Re: finding vertical asymptotes
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Re: finding vertical asymptotes
The derivative of a polynomial function is another polynomial function.
Therefore, the graph of a polynomial function has a (non-vertical) tangent at
every point -- there are no poles.
The typical class of functions where the concept of "poles" arises is the set
of _rational_ functions. A rational function is the quotient of two polynomials
(the zero polynomial not being allowed as a denominator). As with polynomials,
the sum, difference or product of any two rational functions is a rational
function. But also the quotient of two rational functions is a rational
function (zero denominators being excluded).
The graph of a rational function may have a vertical asymptote. This would
correspond to a zero of the denominator, possibly of multiplicity greater than
one. Oftentimes the discussion of poles proceeds in the context of functions of
complex variables, where again rational functions can be defined as the ratio
or quotient of two polynomials, etc. A pole of a complex rational function can
be "removed" by multiplying by an appropriate power of (x-c), where c is the
zero. Some otherwise well-behaved complex functions, such as logarithms,
however have non-removable poles. Etc. etc.
On the TI-89, rational functions can be examined by using the getNum() and
getDenom() fucntions to tear the rational expression apart and anzlyze the num
and denom separately as polynomials.
RWW Taylor
National Technical Institute for the Deaf
Rochester Institute of Technology
Rochester NY 14623
>>>> The plural of mongoose begins with p. <<<<
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