radius of convergence, it will only morph to the function in that region, and will never become a good approximation outside that region. The animations are memory intensive. Since this often causes memory problems the animation will be done as a separate worksheet. Before doing this worksheet, you should have worked through the worksheet on Taylor polynomials in several variables.
<Text-field style="Heading 1" layout="Heading 1">One variable convergence demonstration</Text-field>We start with initial information about the functions and where we want to look at the Taylor polynomial.
function(x) = x0 = , width from center = , height from center = minimal degree = , number of steps = , degrees per step = use DocumentTools in
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use:
# GetProperty( component_name, attribute_name )
# and
# SetProperty( component_name, attribute_name, value )
# to access any of the attributes of the component.
# See ?CustomizingComponents for more information.
f1 := parse(GetProperty(TextAreaFunction, 'value'));
f1x0 := parse(GetProperty(TextAreax0, 'value'));
f1width := parse(GetProperty(TextAreaWidth, 'value'));
f1height := parse(GetProperty(TextAreaHeight, 'value'));
f1lower := parse(GetProperty(TextAreaMinimalDegree, 'value'));
f1numsteps := parse(GetProperty(TextAreaNumSteps, 'value'));
f1degreesteps := parse(GetProperty(TextAreaDegreeSteps, 'value'));
f1yval := eval(f1, x=f1x0);
f1minx := f1x0-f1width: f1maxx := f1x0+f1width:
f1miny := f1yval-f1height: f1maxy := f1yval+f1height:
f1fudge:= (f1maxy-f1miny)/20:
B := textplot([f1maxx,f1maxy-f1fudge,
"graphed against the function (green)"],
align={ABOVE,LEFT}, font = [HELVETICA, BOLD, 12],
view=[f1minx..f1maxx,f1miny..f1maxy]):
for f1step from 0 to f1numsteps do
f1deg := f1lower + f1step * f1degreesteps:
Tf1 := TaylorApproximation(f1,[x]=[f1x0],f1deg);
A := plot([f1,Tf1], x=f1minx..f1maxx,y=f1miny..f1maxy,
color=[green,blue]);
C := textplot([f1minx,f1maxy,
"The Taylor polynomial of degree "||f1deg||
" (blue)"], align={ABOVE,RIGHT}, font = [HELVETICA, BOLD, 12],
view=[f1minx..f1maxx,f1miny..f1maxy]);
f1plot[1+f1step] :=display([A, C, B]):
end do:
SetProperty(Plotf1Taylor, 'value',
display([seq(f1plot[i],i=1..f1numsteps+1)],insequence=true));
end use;
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Desired Taylor polynomial degree = use DocumentTools in
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use:
# GetProperty( component_name, attribute_name )
# and
# SetProperty( component_name, attribute_name, value )
# to access any of the attributes of the component.
# See ?CustomizingComponents for more information.
f1 := parse(GetProperty(TextAreaFunction, 'value'));
f1x0 := parse(GetProperty(TextAreax0, 'value'));
f1degree := parse(GetProperty(TextAreaPolyDegree, 'value'));
Tf1 := TaylorApproximation(f1,[x]=[f1x0],f1degree);
SetProperty(MathContainerTaylorPoly, 'value', sort(Tf1,[x]));
end use;
<Text-field style="Heading 2" layout="Heading 2">Construction Details</Text-field>
<Text-field style="Heading 3" layout="Heading 3">Construction Overview</Text-field>The construction used 8 TextAreas, a MathContainer, a Plot, and 2 Buttons. When these components are added in, Maple named then things like TextArea0, TextArea1, etc. With each component, I opened the component properties item from the edit menu. The components were renamed with names connected to their functions intended use, like TextAreaFunction, TextAreaHeight. I also fixed the size and gave each item a tool tip with its new name so that I could retrieve the name.I then generated an execution group and worked out and debugged the set of commands that would be used in the change on section for each button. Once it was debugged it was copied into the button. The code for each button is given below.
<Text-field style="Heading 3" layout="Heading 3">Change on code for animation button</Text-field>use DocumentTools in # Enter Maple commands to be executed when the specified# action is carried out on the component.# Use: # GetProperty( component_name, attribute_name )# and# SetProperty( component_name, attribute_name, value )# to access any of the attributes of the component.# See ?CustomizingComponents for more information.f1 := parse(GetProperty(TextAreaFunction, 'value'));
f1x0 := parse(GetProperty(TextAreax0, 'value'));
f1width := parse(GetProperty(TextAreaWidth, 'value'));
f1height := parse(GetProperty(TextAreaHeight, 'value'));
f1lower := parse(GetProperty(TextAreaMinimalDegree, 'value'));
f1numsteps := parse(GetProperty(TextAreaNumSteps, 'value'));
f1degreesteps := parse(GetProperty(TextAreaDegreeSteps, 'value'));
f1yval := eval(f1, x=f1x0);
f1minx := f1x0-f1width: f1maxx := f1x0+f1width:
f1miny := f1yval-f1height: f1maxy := f1yval+f1height:
f1fudge:= (f1maxy-f1miny)/20:
B := textplot([f1maxx,f1maxy-f1fudge,
"graphed against the function (green)"],
align={ABOVE,LEFT}, font = [HELVETICA, BOLD, 12],
view=[f1minx..f1maxx,f1miny..f1maxy]):
for f1step from 0 to f1numsteps do
f1deg := f1lower + f1step * f1degreesteps:
Tf1 := TaylorApproximation(f1,[x]=[f1x0],f1deg);
A := plot([f1,Tf1], x=f1minx..f1maxx,y=f1miny..f1maxy,
color=[green,blue]);
C := textplot([f1minx,f1maxy,
"The Taylor polynomial of degree "||f1deg||
" (blue)"], align={ABOVE,RIGHT}, font = [HELVETICA, BOLD, 12],
view=[f1minx..f1maxx,f1miny..f1maxy]);
f1plot[1+f1step] :=display([A, C, B]):
end do:
SetProperty(Plotf1Taylor, 'value',
display([seq(f1plot[i],i=1..f1numsteps+1)],insequence=true));end use;
<Text-field style="Heading 3" layout="Heading 3">Change on code for polynomial button</Text-field>use DocumentTools in # Enter Maple commands to be executed when the specified# action is carried out on the component.# Use: # GetProperty( component_name, attribute_name )# and# SetProperty( component_name, attribute_name, value )# to access any of the attributes of the component.# See ?CustomizingComponents for more information.f1 := parse(GetProperty(TextAreaFunction, 'value'));f1x0 := parse(GetProperty(TextAreax0, 'value'));f1degree := parse(GetProperty(TextAreaPolyDegree, 'value'));Tf1 := TaylorApproximation(f1,[x]=[f1x0],f1degree);SetProperty(MathContainerTaylorPoly, 'value', sort(Tf1,[x]));end use;
<Text-field style="Heading 1" layout="Heading 1">One variable radius of convergence demonstration</Text-field>Next we want to demonstrate the idea of radius of convergence. We want to look at what happens if the series has a finite radius of convergence.For the finite radius of convergence we change: the function to 1/(1+x^2)x0 to 1.5width to 3.5On a second pass through the series of graphs we can raise the degrees per step and the number of steps to 20 so that we are looking at up to a 400 degree polynomial.