Solution of the Final Exam
Problem 1. We say that a set of real numbers is dense if for any open interval , the intersection is
non-empty.
- Give an example of a dense set whose complement
is also dense.
- Give an example of a non-dense set whose complement
is also not dense.
- Prove that if
is an increasing function
( for ) and if the range
of is
dense, then is continuous.
Solution.
- Take for example
, the set of rational numbers. Then is
the set of irrational numbers. We've seen in class that between any two
(different) numbers (i.e., within any open interval) there is a rational
number and there is an irrational number. Hence both and are
dense.
- Take for example
, the set of non-negative
numbers. Then
is the set of negative numbers. The
set is not dense because, for example, it's intersection with the
interval is empty. The set is not dense because, for
example, it's intersection with the interval is empty.
- We have to show that for every
and for every
there is a so that
implies
. So let
be given. By the density of
we know that we can find an element of in
the interval
and another element of in the
interval
. That is, we can find and
so that
and
. It
follows from the monotonicity of that and that . Now
set
(this is a positive number because
and ). Finally if
then is in the
interval
. By
the monotonicity of it follows that is in the interval
, and so
, as required.
Problem 2. Sketch the graph of the function
.
Make sure that your graph clearly indicates the following:
- The domain of definition of .
- The behaviour of near the points where it is not defined (if
any) and as
.
- The exact coordinates of the - and
-intercepts and all minimas and maximas of .
Solution. Our function is defined for all . As
goes to exponentials dominate polynomials, and so certainly
gets much bigger than . So
. Solving the equation
we see that the only intersection of the graph of with the axes is
at . We can compute
and
. Solving
we see that the only critical points are when . That is, at
. As
, the point
is a local max. As
, the point
is a local min. As there are no other
critical points and the behaviour of near the ends of its domain of
deifnition is mute (as determined before),
is actually a
global max and
is actually a global min. Thus overall
the graph is:
Problem and Solution 3. Compute the following derivative
and the following integrals:
- Using the fundamental theorem of calculus in the form
and the chain rule with we
get
- We make the substitution
(and thus and
) to compute
- Integrating by parts twice we get
- We make the substitution (and thus and
) to compute
- We use the factorization
to get
Problem 4. In solving this problem you are not
allowed to use any properties of the exponential function .
- Two differentiable functions, and , defined over
the entire real line
, are known to satisfy
,
, and for all
and also
. Prove that and are the same. That is,
prove that
for all
.
- A differentiable function defined over the entire real
line
is known to satisfy
and for all
and also . Prove that
for all
.
Solution.
- Set
(this is well defined because is
never 0) and compute
So is a constant. But
, so that constant is 1 and
for all . This means that .
- Fix and set
and
. Then
and
and
. All the other conditions of
the first part of this question are even easier to verify, and so the
conclusion of that part holds. Namely, , which means
.
Problem 5. In solving this problem you are not
allowed to use any properties of the trigonometric functions.
- A twice-differentiable function defined over the entire
real line
is known to satisfy
for all
and also
. Write out the degree Taylor polynomial
of at .
- Write a formula for the remainder term
. (To keep the notation simple, you are
allowed to assume that is even or even that is divisible by 4).
- Prove that is the zero function: for all
.
Solution.
- From
it is clear that
and that
. So
and
and hence all the coefficients of
are 0. In other words,
.
- If is divisible by then
and so the remainder
formula says that for any there is a between 0 and for
which
- Factorials grow faster then exponentials, so in the remainder
formula the denominator grows faster then the term
, while the numerator is bounded (by the theorem that a
continuous function on a closed interval is bounded). So the remainder goes
to 0 when goes to , and hence
. But
for all , so
necessarily .
Remark 1. Two alternative forms of the remainder formula are
and
Either one of those could equaly well be used to solve part 3 of the
problem.
Remark 2. There is an alternative approach to the whole
problem; start with part 3 and go backwards. To do part 3, consider the
function
. We have
, so
is a constant function. But
, so
must be the 0 function. But is a sum of squares, and the only way
a sum of squares can be 0 is if each summand is 0. So and
hence as required in part 3. But if is the 0 function then
its Taylor polynomials are all 0 and the remainder terms are also all
0, solving parts 1 and 2 as well. This is not the solution I had in mind
when I wrote the problem, but people who solved the problem this way got
full credit.
Problem 6. In solving this problem you are not allowed
to use the irrationality of , but you are allowed, indeed advised,
to borrow a few lines from the proof of the irrationality of .
Is there a non-zero polynomial defined on the interval
and with values in the interval
so that it and all of its
derivatives are integers at both the point 0 and the point ? In
either case, prove your answer in detail.
Solution. There is no such polynomial. Had there been one,
we would have
but also, by repeated integration by parts (an even number of times, for
simplicity),
For any the first term in this formula involves only integers (as
,
, , , and
are all integers), and if is
larger than the degree of , the second term is 0. So
is an integer. But by the first formula it is
in . That can't be.
The results. 80 students took the exam; the average
grade was 69.33/120, the median was 71.5/120 and the standard deviation
was 26.51. The overall grade average for the course (of
) was
68.5, the median was 71.57 and the standard deviation was 18.64.
Finally, the transformation
was applied to
the grades, with
. This made the average grade 70.41, the
median 73.5 and the standard deviation 17.77. There were 30 A's (grades
higher or equal to 80) and 12 failures (grades below 50).
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Dror Bar-Natan
2004-05-10