[SOLVED] MAT137 Assignment 2SPSS

MAT137 Assignment 2

Due June 9, 2025

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:

❼ T0(x) = 1

❼ T1(x) = x

❼ 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.

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.

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.

Question 4. Find the derivative. If the problem asks for a the nth derivative, give a proof by induction.

(a) Compute  for any n ∈ N.

(b) Compute  for any n ∈ N.

(c) Compute , where f(x) = x(sin x + 1)x

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.

Question 6. We let  denote the curve y2 = x3 − x (Figure 1)

Figure 1: : y
2 = x
3 − x

(a) Find all vertical lines which are tangent to .

(b) Find all horizontal lines which are tangent to .

(c) Find the line which is tangent to at the point .

(d) Find a function f(x) with domain D which represents the branch of lying in the first quadrant. That is, find a function f(x) such that:

{(x, f(x)) : x ∈ D} = {(x, y) : y2 = 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 a function exists. Find the equation of a line which is tangent to g(x) at the point (√6, g(√6)).

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) = √x
3 − 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:

Prove that h(x) is also periodic.