Define and draw its graph
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Control-drag
Context Panel: Assign Function
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Figure 3.2.1(a) can be drawn interactively with the Plots option in the Context Panel.
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The positive zero of certainly lies in the interval , and appears to be slightly less than 2.
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Figure 3.2.1(a) Graph of
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Obtain the Maple solution for the equation
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Type and press the Enter key.
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Context Panel:
Solve≻Numerically Solve from point≻
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Construct the iteration function
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Control-drag
Context Panel: Assign Function
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Implement the Newton iteration
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Start with and compute
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Compute , referencing by its equation label.
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Compute , referencing by its equation label.
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Compute , referencing by its equation label.
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Compute , referencing by its equation label.
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Compute , referencing by its equation label.
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Since , no further iteration at this level of accuracy will produce a more accurate solution. In fact, agrees exactly with Maple's numeric solution computed with the default number of digits, namely, ten.
Table 3.2.1(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.
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Set in the equation of the tangent line at .
Press the Enter key.
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Context Panel: Solve≻Numerically Solve
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Set in the equation for the tangent line at the new .
Press the Enter key.
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Context Panel: Solve≻Numerically Solve
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Set in the equation for the tangent line at the new .
Press the Enter key.
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Context Panel: Solve≻Numerically Solve
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Set in the equation for the tangent line at the new .
Press the Enter key.
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Context Panel: Solve≻Numerically Solve
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Set in the equation for the tangent line at the new .
Press the Enter key.
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Context Panel: Solve≻Numerically Solve
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Table 3.2.1(a) Newton iterates generated as -intercepts of successive tangent lines
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The Newton's Method tutor
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Figure 3.2.1(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.
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Notice at the bottom of the tutor that there is the NewtonsMethod command from the Student Calculus1 package.
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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.
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Figure 3.2.1(b) Newton's Method tutor
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The NewtonsMethod command
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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.