York University
MATH 5411 3.00AF – Analysis for Teachers
Problem Set 1 – Solutions
Due Monday September 19, 2016
1. In class (and in the notes I posted) we had an argument that allows us calculate
the Hausdorff dimension of a fractal A. In the form I gave it, it depends on knowing
that there is some δ > 0 such that 0 < Hδ (A) < ∞. You may assume that this
holds for the regular fractals below. Assuming this, use the argument to figure
out what the value of δ must be in each case.
(a) The Sierpinski carpet
(b) The Koch snowflake
(c) The Cantor-like set in which you first delete the middle fifth of [0, 1], then
the middle fifth of both intervals left, then . . .
Solution: Recall from class that if A is made of up n copies of a set that is
log n
congruent to αA, then the dimension δ satisfies nαδ = 1. In other words, δ = log
1 .
α
log 8
log 3
1
3
≈ 1.89
(a) For the carpet, n = 8 and α = which implies δ =
log 4
1
(b) For the Koch snowflake, n = 4 and α = 3 which implies δ = log
≈ 1.26
3
log 2
(c) For modified Cantor set, n = 2 and α = 25 which implies δ = log
≈ 0.76
5/2
2. Review the fundamental theorem of calculus, paying close attention to its
hypotheses. Recall that we saw in class an example of a continuous function
R x 0 F (x)
whose derivate existed at enough points that Riemann integrals like 0 F (t) dt
make sense, yet these look nothing like F (x) − F (0). In fact, there is a version
of the FTC that says
R x that if F (x) is differentiable at EVERY point in [0, 1] then
F (x) − F (0) = 0 F 0 (t) dt for every x. Our example fails to contradict this,
because it turns out that our F is not differentiable at points x ∈ C.
(a) Show by example that the conclusion of the above FTC needn’t hold, if we
just assume that F is differentiable at all but finitely many points in [0, 1]
(b) It is however true that if F is continuous on [0, 1] and differentiable at all
but finitely many points, then the conclusion of the FTC holds. Prove this
in the special case that F is continuous, and differentiable at every point
of [0, 1] \ { 21 }. [Hint: consider the two cases x ≤ 12 and x > 12 .]
Solution: The point of this exercise is to explore when the FTC holds, and when
it fails. For its first 150 years, Calculus was done quite informally, without precise
definitions or proof that we would call rigorous. In the 19th century, people like
Cauchy cleaned the theory up and added rigour. The way we teach calculus today
is very much in the 19th century tradition, and would have seemed foreign to the
1
2
people who invented Calculus in the first place. No wonder our students struggle
to learn our way of thinking about it!
Analysis has two things to contribute here: a formal proof of the FTC (which we
won’t go into), and an understanding of what goes wrong and how far the result
can be pushed. That is the point of this exercise.
(a) Without continuity, there is no hope of the FTC holding.
example, if
R x For
0
0
F is a step function then F = 0 except at the steps, so 0 F (t) dt = 0 for
every x. Which does not agree with F (x) − F (0).
(b) So assume that F (x) is continuous. If F 0 exists nowhere then we can’t
integrate F 0 (t) so the FTC fails completely. If F 0 exists everywhere then
we know the FTC holds. It is natural to wonder how big the exceptional set
[of points where F 0 (t) doesn’t exist] can be before the FTC breaks down.
We might guess that if it has zero length (so we can integrate F 0 ) then the
FTC would hold. But our example in class shows that this is not the case.
In other words, having C for the exceptional set is too big. But having the
exceptional set be finite turns out to be OK. You’re asked to prove this
when the exceptional set is simply { 12 }.
1
1
0
First
R x 0 consider x ≤ 2 . Since F exists on (0, 2 ), FTC applies there and so
F (t) dt = F (x) − F (0).
0
Second, consider x > 12 . FTC applies to both [0, 12 ] and [ 12 , 1]. So
Z 1
Z 1
Z x
2
1
1
0
0
F (t) dt+
F (t) dt =
F 0 (t) dt = F ( )−F (0)+F (x)−F ( ) = F (x)−F (0).
1
2
2
0
0
2
3. You don’t have to hand anything in here – this is a warmup for the next class,
in which we’re going to look at something called the “Chaos game”. Try reading
over the Wikipedia article with that title. Next week I’m going to get you to
code up a variant of the Chaos game, so this week I’d like you to prepare. If you
have a favourite programming language, try implementing the Chaos game for
an triangle, and see if that draws the Sierpinski triangle for you. If you’ve never
programmed, or don’t have a programming language available on your computer,
try downloading an implementation of R, and see if you can run the program I’ve
posted on the course webpage.