Unformatted text preview: zero (tan θ = 0 ) then that point must be a maxim um
maximum
um,
minimum or inflexion point.
We mentioned earlier that pol ynomials ar e to functions w ha t r a tionals
polynomials are
wha ra
hat
p ol
ar e to r eal n umber s . We can do all our calculations of reals to any degree
are real number
umbers
of accuracy using the rationals. Likewise we may approximate any singlevalued
function by a suitable polynomial. In fact, thinking more deeply on what we
just covered in the last few chapters, we may be more specific. We may
piecewise approximate any singlevalued function by a suitable choice of
polynomials of maximum degree 3 or cubic.
We just saw how a function is either increasing, decreasing or changing
direction. Polynomials of degree 0,1,2, and 3 are sufficient to cover all
these variations. We only need to identify the maxim um minim um and
maximum minimum
um,
um,
infle
lexion
inf lexion points of the function thru which the approximating polynomials
must interpolate.
Note : Later the student will learn ROLLE’S THEOREM, which is a special case of
the MEAN VALUE THEOREM. In the special case when f(a) = f(b) then there exists
_
_
some MEAN point x in the interval [a, b] such that f ’( x ) = 0. Is it possible to find
_
the point x now ? (see note page 64) 120 2 4. Po
Po i n t s o f I N F L E X I O N A point at which a curve changes its shape in known as a point of inf lexion.
infle
lexion.
infle
lexion
inf lexion = a bending in the curve, a change in curvature or shape, a
change in direction, modulation of sound. (flexible, inflexible)
We have seen in the previous sections that at a maxim um or minim um
maximum
minimum
point a cur ve changes its shape.
maximum : concave down and going left to right from rising to falling.
minimum : concave up and going left to right from falling to rising.
There is another type of change in shape possible: concave up to concave
down or vice versa concave down to concave up.
Consider the function f(x) = x 3 + 1 . Please see the graph.
f ’(x) = 3x 2 a n...
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 Fall '09
 TAMERDOğAN
 Limit, Δx

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