Review of the Answers to Assignment # 5 about limits and
asymptotes
Theorem of limit of
rational functions in infinity If f(x) is a fraction of
two polynomials A(x) and B(x) then the limit of f(x) in infinity is the
same as the ratio of their higher degree terms.
Study of SINGULAR POINTS on the curve of a function :
Angular point (two half-tangents)
Right derivative ≠ Left derivative
e.g. : function with absolute value
Vertical Tangent line : Limits of Growth rate =
±∞
e.g. : functions with
radicals and absolute value (鸟function...)
Assignment # 6 : four new functions to
study specifically with their asymptotes and their singular points. For
each function show the proofs of the asymptotes and of the
singularities of some points.
In each point where the derivative cannot be defined by the
usual formulas, find the limits of the growth rate on the right and on
the left sides of the point :
Right derivative
Left derivative
f is not derivable in a if ß ≠ alpha
(two half tangents : angular point)
Detailed correction of the complete study of the rationnal
function defined by :
set of definition,limits, asymptotes
derivative, sign and zeroes
oblique asymptote, zero of f
curve representing f.
Theorem of determination
of an oblique asymptote : (D)
y=ax+b is an asymptote to (Cf) if and
only if the limit of the difference between f(x) and (ax+b) is
0.
The sign of that difference determines the position of
(Cf) with respect to (D).
Theorem of intermediate
values : how to determine an approximate value of the zeroes of
a function : If f is continuous and monotonous
on [a ; b ] and if f(a).f(b) < 0 then
there is one - and only one - number c between a and b
such that f(c) = 0.
examples in the above function of which the zeroes of the
derivative and of itself are the zeroes of a 3° polynomial.
Theorem and definition of
an Inflexion point : a point (a ; f(a)) of a
curve (Cf) is an inflexion point if and only if :
f "(a) = 0
f "(x) changes of sign in a.
On an inflexion point the
curve (Cf) goes through the tangent line.
Oblique asymptote
Inflexion Point
Assignment 6 will be collected only on Nov.24
to allow a better preparation in view of the test which will take place
on dec.1st.
after a complete review of all topics studied since September.
Assignment
#
7
is
designed
to
give
supplementary
exercises
to
practice
in
view
of
the general test
(90 minutes)
Tuesday Dec.1st Program : Rational and irrational
functions
limits
and asymptotes
derivatives
and
tangent
lines
variations,
maxi,
mini
zeroes
of f and f '
inflexion
point
(0
and
sign
of
f
")
symmetry
curve
students who want to have their work checked before the test should
return Assignment 7
to Wu LaoShi before Friday 27 at 5 pm
