Articles of calculus

Intermediate Value Theorem and Continuity of derivative.

Suppose that a function $f(x)$ is differentiable $\forall x \in [a,b]$. Prove that $f'(x)$ takes on every value between $f'(a)$ and $f'(b)$. If the above question is a misprint and wants to say “prove that $f(x)$ takes on every value between $f(a)$ and $f(b)$”, then I have no problem using the intermediate value theorem here. […]

It is an easy question about integral,but I need your help.

How to compute this integral? $$ \int^{\pi}_{0} \frac{\sin(nx)\cos\left ( \frac{x}{2} \right )}{\sin \left ( \frac{x}{2} \right ) } \, dx$$ I need your help.

Show $\sum\limits_{k=1}^{\infty} \frac {1}{(p+k)^2} = -\int_0^1 \frac{x^p \log x}{1-x}\,dx$ holds

Prove that $$\sum\limits_{k=1}^{\infty} \frac {1}{(p+k)^2} = -\int_0^1 \frac{x^p \log x}{1-x}\,dx$$ for $p>0$. I tried to transform LHS as Riemann sum form but failed. Can anyone give some idea? Many Thanks!

$\displaystyle\sum_{k=0}^n \frac{\cos(k x)}{\cos^kx} = ?$

Prove that (not use induction) $\displaystyle\sum_{k=0}^n \frac{\cos(k x)}{\cos^kx} = \frac{1+(-1)^n}{2\cos^nx} + \dfrac{2\sin\big(\lfloor\frac{n+1}{2}\rfloor x\big) \cos\big(\lfloor\frac{n+2}{2}\rfloor x\big)} {\sin x\cos^n x} \qquad\qquad (\frac{2x}{\pi}\not\in \mathbb Z)$

$f: \to \mathbb{R}$ is differentiable and $|f'(x)|\le|f(x)|$ $\forall$ $x \in $,$f(0)=0$.Show that $f(x)=0$ $\forall$ $x \in $

$f:[0,1] \to \mathbb{R}$ is differentiable and $|f'(x)|\le|f(x)|$ $\forall$ $x \in [0,1]$,$f(0)=0$.Show that $f(x)=0$ $\forall$ $x \in [0,1]$ I used the definition of derivative: $f'(x)=|\lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}| \le |f(x)|$ Now, checking differentiability at $x=0$, $|\lim_{h \rightarrow 0} \frac{f(h)}{h}| \le 0$ which should give a contradiction as modulus of something is negative So, $|f(h)| \le 0 […]

Prove that if $f:A\to B$ is uniformly continuous on $A$ and $g$ is uniformly continuous on $B$, then $g(f(x))$ is uniformly continuous on $A$

Suppose that $f\colon A \to B$ is uniformly continuous on $A$ and $g$ is uniformly continuous on $B$. Show that $g \circ f$ is uniformly continuous on $A$. I tried to use the definition of uniformly continuous but it doesn’t work.

Comparison of the change of variable theorem

I would like to compare the change of variable theorem for 1 variable and more. What are the differences, in which case we need stronger assumptions? How do they differ? What is the best way to write the theorems for comparison? Multivariable: Let $\varphi : \Omega \rightarrow \mathbb{R}^n$ (where $\Omega \subset \mathbb{R}^k$ and $k\leq n)$ […]

Rectifiability of a curve

Let $f$ be a function defined on $[0,1]$ by $$f(x) = { 0, \text{ if } x = 0} $$ $$f(x) = { x \sin \frac 1 x , \text{ if } 0 < x \leq 1} $$ Prove that the curve $\{(x, f(x)) : x \in [0,1]\}$ is not rectifiable. I’m not sure how […]

Closed form of $\int_0^\infty \frac{dt}{\left(i t\sqrt{3}+\ln(2\sinh t)\right)^2}$

While evaluating the integral $$ I_1=\int_{0}^\infty\frac{\sin\pi x~dx}{x\prod\limits_{k=1}^\infty\left(1-\frac{x^3}{k^3}\right)}, $$ I came to this integral of elementary function $$ I_2=\int_0^\infty \frac{dt}{\left(i t\sqrt{3}+\ln(2\sinh t)\right)^2}. $$ In fact $I_2$ is real and $$ I_1=-2\pi I_2. $$ Brief outline of proof is as follows. Write the infinite product in terms of Gamma functions, apply reflection formula for Gamma function to […]

Rate of divergence of the integral of an $L^q$ function

Let $p,q\in(1,\infty)$ be conjugate exponents (i.e., $1/p+1/q=1$) and let $f:(0,\infty)\to\mathbb R_+$ be a Lebesgue-measurable function such that $\int_0^{\infty} f(x)^q\,\mathrm dx<\infty$. It follows fairly easily from Hölder’s inequality that $$\int_0^x f(t)\,\mathrm dt\leq x^{1/p}\cdot\|f\|_q\quad\forall x>0,$$ so that the (continuous) function $x\mapsto \int_0^x f(t)\,\mathrm dt$ must diverge no faster than $x^{1/p}$. (The notation $\int_0^x$ means Lebesgue integral on […]