LHospitals Rule

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LHospitals Rule
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To Tell the Truth
My name is LHospital! And that first guy cant
spell.
Hello, my name is LHopital.
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Question Messieurs, can you tell us
something about your famous
rule?
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Will the real LHospital please stand up!!!
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Heres a correct statement of LHospitals Rule
This just takes care of one-sided limits and
limits at infinity all at once.
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The many different cases of LHospitals Rule can
all be proven using Cauchys Mean Value Theorem
Extra letters and limits of rachieauxs seem to be
French things. Maybe its something in the
Perrier. Eau well!
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Cauchys Mean Value Theorem can be proven from
Rolles Theorem
See Handout!
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Examples where LHospitals Rule doesnt apply
See Handout!
See Handout!
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Examples where LHospitals Rule doesnt
apply(cont.)
See Handout!
See Handout!
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Examples where LHospitals Rule doesnt
apply(cont.)
See Handout!
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Surprising examples where LHospitals Rule
applies
See Handout!
See Handout!
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More surprising examples where LHospitals Rule
applies
See Handout!
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More surprising examples where LHospitals Rule
applies
See Handout!
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See Handout!
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See Handout!
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Hint
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7. Use LHospitals Rule to evaluate Where
the numbers are arbitrary
real numbers.
See Handout!
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The Nth Derivative Test
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A common test for determining the nature of
critical numbers in a first semester Calculus
course is the 2nd Derivative Test. Here is a
list of three common hypotheses from six Calculus
textbooks
I. Suppose that is continuous in an open
interval containing c.
II. Suppose that exists in an open interval
containing c.

III. Suppose that exists in an open interval
containing c, and exists.
The weakest hypothesis is III.
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The following conclusions are common to all
versions of the 2nd Derivative Test
If and , then f has a
local maximum at .
If and , then f has a
local minimum at .
If and , then the 2nd
Derivative Test fails.
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Suppose that exists in an open interval
containing c, exists, and
.
See Handout!
If , then the sign chart of
looks like
If , then the sign chart of
looks like
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If you ask a veteran Calculus student about the
2nd Derivative Test, youll probably get a
positive response, but if you ask about the Nth
Derivative Test, youre likely to get a puzzled
look.
Typically, the Nth Derivative Test is proved
using Taylors Theorem along with the following
hypotheses in second semester Calculus or higher
For , suppose that
are continuous in an open interval containing
and that ,
but .
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We can prove an Off-the-rack Nth Derivative Test
(without using Taylors Theorem) and with weaker
hypotheses, i. e. first semester Calculus style.
First, lets find some general hypotheses on
and its derivatives.
Beginning of the Off-the-rack Nth Derivative
Test
For , suppose that
exist in an open interval containing ,
,
exists and .
Now well investigate four cases

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Case I is odd and
The weakest 2nd Derivative Test applied to
along with the Mean Value Theorem yield the
following
Odd derivative
Even derivative
Odd derivative
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Case II is odd and
The weakest 2nd Derivative Test applied to
along with the Mean Value Theorem yield the
following
Odd derivative
Even derivative
Odd derivative
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Case III is even and
The weakest 2nd Derivative Test applied to
along with the Mean Value Theorem yield the
following
Even derivative
- -

0
Odd derivative
c
Odd derivative
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Case IV is even and
The weakest 2nd Derivative Test applied to
along with the Mean Value Theorem yield the
following
Even derivative
- -

0
Odd derivative
c
Odd derivative
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From the sign patterns in the previous four
cases, we can now state an Off-the-rack Nth
Derivative Test
For , suppose that
exist in an open interval containing ,
,
exists and .
If is odd, then f has neither a maximum or
minimum at .
If is even and , then f
has a local minimum at .
If is even and , then f
has a local maximum at .
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Determine whats going on at zero for the
following functions
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Suppose that f has derivatives of all orders in
an open interval containing and theyre
all equal to zero at . Can we conclude
anything about the nature of f at ?
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In the case of n being odd, can we conclude
anything about the graph of f at x c?
See Handout!
In the case of , we can
conclude that the sign chart for f near x c is
as follows
In the case of , we can
conclude that the sign chart for f near x c is
as follows
The Nth Derivative Test is fairly definitive.
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Suppose that g has a (piecewise) continuous
derivative on the interval and
on . By considering the
formula for the length of the graph of g on the
interval ,
Determine the maximum possible length of the
graph of g on the interval .
Determine the minimum possible length of the
graph of g on the interval .
See Handout!
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1
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If , then complete the graph of
the function g on the interval that
has the maximum length.
See Handout!

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If , then complete the graph of
the function g on the interval that
has the minimum length.
See Handout!

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Do the same under the assumptions
or
See Handout!
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Suppose that f and g have (piecewise) continuous
derivatives, ,
, and ,
then use the surface area of revolution about
the y-axis formula
to find a decent upper bound on the surface area
of revolution about the y-axis of the curve
See Handout!
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to find the minimal surface area of revolution
about the y-axis of the curve
See Handout!
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Fermat/Steiner Problems
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Fermats problem Given three points in the
plane, find a fourth such that the sum of its
distances to the three given ones is a minimum. 
Euclidean Steiner tree problem Given N points in
the plane, it is required to connect them by
lines of minimal total length in such a way that
any two points may be interconnected by line
segments either directly or via other points and
line segments.
The Euclidean Steiner tree problem is solved by
finding a minimal length tree that spans a set of
vertices in the plane while allowing for the
addition of auxiliary vertices (Steiner
vertices). The Euclidean Steiner tree problem has
long roots that date back to the 17th century
when the famous scientist Pierre Fermat proposed
the following problem Find in the plane a point,
the sum of whose distances from three given
points is minimal.
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power supply
Steiner point or vertex
factory
factory
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The length of the power line as a function of x
with the parameters a and h is given by
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Here are the possible sign charts for L
depending on the values of the parameters a and h.
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Compare this with the soap film configuration
using the frames.
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Vary the height of this suction cup and compare
Natures minimization to the Calculus predictions.
See Handout!
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A four point example We want to link the four
points , ,
, and with in a
minimal way.
Steiner points
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There are two competing arrangements for the
position of the Steiner points horizontal or
vertical.
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The length of the connection as a function of x
with the parameters a and h in the vertical case
is given by
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Here are the possible sign charts for LV,
depending on the values of the parameters a and h.
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From the symmetry of the problem, we can quickly
get the results for the horizontal case by
switching a and h.
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Here are the possible sign charts for LH,
depending on the values of the parameters a and h.
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Here are the possible relationships between a and
h in both the horizontal and vertical
arrangements
Vertical
Horizontal
There are four combinations of the inequalities
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Not possible.
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Heres the Phase Transition diagram in the ha
parameter plane
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minimum
minimum
minimum
minimum
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Compare this with the soap film configuration
using the frames.
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Vary the distance between pairs of suction cups
and compare Natures minimization to the Calculus
predictions.
See Handout!
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The Blancmange Function
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In the 19th Century, mathematicians gave examples
of functions which were continuous everywhere on
there domain, but differentiable nowhere on their
domain. One such example is constructed as
follows Start with a function defined on the
interval with a single corner at ,

Heres its graph
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Now extend it periodically to all the nonnegative
real numbers to get the function .
Heres a portion of its graph
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Now we can define the continuous,
nondifferentiable function on the interval
,
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Heres an approximate graph of the function f
known as the Blancmange Function
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It is an example of a fractal, in that it is
infinitesimally fractured, and self-similar. No
matter how much you zoom in on a point on the
graph, the graph never flattens out into an
approximate non-vertical line segment through the
point. The number of points of
nondifferentiability in the interval of
the component functions increases with n.

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Here are some selected plots of on
the interval

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Show that the formula for the function f actually
makes sense.
, which means that for each x in
. If you can show that
,
is bounded from above
for each x
and is nondecreasing in N, then
must exist as a number.
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Bounded above
Nondecreasing
See Handout!
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Show that the function f is continuous on
.
So
for every x in . Use this
to show that you can make

for every x in
.
by choosing N large enough.
See Handout!
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So if , then
and .
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Since is a
continuous function, there is a so
that if , then
Choose x in , let , and consider
See Handout!
Finish the proof of the continuity of f.
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Show that the function f is nondifferentiable on
.
Consider the sequence of points .
Show that .
For values of n greater than or equal to m,
for some positive whole number p, but
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so we get that
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What does this imply about ?
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Since as , if
exists, then
, but
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Consider the sequence of points
. Examine .
For values of n greater than or equal to m,
for some positive whole numbers p and k, but as
before,
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So we get that
Since g is periodic of period 1, we get that
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What does this imply about ?
See Handout!
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Consider the sequence of points
.
Examine
What does this imply about ?
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Try similar thinking to show that
doesnt exist for any
where p and k are whole numbers.
These xs are called dyadic rational numbers.
If x is in

and is not a dyadic rational, then for a fixed
value of m, x falls between two adjacent dyadic
rationals, .
Let and , for
each whole number m to get two sequences
and so that
and .




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For example, let , and , then
and
For example, let , and ,
then and
For example, let , and ,
then and
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For , if x is not
a dyadic rational, then show that for every
value of m, .
See Handout!
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Prove that
.
First well show that
.
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Since is a linear function on the
interval .
For example, let , and , then
and

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For example, let , and ,
then and
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For example, let , and ,
then and
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You can do a similar argument to show that

.
If exists, then
and
, but the
Squeeze Theorem would imply something about
.
Using the previous results, show why
doesnt exist.


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Differentiability of Powers of the Popcorn
Function
Consider the function
This function was originally defined by the
mathematician Johannes Thomae.
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Its called the popcorn function, ruler function,
raindrop function,
Here is a portion of its graph
Popcorn/Raindrop
Ruler
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Outline of the Proof of the continuity of the
Popcorn Function at the irrationals and the
discontinuity at the rationals.
Lets begin with a basic fact about rationals and
irrationals. Both types of numbers are dense in
the real numbers meaning that every interval of
real numbers contains both rational and
irrational numbers. Since the Popcorn Function
value at any irrational number is zero, to show
that the Popcorn Function is continuous at an
irrational number, , we just have to show
that
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Remember, to show that for
any function f, we have to show that for every
, there is a so that if
, then
.
So lets begin the proof by letting . Now
we will choose so that .
Now consider the finitely many rational numbers
in whose denominator is less than or
equal to
.

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Let
If and
is irrational, then

And if and is
rational with , then
, and
.
See Handout!
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To show that the Popcorn Function is
discontinuous at a rational number,
, we have to show that for some there is no
with the property that if
, then . In other
words, we have to show that for every
, there is at least one with
, but

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To accomplish this, let . For
every, there is an irrational number
with , but
See Handout!
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Is the darn thing differentiable?
Lets look at the difference quotient for an
irrational number a
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Hurwitzs Theorem
So Hurwitzs Theorem implies that


So its not differentiable anywhere.
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What about powers of the popcorn function?
is not differentiable at , but its
square is.
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Lets look at the difference quotient for an
irrational number a
So Hurwitzs Theorem implies that
So it square is not differentiable anywhere.
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Liouvilles Theorem
An algebraic number, , of degree , has
the property that for each positive number ,
there are only finitely many reduced rationals
with for
.
So for there are only finitely many
rationals with
,then for the remaining rationals in the
reduced interval wed have
. This means that in this case,

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Lets look at the difference quotient of the
popcorn function raised to the power at an
irrational algebraic number a of degree k
If then
. In other words, the
popcorn Function raised to the power is
differentiable at all algebraic irrationals of
degree less than or equal to . For
example, is differentiable at the
second degree algebraic irrational , but
not necessarily at the third degree irrational
.
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So eventually, every algebraic irrational number
will be a point of differentiability of some
power of the popcorn function. Furthermore,
is differentiable at every irrational
algebraic number for , using the
Thue-Siegel-Roth Theorem for which Klaus Roth
received a Fields Medal in 1958. What about
the transcendental numbers? Some
transcendental numbers are not points of
differentiability for any power of the popcorn
function,the Liouville transcendentals. Other
transcendental numbers are eventually points of
differentiability for some power of the popcorn
function
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