Define and draw its graph
|
•
|
Control-drag
Context Panel: Assign Function
|
|
|
•
|
Figure 3.2.2(a) can be drawn interactively with the Plots option in the Context Panel.
|
•
|
The positive zero of certainly lies in the interval , and appears to be slightly less than .
|
|
|
Obtain the Maple solution for the equation
|
•
|
Type and press the Enter key.
|
•
|
Context Panel:
Solve≻Numerically Solve from point≻
|
|
|
Construct the iteration function
|
•
|
Control-drag
Context Panel: Assign Function
|
|
|
Implement the Newton iteration
|
|
|
•
|
Compute , referencing by its equation label.
|
|
|
•
|
Compute , referencing by its equation label.
|
|
|
•
|
Compute , referencing by its equation label.
|
|
|
•
|
Compute , referencing by its equation label.
|
|
|
•
|
Compute , referencing by its equation label.
|
|
|
|
|
Table 3.2.2(a) implements the Newton iteration by explicitly obtaining the equations determining the -intercepts of successive tangent lines, and solving the equations for the Newton iterates.
•
|
Set in the equation of the tangent line at .
Press the Enter key.
|
•
|
Context Panel: Solve≻Numerically Solve
|
|
|
•
|
Set in the equation for the tangent line at the new .
Press the Enter key.
|
•
|
Context Panel: Solve≻Numerically Solve
|
|
|
•
|
Set in the equation for the tangent line at the new .
Press the Enter key.
|
•
|
Context Panel: Solve≻Numerically Solve
|
|
|
•
|
Set in the equation for the tangent line at the new .
Press the Enter key.
|
•
|
Context Panel: Solve≻Numerically Solve
|
|
|
•
|
Set in the equation for the tangent line at the new .
Press the Enter key.
|
•
|
Context Panel: Solve≻Numerically Solve
|
|
|
Table 3.2.2(a) Newton iterates generated as -intercepts of successive tangent lines
|
|
|
The Newton's Method tutor
|
•
|
Figure 3.2.2(b) contains an image of the
tutor used to solve the equation starting from . The tutor shows the ensuing iterates, and can display an animation of the successive tangent lines.
|
•
|
Notice at the bottom of the tutor that there is the NewtonsMethod command from the Student Calculus1 package.
|
•
|
The output from this command can be one of plot, animation, sequence, or value. With the option sequence, the command returns a sequence of iterates. The default is value, for which the return is just the "final" iterate.
|
|
|
Figure 3.2.2(b) Newton's Method tutor
|
|
|
|
|
|
By wrapping the command in angle-brackets (inequality symbols), the output of the NewtonsMethod command is converted from a horizontally-written sequence, to a column vector, a much easier display to read.