MAT137 Assignment 2
Due June 9, 2025
1
Before you begin the problem set, the teaching team has a few suggestions:
ˆ Start early! Some of the problems you may not be able to start yet, but the first four
problems are doable with all the material covered so far.
ˆ Do your scratch work separately. Your final writeup should be a clear, legible proof, written
with full sentences.
ˆ Our hope is that none of the proofs should take more than a page, written with 12 pt font
size. Please try to stay under one page, at most two. You should use a new page for your
solution, not the pdf provided.
ˆ Please make sure your handwriting is not too small to read when viewing the full page - this
will make grading much easier, and allow the TAs to leave better feedback.
ˆ If you’re stuck, don’t forget to attend office hours and ask questions on Piazza. You can’t
give or get full solutions, but someone may be able to help you understand the problems
better.
Best of luck!
-The MAT137 Teaching Team
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MAT137 Assignment 2
Due June 9, 2025
2
Question 1. In this question you prove a few important facts about polynomials. The Chebyshev
polynomials are an important collection of polynomials which are particularly well behaved on the
region [−1, 1] and are used widely in function approximation and numerical methods. The nth
Chebyshev polynomial Tn (x) is defined recursively as:
ˆ T0 (x) = 1
ˆ T1 (x) = x
ˆ Tn+1 (x) = 2xTn (x) − Tn−1 (x)
(a) Prove by induction that Tn (cos θ) = cos(nθ).
(b) Where are the extreme values of Tn (x) over the region [−1, 1]? What are they? How many
are there?
(c) Let p(x) be a degree n polynomial. Give a proof by induction that p(x) = 0 has at most n
solutions.
Hints/comments:
⋆ Throughout this problem, you may use the following fact without proof:
Fact: If p is a degree n polynomial and p(a) = 0, then p(x) = q(x)(x − a), where q is a degree n − 1 polynomial.
⋆⋆ You can use without proof trigonometric identities for this question.
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MAT137 Assignment 2
Due June 9, 2025
3
Question 2. Let Tn (x) be the nth Chebyshev polynomial. Let p(x) be a polynomial of degree n
such that ∀x ∈ [−1, 1], |p(x)| < 1. Prove that the following statement holds true:
∀x ∈ R s.t. x ̸∈ [−1, 1], |p(x)| ≤ |Tn (x)|
Roughly speaking, this theorem characterizes a tradeoff with Chebyshev polynomials: They are in
some sense the “best” polynomials over [−1, 1], and so they must be the “worst” outside.
Hints/comments:
⋆ It will be helpful for this problem to use facts from the previous problem.
⋆⋆ Using the intermediate value theorem is necessary for this question.
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MAT137 Assignment 2
Due June 9, 2025
4
Question 3. Recall that for D ⊆ R and a function f (x) defined on D, we say that f (x) is
continuous on D if:
∀a ∈ D, ∀ϵ > 0, ∃δ > 0 s.t. ∀x ∈ D, |x − a| < δ =⇒ |f (x) − f (a)| < ϵ
We say that f (x) is trouble-free on D if:
∀ϵ > 0, ∃δ > 0 s.t. ∀a, x ∈ D, |x − a| < δ =⇒ |f (x) − f (a)| < ϵ
We say that f (x) is bounded on D if:
∃C ∈ R s.t. ∀x ∈ D, |f (x)| ≤ C
(a) Statement 1: Let a < b and let f (x) be a function which is continuous on (a, b). Then f (x)
must be bounded on (a, b).
Is this statement true? If yes, prove it. If not, give a counterexample.
(b) Statement 2: Let a < b and let f (x) be a function which is trouble-free on (a, b). Then
f (x) must be bounded on (a, b).
Is this statement true? If yes, prove it. If not, give a counterexample.
Hints/comments:
⋆ In order to guarantee full marks, you must prove that any examples you use have the desired properties.
⋆⋆ You can freely use IVT/EVT as well as anything else covered in Unit 1/Unit 2
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MAT137 Assignment 2
Due June 9, 2025
5
Question 4. Find the derivative. If the problem asks for a the nth derivative, give a proof by
induction.
n
d
1
(a) Compute dx
n 2x+1 for any n ∈ N.
n
d
2x
for any n ∈ N.
(b) Compute dx
n xe
(c) Compute f ′ ( 3π
), where f (x) = x(sin x + 1)x
2
Hints/comments:
⋆ Throughout this question you may freely use differentiation rules.
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MAT137 Assignment 2
Due June 9, 2025
6
Question 5. Let a ∈ R
(a) If f + g is differentiable at a, must f and g be differentiable at a?
(b) Suppose f is differentiable at a, and f (a) = 0. Show that f = (x − a)g(x), where g is a
function which is continuous at a.
(c) Suppose that f (x) = (x − a)g(x) for some function g which is continuous at a. Prove that f
is differentiable at a and find f ′ (a) in terms of g.
Hints/comments:
⋆ Your arguments should make use of the formal definition of a derivative.
⋆⋆ You should prove your answer for part (a)
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MAT137 Assignment 2
Due June 9, 2025
7
Question 6. We let E denote the curve y 2 = x3 − x (Figure 1)
Figure 1: E : y 2 = x3 − x
(a) Find all vertical lines which are tangent to E.
(b) Find all horizontal lines which are tangent to E.
√
(c) Find the line which is tangent to E at the point (− 12 , − 2√32 ).
(d) Find a function f (x) with domain D which represents the branch of E lying in the first
quadrant. That is, find a function f (x) such that:
{(x, f (x)) : x ∈ D} = {(x, y) : y 2 = x3 − x, x > 0, y > 0}
(e) Let g(x) be a function such that g(f (x)) = x for all x ∈ D. You do not need to prove that
such
function exists. Find the equation of a line which is tangent to g(x) at the point
√ a√
( 6, g( 6)).
Hints/comments:
⋆ You do not need any proofs for this question, however you should still show some work. Just writing answers will
not receive full marks.
⋆⋆ You do not need to find an explicit equation for g(x) in order to answer this question.
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MAT137 Assignment 2
Due June 9, 2025
8
Question 7.
(a) Let f (x) be an increasing function defined on a domain D. Prove there exists a function g(x)
such that g(f (x)) = x for all x ∈ D.
(That is, prove all increasing functions are invertible.)
√
(b) Let f (x) = x3 − x be defined with domain D = (1, ∞). Prove that f (x) is increasing on
D.
(c) Let t > 0. Recall that if a function f (x) defined on R is periodic with period t, then the
following holds:
∀x ∈ R, f (x + t) = f (x)
Show that if f (x) is periodic with a period t, then f (x) is not invertible.
(d) Let t1 , t2 , t3 , ..., tn be positive integers. Let f1 , ..., fn be functions defined on R such that fi
is periodic with period ti for each 1 ≤ i ≤ n. Define:
h(x) :=
i=n
X
fi (x)
i=1
Prove that h(x) is also periodic.
Hints/comments:
⋆ For part (a) be sure to show that the g(x) you have defined is indeed a function
⋆⋆ You may not freely use MVT or the fact that positive derivative implies increasingness, as this has not yet been
covered in course material.
⋆ ⋆ ⋆ There is a reason ti are chosen to be integers.
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