1. L’Hospital’s Rule

2. To Tell the Truth
I’m the real L’Hopital! And they’re both bad spellers. ^
My name is L’Hospital!
And that first guy can’t spell.
Hello, my name is L’Hopital.

3. Question: Messieurs, can you tell us something about your famous rule?
L’Hospital #1
L’Hospital #2
L’Hospital #3

4. doesn’t exist, then doesn’t
If has the form or and
doesn’t exist, then doesn’t
exist.
L’Hospital #1

5. My rule can only be directly applied to the limit forms and .
L’Hospital #2

6. My rule can be directly applied to the limit forms and *, but
if doesn’t exist, then my rule
can’t be applied.
L’Hospital #3
* can also be replaced with

7. Will the real L’Hospital please stand up!!!
Johann Bernoulli
L’Hospital #3
It’s true, L’Hospital’s Rule can be directly applied to the
limit forms and
L’Hospital #1
L’Hospital #2
L’Hospital #3
But the rule says nothing if doesn’t exist.

8. Here’s a correct statement of L’Hospital’s Rule:
Let a be a real number or and I an open interval which contains a or has a as an endpoint.
This just takes care of one-sided limits and limits at infinity all at once. or b a c
[Suppose that and for all x in I.]
This condition is typical, but not needed if we assume
exists.
If or ,
and ,
then

9. Let f and g be continuous on and differentiable on . Then there is a c
The many different cases of L’Hospital’s Rule can all be proven using Cauchy’s Mean Value Theorem:
Extra letters and limits of rachieauxs seem to be French things. Maybe it’s something in the Perrier. Eau well!
Let f and g be continuous on and
differentiable on Then there is a c
in with or

10. Cauchy’s Mean Value Theorem can be proven from Rolle’s Theorem:
Apply Rolle’s Theorem to the function
on the interval
See Handout!

11. The case is covered in most textbooks, but the
case isn’t mentioned.
From Cauchy, we get that

12. Case #1:
If and are sufficiently large, then we
can make and arbitrarily close to
zero, since , and arbitrarily close to
So we get that

13. Case #2:
If and are sufficiently close to a, then we
can make and arbitrarily close to
zero, since , and arbitrarily close to
So we get that

14. Examples where L’Hospital’s Rule doesn’t apply:1.
See Handout! 2.
See Handout!

15. Examples where L’Hospital’s Rule doesn’t apply(cont.):3.
See Handout! 4.
See Handout!

16. Examples where L’Hospital’s Rule doesn’t apply(cont.):5.
See Handout!
An example where you can’t get away from the zeros of

17. Surprising examples where L’Hospital’s Rule applies:1.
See Handout! 2.
See Handout!
Watch out for !!!

18. More surprising examples where L’Hospital’s Rule applies:
3. If exists on and ,
then find
If L is a number, then find
See Handout!
{Hint: Apply L’Hospital’s Rule to , and
then observe that }
What’s if ? Is this a problem?

19. More surprising examples where L’Hospital’s Rule applies:
4. If exists on and , and
exists as a number, then what must
be the value of L?
See Handout!
{Hint: Apply L’Hospital’s Rule to , and use it to
determine Determine the limit from
the fact that exists as a number .}

20. is a number. Show that there is a sequence with and .
5. If exists on and , where A
is a number. Show that there is a sequence
with and
See Handout!
{Hint: The Mean Value Theorem implies that for each n
for some So .}
You might think that , by applying L’Hospital’s Rule in reverse, but consider

21. 6. You can see that doesn’t exist. If we
write the limit as and try
L’Hospital’s Rule, we get
. What’s wrong?
See Handout!

22. Hint:

23. 7. Use L’Hospital’s Rule to evaluate
Where the numbers are arbitrary
real numbers.
See Handout!
Hint:

24. The Nth Derivative Test

25. 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.

26. 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.

27. Suppose that exists in an open interval containing c,
exists, and
See Handout!
If , then the sign chart of looks like: c
If , then the sign chart of looks like: c

28. If you ask a veteran Calculus student about the 2nd Derivative
Test, you’ll probably get a positive response, but if you ask about
the Nth Derivative Test, you’re likely to get a puzzled look.
Typically, the Nth Derivative Test is proved using Taylor’s 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 .

29. We can prove an Off-the-rack Nth Derivative Test (without using
Taylor’s Theorem) and with weaker hypotheses, i. e. first semester
Calculus style.
First, let’s 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 we’ll investigate four cases:

30. Case I: is odd and :
The weakest 2nd Derivative Test applied to along with the
Mean Value Theorem yield the following: c
+ +
Odd derivative: c
+ +
- -
Even derivative: c
+ +
Odd derivative:

31. Case II: is odd and :
The weakest 2nd Derivative Test applied to along with the
Mean Value Theorem yield the following: c
- -
Odd derivative: c
- -
+ +
Even derivative: c
- -
Odd derivative:

32. Case III: is even and :
The weakest 2nd Derivative Test applied to along with the
Mean Value Theorem yield the following: c
+ +
Even derivative:
- -
+ +
Odd derivative: c c
+ +
- -
Odd derivative:

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

34. 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 .

35. Determine what’s going on at zero for the following functions:

36. Suppose that f has derivatives of all orders in an open interval
containing and they’re all equal to zero at Can we
conclude anything about the nature of f at ?
Consider the functions:

39. 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: c
+ +
- -
In the case of , we can conclude that the sign chart for f” near x = c is as follows: c
- -
+ +
The Nth Derivative Test is fairly definitive.

40. 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!

41. 1

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

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

44. Do the same under the assumptions:or
See Handout!

45. 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!

46. to find the minimal surface area of revolution about the y-axis of the curve
See Handout!

47. Fermat/Steiner Problems

48. Fermat’s 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.

49. Steiner point or vertex
A practical example:
Two factories are located at the coordinates and with their power supply located at ,
. Find x so that the total length of power line from the power supply to the factories is a minimum.
power supply
Steiner point or vertex
factory
factory

50. The length of the power line as a function of x with the parameters a and h is given by

51. Here are the possible sign charts for L’ depending on the values of the parameters a and h.

52. So in the case of the Steiner point should
be located units above the factories. If ,
then the power lines should go directly from the factories to the power supply without a Steiner point.

53. Compare this with the soap film configuration using the frames.

54. Vary the height of this suction cup and compare Nature’s minimization to the Calculus predictions.
See Handout!

55. A four point example: We want to link the four points , ,
, and with in a minimal way.
Steiner points

56. There are two competing arrangements for the position of the Steiner points: horizontal or vertical.

57. The length of the connection as a function of x with the parameters a and h in the vertical case is given by

60. Here are the possible sign charts for LV’, depending on the values of the parameters a and h.

61. So in the case of the two vertical Steiner
points should be located units above and
below the origin. If then there should
only be one Steiner point at the origin.

62. From the symmetry of the problem, we can quickly get the results for the horizontal case by switching a and h.

63. Here are the possible sign charts for LH’, depending on the values of the parameters a and h.

64. So in the case of the two horizontal Steiner
points should be located units left and
right of the origin. If then there should
only be one Steiner point at the origin.

65. 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:

69. Not possible.

70. Here’s the Phase Transition diagram in the ha parameter plane
One vertical Steiner Point and two horizontal Steiner Points
Two Steiner Points for the vertical and horizontal
One horizontal and two vertical

71. minimum
minimum
minimum
minimum

72. Compare this with the soap film configuration using the frames.

73. Vary the distance between pairs of suction cups and compare Nature’s minimization to the Calculus predictions.
See Handout!

74. The Blancmange Function

75. 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 ,
Here’s its graph:

76. Now extend it periodically to all the nonnegative real numbers to get the function .
Here’s a portion of its graph:

77. Now we can define the continuous, nondifferentiable function on the interval ,

78. Here’s an approximate graph of the function f known as the Blancmange Function:

79. 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.

80. Here are some selected plots of on the
interval

82. Show that the formula for the function f actually makes sense.
, which means that for each x in
. If you can show that ,
for each x
is bounded from above
and is nondecreasing in N, then
must exist as a number.

83. Bounded above:
Nondecreasing:
See Handout!

84. 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!

85. So if , then and

86. 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.

87. 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

88. so we get that

89. What does this imply about ?

90. Since as , if exists, then
, but

91. 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,

92. So we get that
Since g is periodic of period 1, we get that

94. What does this imply about ?
See Handout!

95. Consider the sequence of points .
Examine
What does this imply about ?

96. Try similar thinking to show that doesn’t exist for any
where p and k are whole numbers.
These x’s 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 .

97. For example, let , and , then and

98. For , if x is not a dyadic rational, then
show that for every value of m,
See Handout!

99. Prove that
First we’ll show that

101. Since is a linear function on the interval .
For example, let , and , then and

102. For example, let , and , then and

103. For example, let , and , then and

105. 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 doesn’t exist.

106. Differentiability of Powers of the Popcorn Function
Consider the function
This function was originally defined by the mathematician Johannes Thomae.

107. It’s called the popcorn function, ruler function, raindrop function,…
Here is a portion of its graph:
Popcorn/Raindrop
Ruler

108. Outline of the Proof of the continuity of the Popcorn Function
at the irrationals and the discontinuity at the rationals.
Let’s 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

109. Remember, to show that for any function f, we
have to show that for every , there is a so that if
, then
So let’s 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 :

110. If and is irrational, then
Let
If and is irrational, then
And if and is rational with , then
, and
See Handout!

111. 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

112. To accomplish this, let . For every, there is an
irrational number with , but
See Handout!

113. Is the darn thing differentiable?
Let’s look at the difference quotient for an irrational number a:

114. Hurwitz’s Theorem:
Any irrational number has an infinity of rational
approximations which satisfy
So Hurwitz’s Theorem implies that
So it’s not differentiable anywhere.

115. What about powers of the popcorn function?
is not differentiable at , but its square is.

116. Let’s look at the difference quotient for an irrational number a:
So Hurwitz’s Theorem implies that
So it square is not differentiable anywhere.

117. Liouville’s 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 we’d have This means that in this case,

118. Let’s 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

119. 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 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:

120. is differentiable at for (1953)