HOMEWORK 6 — DUE OCT 31ST

MATH 25

There are two sections: A and B. Please return each part to the proper CA. Remember that you are not allowed to use a calculator nor a computer (in problem B.4 for example!)

A. Problems graded by Benjamin

A.1. Filling holes. Let f, g be two continuous functions on the interval [a, b] and fix c ∈ (a, b). Suppose that f is C^{1} on [a, b] \ {c} and that f′ = g on [a, b] \ {c}.

(1) Prove that f is differentiable at c and that f′ = g on [a, b]. Hint: consider the function x ↦ f(x) − ∫_{c}^{x} g(t)dt.

(2) Prove that the function f defined by f(0) = 1 and f(x) = x/ sin(x) is C^{1} on [−π/2, π/2].

A.2. Some Riemann sums I. Let I = [0, 1] and let f : I → R be a continuous function. For n ≥ 1, define a step function f_{n} by:

f_{n}(x) = \begin{cases}f(1/n) & if 0 ≤ x ≤ 1/n \\ f(k/n) & if 1 ≤ k ≤ n − 1 and k/n < x ≤ (k + 1)/n. \end{cases}

(1) Prove that |f_{n} − f|_{I} → 0 as n → +∞;

(2) use this to show that

lim_{n→+∞} \frac{1}{n} ∑_{k=1}^{n} f (\frac{k}{n}) = ∫_{0}^{1} f

(3) prove that the above formula still holds if f is a ruled function on I.

A.3. Some Riemann sums II. Using the results of the previous exercise, compute the following limits:

lim_{n→+∞} ∑_{k=1}^{n} \frac{1}{k + n}, lim_{n→+∞} ∑_{k=1}^{n} \frac{n}{k^{2} + n^{2}} , lim_{n→+∞} ∑_{k=1}^{n} \frac{k^{3}}{n^{4}} cos (\frac{4k}{n} ) .

2 MATH 25

A.4. Some Riemann sums III. Let n be an integer ≥ 2. If 1 ≤ i ≤ n, we write n = q_{i,n}i + r_{i,n} with 0 ≤ r_{i,n} ≤ i − 1. Let a_{n} be the number of i’s with 1 ≤ i ≤ n such that r_{i,n} ≥ i/2. Let ϕ : I → R be defined by ϕ(0) = 0 and ϕ(x) = ⌊2/x⌋ − 2⌊1/x⌋, where ⌊y⌋ denotes the integer part of y.

(1) prove that q_{i,n} and r_{i,n} above are uniquely determined;

(2) prove that ϕ is a ruled function;

(3) compute ∫_{1/k}^{1} ϕ if k ≥ 2 and use this to compute ∫_{0}^{1} ϕ;

(4) prove that a_{n} = ∑_{k=1}^{n} ϕ(k/n);

(5) prove that lim_{n→+∞} a_{n}/n exists and compute it.

B. Problems graded by Inna

B.1. Euler’s constant.

(1) prove that there exists a constant M ≥ 0 such that 0 ≤ x − log(1 + x) ≤ Mx^{2} if 0 ≤ x ≤ 1;

(2) use this to show that

lim_{n→+∞} (∑_{k=1}^{n} \frac{1}{k}) − log(n)

exists. It is called Euler’s constant.

B.2. Cauchy-Schwarz’ inequality. Let f, g : [a, b] → R be two ruled functions. Prove that

(∫_{a}^{b} fg)^{2} ≤ ∫_{a}^{b} f^{2} ∫_{a}^{b} g^{2}.

Hint: obviously, ∫_{a}^{b}(f − gt)^{2} ≥ 0. Now judiciously choose t.

B.3. Four inequalities.

(1) Prove that if f, g : [0, 1] → R are non-decreasing ruled functions, then

∫_{0}^{1} fg ≥ (∫_{0}^{1} f) (∫_{0}^{1} g) .

(2) prove that if f : [0, 1] → R \ {0} is a continuous function, then

(∫_{0}^{1} f) (∫_{0}^{1} 1/f) ≥ 1.

(3) Let f : [0, 1] → R be a C^{1} function such that f(0) = f(1) = 0. Prove that

|∫_{0}^{1} f| ≤ \frac{1}{4}|f′|_{[0,1]}.

HOMEWORK 6 — DUE OCT 31ST 3

Can one replace 1/4 by a smaller number?

(4) Let f : [0, π] → R be a C^{1} function such that f(0) = f(π) = 0. Prove that

∫_{0}^{π} f(t)f′(t) cotan(t)dt = \frac{1}{2} ∫_{0}^{π} f^{2}(t)(1 + cotan^{2}(t))dt.

and use this to show that

∫_{0}^{π} f^{2} ≤ ∫_{0}^{π}(f′)^{2}.

B.4. Lots of derivatives. Recall that sinh(x) = (e^{x} − e^{−x})/2. Let f(x) = sin(sinh(x)) − sinh(sin(x)). Find the least integer n ≥ 0 such that f^{(n)}(0) ≠ 0 and compute f^{(n)}(0). Hint: you can use Taylor’s expansion for sin and sinh.