Unformatted text preview: zero (tan θ = 0 ) then that point must be a maxim um
minimum or inflexion point.
We mentioned earlier that pol ynomials ar e to functions w ha t r a tionals
ar e to r eal n umber s . We can do all our calculations of reals to any degree
are real number
of accuracy using the rationals. Likewise we may approximate any single-valued
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
piece-wise approximate any single-valued 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
inf lexion points of the function thru which the approximating polynomials
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.
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
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
- Limit, Δx