For basic questions about limits, derivatives, integrals, and applications, mainly of one-variable functions.

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2
votes
1answer
39 views

Difference by a constant when evaluating $I = \int x\ln\sqrt{1+x^2}$ in two ways

Evaluate $I = \int x\ln\sqrt{1+x^2}$ Integration by parts yields that $$I = \frac{1}{4}[(1+x^2)\ln(1+x^2)-x^2]+C_1$$ while integration by substitution $(u=x^2+1)$ shows that $$I = ...
2
votes
1answer
84 views

Are those Locally Lipschitz definitions equivalent?

Let $f:\mathbb{R}^{+}\times \mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ be locally Lipschitz in the sence that there exists a positive $C^{0}$ function $\ell :\mathbb{R}^{+}\times \mathbb{R}^{+} ...
2
votes
2answers
154 views

Uniform Continuity: $(\ln x)^2$

Determine if $f(x)=(\ln x)^2$ is uniformly continuous on $(0,\infty)$. I think I need to use the definition on this one. im not sure how tho, so any tips/solutions? $|y-x|< \delta$ => ...
2
votes
1answer
56 views

Is this expression bounded?

I wonder: is $$ \left( 1 + \frac{n}{a} \right)^{-a} \prod_{k = 1}^n \left( 1 + \frac{a}{k} \right) $$ uniformly bounded in $n \in \mathbb{N}$ and $0 < a \leq n$? Following Jack's answer, I have ...
2
votes
1answer
26 views

Proving an inequality involving reduction sine

I'm trying to prove that $I_{2n + 2} \leqslant I_{2n + 1} \leqslant I_{2n}$ where $$ I_k = \int_0^{\pi/2} \sin^k x\, dx = \frac {k - 1} {k}\int_0^{\pi/2} \sin^{k - 2} x \, dx $$ From this I have ...
2
votes
2answers
51 views

Integral Issues.

$\displaystyle \int \cosh ^2t\,\sinh ^5t \; \textrm{d}t \,$ Can't for the life of me figure this one out. I have tried various substitutions. The pythagorean hyperbolic identity, the double variable ...
2
votes
1answer
315 views

Probability of intersection of line segments

A pair of points is selected at random inside a unit circle and a line segment is drawn joining the two. Another pair is selected and a second line segment is drawn. Find probability that the two ...
2
votes
2answers
60 views

Are there restrictions I have forgotten for Integration by Trigonometric Substitution or am I making some other mistake?

I have been playing around with some integration problems I had previously solved correctly. I attempted an approach that was a bit different on one in particular, and I am getting what seems to be a ...
2
votes
1answer
59 views

Arc length of curve (regular condition)

I have a question regarding the defintion of arc length of a curve in $\mathbb{R}^n$. If $\gamma$ is a regular curve, the define arc length as $S(t)=\int_{t_0}^t|\gamma'(t)|dt$. Since $\gamma$ is ...
2
votes
2answers
62 views

Prove using Rolle's theorem that $x^2$ has $1$ root

I'm having a hard time using Rolle's theorem in this proof and my professor said that it will likely be an exam question similar to the one above.
2
votes
2answers
44 views

Proof that the sequence $a_{n} = n(-1)^n, n \in \mathbb{N} $ diverges

The sequence is $-1,2,-3,4,-5,...$ I understand that it will oscillate between arbitrarily large positive and negative values as $n \rightarrow \infty $. How could I formalize that argument?
2
votes
1answer
138 views

Loss functions for regression

[From PRML Bishop, p:46] The average or expected loss function is given by $$E[L] = \int\int (y(x)-t)^2 p(x,t)\ \ dx\ \ dt$$, where, the loss function $L = (y(x)-t)^2$, given x and the ...
2
votes
4answers
247 views

Find all continuous functions satisfying $\int_0^xf=(f(x))^2+C$ for some constant $C \neq 0$.

Find all continuous functions $f$ satisfying $$\int_0^xf=(f(x))^2+C$$ for some constant $C \neq 0$, assuming that $f$ has at most one $0$. I have a question about the solution to this problem. It ...
2
votes
2answers
114 views

Solving inequality involving square root and division by logarithm: $\sqrt n<\frac{n}{\log(n)}-2$

I would like to solve the inequality $\sqrt n<\frac{n}{\log(n)}-2$. for some reason I had never done this before. This is clearly the same as $\frac{n}{\log(n)}-\frac{n}{\sqrt{n}}>2$. Which is ...
2
votes
1answer
119 views

Integral Power Rule Step-by-step

Real quick, two things: I'm sorry if my notation or terminology is incorrect, and I know what I'm asking isn't strictly necessary for my studies, but writing something out step by step helps me to ...
2
votes
2answers
174 views

Chain rule proof doubt

I was reading this pdf document that shows a proof of the chain rule. My doubt is in the second slide I dont understand why the $k$ value is equal to $g'(x)$ plus $v$ all of this plus $h$. Sorry I ...
2
votes
4answers
58 views

Show that the series $\sum_{k=o} ^\infty (-1)^k \dfrac{x^{2k+1}}{2k+1}$ converges for $|x|<1$ and that it converges to $\arctan x$

Show that the series $\sum_{k=o} ^\infty (-1)^k \dfrac{x^{2k+1}}{2k+1}$ converges for $|x|<1$ and that it converges to $\arctan x$ I tried using the ratio test but I got that it equals 1, so it is ...
2
votes
2answers
128 views

Complicated integration

How can this be integrated? : $$\int_{b}^{a} x \left ( \frac{D}{a-b} \right ) \left ( \frac{a-x}{a-b} \right )^{D-1}dx$$ The solution is : $$\frac{a+(D)(b)}{D+1}$$
2
votes
2answers
110 views

Proving that a polynomial has a positive root

So I want to prove that a polynomial $ P(x)=a_nx^n+a_{n−1}x^{n−1}+.....+a_1x+a_0 $ has a positive root. I'm given that $ a_n $ is positive and $ a_0 $ is negative. I want to know how to apply the ...
2
votes
1answer
49 views

Proof that the equation $x^2=\sin x $ has only one real solution different than $0$

I started doing it as following: Let $f(x) = \sin x - x^2$ Using the fact that $\sin x> x-\frac{x^3}{3!}$, I got that $f(\frac{1}{2})>0$ and, as $0<\sin 1< 1 , f(1)<0$. So, as ...
2
votes
2answers
118 views

Find $\int\frac{dx}{2+\sqrt{x}}$ (using Integration by Substitution)

I used the substitution: $u=x$ $du=dx$ $2+\sqrt{u}=2+\sqrt{x}$ I then substituted the u into the equation: $\int\frac{1}{2+\sqrt{u}}du$ $=\int{(2+\sqrt{u})^{-1}du}$ I'm not too sure how to ...
2
votes
1answer
62 views

Find $F'(t)$, where F is an integral

I need to find $F'(t)$, where $F(t)=\int_{[0,t]^2}e^{\frac{tx}{y^2}}dxdy$. My first approach: Let's observe that $\int e^{\frac{tx}{y^2}}dx=\frac{y^2}{t}e^{\frac{tx}{y^2}}+C$. So I get: ...
2
votes
1answer
98 views

Evaluating $\int_0^\infty dn \, \frac{x^n}{(3n+1)(3n+2)}$

I'm trying to prove a particular series is convergent, and I would like to use the Cauchy integral test for fun, even though it's not the most convenient. I need to evaluate, $$\int_0^\infty dn \, ...
2
votes
1answer
55 views

Convergence of $\sum_{n=1}^\infty n^s(\sqrt {n+1} - 2 \sqrt n + \sqrt {n-1})$

Convergence of $$\sum_{n=1}^\infty n^s(\sqrt {n+1} - 2 \sqrt n + \sqrt {n-1})$$ Attempt: $\sum_{n=1}^\infty n^s(\sqrt {n+1} - 2 \sqrt n + \sqrt {n-1}) \sim \sum_{n=1}^\infty n^s( \sqrt n )$ As ...
2
votes
1answer
114 views

If $f,g$ are unifromly continuous, then $\alpha f+\beta g$ is uniformly continuous?

If $f,g$ are uniformly continuous, then is $\alpha f+\beta g$ uniformly continuous? So far, I looked at here If $f,g$ are uniformly continuous prove $f+g,fg$ are uniformly continuous, but I didn't ...
2
votes
1answer
27 views

Composed of non differentiable functions

It will be possible to find a function $f:\mathbb{R}\rightarrow \mathbb{R}$ non-differentiable at zero such that $f\circ g$ is differentiable at zero where $g:\mathbb{R}\rightarrow \mathbb{R}$ is ...
2
votes
1answer
28 views

Prove the series converges uniformly at $[x_0, \infty)$

Let $\sum_{n=0}^\infty a_ne^{-\lambda_n x}$, where $0 < \lambda_n < \lambda_{n+1}$. It is given that the series converges at $x_0$. Prove that the series converges uniformly at $[x_0,\infty)$. ...
2
votes
4answers
87 views

the choice of 2 when proving the limit when $x\to\pm\infty$

Suppose that $f$ is a continuous function on $\mathbb{R}$ and $\lim_{x\to -\infty}f(x)$ and $\lim_{x\to -\infty}f'(x)$ exist. Show that $\lim_{x\to -\infty}f'(x)=0$ A common way to show this is ...
2
votes
1answer
79 views

Why is $\int_\varepsilon^{1/e}-\frac{1}{\\\log(t)}dt = \int_1^a \frac{e^{-x}}{x}dx$

Let $\varepsilon = e^{-a} \in [0,e)$, why does the following hold (including $\varepsilon=0$)? $$ \int_\varepsilon^{1/e}-\frac{1}{\\\log(t)}dt = \int_1^a \frac{e^{-x}}{x}dx $$
2
votes
1answer
66 views

How do I find the derivative of an integral?

$$g(x) = \int_{x^2 + 11}^{3} \frac{dt}{t} $$ I am not sure how to approach this, do I take the antiderivative and then minus the lower limit from upper limit plugged into the antiderivative? I think ...
2
votes
3answers
49 views

Taylor series of $\ln(1+x)$

So let's say we want to obtain the Taylor series for $\ln(1+x)$. We know that its derivative is $\dfrac{1}{1+x}$, which has the series $\sum_{n=0}^{\infty} (-1)^nx^n$. Can we take the antiderivative ...
2
votes
1answer
318 views

How to prove the limit of Thomae function?

Given the Thomae Function: $$t(x)= \begin{cases} 1 & \text{if }x=0, \\ 1/n & \text{if }x=m/n \in \mathbf Q\setminus\{0\}\text{ is in lowest terms with }n>0,\\\ 0 & ...
2
votes
1answer
92 views

Clarifying the elementary calculus used in this statistics problem

Let $X \sim N(\mu, \sigma^{2})$ and $Y = \alpha X + \beta$ for $\alpha > 0$. I'm looking at a demonstration that $Y = \alpha X + \beta \sim N(\alpha\mu + \beta, (\alpha\sigma)^{2})$, and find ...
2
votes
1answer
42 views

Difficult Limit involving digamma function

Evaluate: $$\lim_{z \to 0} \psi(-z)\cdot \bigg ( 1 - 2z(z+1) \bigg) - z\cdot\psi'(-z) $$ If we simply substitute in $0$ that gets us infinity, and problems. The answer is $-2 - \gamma$ How do we ...
2
votes
1answer
95 views

quick into 'Function Analysis', ‘Measure Theory’

Can someone suggests some quick introduction document?
2
votes
1answer
122 views

Derivative with respect to another function

I stumbled on this calculus problem here: Let $f(x) = \ln|\sec x + \tan x|$ and $g(x) = \sec x + \tan x.$ Find the fourth derivative of $g(x)$ taken with respect to $f(x)$ A)$\\$ $f'(x)$ ...
2
votes
1answer
41 views

Understanding classification of discontinuities

$\forall a,b\in \mathbb{R}$ , let $\displaystyle f\left(x\right)\:=\:\begin{pmatrix}\frac{x^2+ax+b}{x^2-8x+15} & x\ne 3,5 \\0 & x=3,5\end{pmatrix}$. I need to Find all the discontinuities ...
2
votes
3answers
33 views

Continuity of basic functions

Why is the function $y=b^x$ continuous on the real line (for$\quad b>0, b\neq 1$). I understand why $b$ has to be greater than $0$, but not why it cannot be $1$ When I graph on a graphing ...
2
votes
2answers
507 views

How to prove this function is monotonic increasing?

Let $\:f$ be a continuous function in $\mathbb{R}$,and $\forall q_1,q_2\in \mathbb{Q},\:if\:\:q_1<q_2$ then $f\left(q_1\right)<f\left(q_2\right)$. Show that $\forall x_1,x_2\in \mathbb{R}\:$, if ...
2
votes
2answers
137 views

Integral $\int\frac{(\sin(x))^2}{x^2+1} dx$

I have no idea of variable changement to use or other to calculate this integral : $$ \int_0^{\infty}\frac{(\sin(x))^2}{x^2+1}\,dx $$ Wolfram alpha gives me the result but really no idea ... I ...
2
votes
1answer
64 views

For $0<a<b$, prove that $1-\frac{a}{b}<\ln (\frac{b}{a})<\frac{b}{a}-1$ [closed]

I tried to answer this question for a few hours, without any success. I would appreciate if you helped me with the following task: $$\text{For }\,0<a<b,\, \text{ prove that ...
2
votes
2answers
97 views

Super convergent cos(x)

could you find where it come from? $$\cos (x)=\sum _{n=0}^{\infty } \frac{\left((-1)^n+1\right) e^{\frac{i \pi n}{2}} (n+1)^2 J_{n+1}(x)}{x}$$ it is seem as bessel series?
2
votes
1answer
94 views

Approximating $ e^{\sin(x)} $

I know that $$ e^{\sin(x)} = 1 + \sin(x) + {\sin^2x \over 2} + {\sin^3x \over 6} + B\sin^4x $$ I also know that $$ \sin(x) = x-{x^3 \over 6}+Bx^5 $$ I know I should probably just plug in the ...
2
votes
3answers
311 views

Find the volume of the region of a sphere bounded by two planes

Calculate the volume of a sphere $x^2+y^2+z^2=R^2$ which is bounded by $z=a$ and $z=b$, where $0\leq a<b<R$ using double integral. I can imagine the picture but I don't know how to set it up.
2
votes
1answer
36 views

Does $g'$ need to be continuous for $g(x_0) = 0$, $g'(x_0) \neq 0$ to imply $g$ changes sign in a neighborhood of $x_0$

The following theorem holds: Theorem: Let $g:\mathcal{A} \rightarrow \mathbb{R}$ be differentiable and let $x_0 \in \mathcal{A} $. If $g(x_0)=0, \; g'(x_0)\neq 0$ then $g$ changes sign at a ...
2
votes
1answer
33 views

The system has a unique solution if and only if $w^{T}A^{-1}v\ne 0$.

I have the following problem: $$\left\{ \begin{array}{ll} Ax+\lambda v=b \\ w^{T}x=\rho \end{array} \right.$$ where $A\in Gl_n(\Bbb{R})$, $u,v$ are column vector in ...
2
votes
1answer
205 views

Integrability of Composition of continuous and Lebesgue integrable functions

Suppose $g:[a,b]\to \mathbb{R}$ is Lebesgue-integrable and $f:\mathbb{R} \to \mathbb{R}$ is continuous, is $f\circ g$ Lebesgue-integrable? I am trying to prove this by obtaining simple functions ...
2
votes
2answers
136 views

Understanding the definition of a pullback of a differential $k$-form and applying it in $1-d$

I am having trouble understanding the definition of a pullback of a differential k-form in a basic course in differentiable geometry. This is the definition I am given. I believe it is easier to ...
2
votes
2answers
59 views

Partial differentiation of double integrals.

Let \begin{equation} H(x,y) = \int_0^{f(x)} \int_0^{g(y)} h(s,t) \, ds\,dt \end{equation} where $f,g$ and $h$ are continuously differentiable. Is there a standard way of computing the partial ...
2
votes
3answers
63 views

$a_n$ is bounded and decreasing

my second question from [An inequality for the product $\prod_{k=2}^{n}\cos\frac{\pi }{2^{k}}$ Let $n\geq 2\quad a_{n}=\prod\limits_{k=2}^{n}\cos\left(\dfrac{\pi }{2^{k}}\right)$ and ...