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| We have techniques for finding solutions to linear and quadratic equations. For quadratic equations, we could factor, complete the square, or use the quadratic formula. But what about higher degreed equations? How would we find the x-intercept to the curve below? | ||
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Newton's
Method says that you can: |
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| This method does not always work. It is possible that the first estimate will be such that after following the above procedure, one will be led away from the zero of the function, instead of towards it. This can be overcome with a better first estimate. Following is an example of this type of failure: | ||
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Example Problem.
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Use Newton's method with the
specifiied initial approximation,
 to find
 , the third approximation to
the root of the given equation. (Give your answer to four decimal
places.)
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Example Problem. |
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We first differentiate so that we will be able to
calculate the slope of the tangent line at the Now using the slope of 1, and the point (1,-1), we find
the equation of the tangent line to our function at |
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Now we find where this tangent line
intersects the x-axis, by setting equal to zero and
solving. Now we have our second estimate,  . |
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We again go through the same
procedure. We find the coordinates for the point on the curve at , we claculate the slope of the tangent line at
this point on the curve, and then using the coordinates of the point on
the curve, find the equation of the tangent line. |
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Finally, we find where this tangent
line intersects the x-axis, giving us our third estimate, . |
Whenever a process is to take its previous answer and repeat
calculations based on the previous answer, this is called iteration.
A formula which summarizes this iterative procedure is:
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