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

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4
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98 views

In the mean value theorem, we are guaranteed $c$ such that $f'(c) = (f(b)-f(a))/(b-a)$. Does $c$ have a name?

The Mean Value Theorem says approximately that for differentiable $f$, there is a $c \in (a,b)$ such that $$ f'(c) = \frac{f(b)-f(a)}{b - a}. $$ I presume that the number $f'(c)$ is the mean value. My ...
4
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0answers
59 views

Find the slope at $t=16$ for $s(t) = $arctan$(\sqrt{t})$

A particle moves along the x axis so that its position at any time when t is greater than or equals zero is $s(t) = $arctan$(\sqrt{t})$. Find the velocity of the particle at $t=16$. The point of ...
4
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0answers
95 views

What is a good conceptual interpretation of a differential?

I'm having trouble with understanding what exactly a differential really is. For example, if we have the following function, $f(x,y)=x^2+xy+\frac{37}{x} +5$, does this differential, ...
4
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109 views

Graphs of interesting integrals of the form: $\int \sin^a(x^a)\cos^a(x^a)$

Here are a few graphs of the form:- $$\int \sin^a(x^a)\cos^a(x^a)dx$$ Where $a$ is an even, positive integer. $a = 2$ $a = 4$ $a = 6$ Now, a few graphs of the form:- $$\int ...
4
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136 views

Integral identity related with cubic analogue of arithmetic-geometric mean

Let $a,b$ be positive real numbers and we define two sequences $\{a_{n}\},\{b_{n}\}$ as follows: ...
4
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0answers
58 views

Area under a curve with polar coordinates. Seems to be too simple?

Curve is given by equation: $$r^2 = 2a^2|\cos \phi|$$ I would like to use the formula: $$A = \frac{1}{2}\int_a^b (f(\phi))^2 \, d\phi$$ So, since equation is already squared, i can put the right ...
4
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71 views

Non-analytic smooth function

The Wikipedia page (http://en.wikipedia.org/wiki/Non-analytic_smooth_function) proves that $$f(x) = \begin{cases} \exp(-1/x), & \mbox{if }x>0 \\ 0, & \mbox{if }x\le0 \end{cases}$$ is a ...
4
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0answers
89 views

how to calculate this line integral $\int_{0}^{2\pi} (16\sin^2 3t +16\cos^2 4t)\sqrt{(144\cos^2 3t +256\sin^2 4t)}dt$

I am working on a line integral to calculate the amount of chocolate to cover a pretzel. the density of the pretzel is given by this formula $\lambda=3(x^2+y^2)$ and the parameter equation of a ...
4
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97 views

Tools to bound the singular values of a finite sum of random matrices from below?

Matrix Chernoff bounds (see also this arXiv paper) are usually used to give upper bounds on the largest eigenvalue of a finite sum of random matrices. Sometimes it can also be used to give a lower ...
4
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174 views

Infinite Double Exponential Sum, with Functional Equation $g(x) = g(\sqrt{x})$

What is a closed form for $$ \lim_{n\to-\infty}\sum_{i=n}^{\infty}\frac{x^{2^i}(x^{2^i}-1)}{(x^{2^{i+2}}+1)} $$ The series has the form: $$... \frac{x^{\frac{1}{4}}(x^{\frac{1}{4}}-1)}{x+1} + ...
4
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51 views

Elliptic integration with exponential numerator.

I was wondering if someone could help me with the evaluation of an integral: \begin{equation} \int_{x_1}^{x_2} \frac{e^{ax}}{\sqrt{1-b\cos(x)}}dx \end{equation} I'm familiar with elliptic integrals ...
4
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81 views

Seperating single integral into an double integral.

Please refer to : How to prove that $\int_{0}^{\infty}\sin{x}\arctan{\frac{1}{x}}\,\mathrm dx=\frac{\pi }{2} \big(\frac{e-1}e\big)$ The answer by @Venus. What is the procedure in converting that ...
4
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159 views

Solving $\int\frac{x}{(x^2+x+1)^{\frac{1}{12}}}$

I'm trying to solve $$\int_0^1\frac{x}{(x^2+x+1)^{\frac{1}{12}}}\mathrm dx$$ To calculate it I first tried to calculate the primitive function. So let $$\int\frac{x}{(x^2+x+1)^{\frac{1}{12}}}\mathrm ...
4
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258 views

Differences among Cauchy, Lagrange, and Schlömilch remainder in Taylor's formula: why is generalization useful?

I would like to know what really are the main differences (in terms of "usefulness") among Cauchy, Lagrange, and Schlömilch's forms of the remainder in Taylor's formula. Could you provide examples ...
4
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64 views

a definite integral related to mean Gamma function

I am looking for a closed form of the following integral $$\int_0^1\frac{x^{a-1}\sin \theta dx}{x^2 - 2x\cos\theta + 1}$$ where $a>0$ and $0<θ<π$. I believe the answer is similar to the ...
4
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129 views

Integrate $\int\tan (x)\sin (ax)dx$

$$\int\tan (x)\sin (ax)dx$$ If $a$ integer number I used the wolfram.Alpha.com site to give me the following How can I use this result if I need to compute some values by using a program in ...
4
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0answers
108 views

Limit based on bounds

Let $k\in(0,1)$ be fixed. If $\limsup_{x\to\infty}f(kx)\leq \epsilon$ and $\liminf_{x\to\infty}f(\frac{x}{k})\geq -\epsilon$ for all $\epsilon>0$ then is it possible to say ...
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43 views

Is there any relation between summation and indefinite integration?

Indefinite integral is the family of all its primitives or antiderivatives. It represents geometrically a family of curves having parallel tangents at their points of intersection with the lines ...
4
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156 views

Exact values of error function

The error function is defined as $$\operatorname{erf}(z)=\frac{2}{\sqrt{\pi}} \int_0^z e^{-t^2} \, dt.$$ We know that the Gaussian integral is $$\int_{-\infty}^{\infty} e^{-x^2}\,dx=\sqrt{\pi}.$$ ...
4
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43 views

Help with an ODE

I need some help, I have this ODE but can't solve it for $y(x)$, I try every method I know, but with no succes,please, somebody can help me? $(\varepsilon-x)y=y'(-x+y^2-2x^2)$ Thanks.
4
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103 views

Closed form of arctanlog series

What tools would you recommend me for $$\operatorname{arctan}\left( \frac{1}{1}\right)\log\left(1+\frac{1}{1}\right)+\operatorname{arctan}\left( ...
4
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0answers
64 views

When do Leibniz-like rules lead to unique linear operators

Background Usually one defines differentiation in terms of limits, and then shows that differentiation satisfies the Leibniz (product) rule, $$\frac{d}{dx}(f \cdot g) = f\frac{dg}{dx} + g ...
4
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97 views

If Graham's number used $4$s instead of $3$s, at which $G$ would that number be bigger than Graham's number?

If $3\uparrow \uparrow\uparrow\uparrow3=G_1$, $ \quad G_2=\underbrace{3 \uparrow \ldots\uparrow3}_{G_1 \ \text{times}}, \quad G_3=\underbrace{3 \uparrow \ldots\uparrow3}_{G_2 \ \text{times}}$ , $ ...
4
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0answers
84 views

If a Sequence of Polynomials Converge to Another Polynomial Then the Roots Also Converge.

Proposition 5.2.1 in Artin states that: THEOREM. Let $p_k(t)\in \mathbf C[t]$ be a sequence of monic polynomials of degree $\leq n$, and let $p(t)\in \mathbf C[t]$ be another monic polynomial ...
4
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0answers
265 views

Is there a book only about epsilon delta proofs?

I want to know if there is such book, with beautiful epsilon delta proofs of all kind.
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0answers
547 views

Absolute Value in Linear Differential Equation

I have a question about dropping the absolute value sign when solving a linear differential equation. If $y'-y/x=1$ Integrating Factor $=e^{\int{-1/xdx}}=e^{-lnx}=1/x$ $y/x-y/x^2=1/x$ $[y/x]'=1/x$ ...
4
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0answers
104 views

Is it possible to find $\int \frac{1}{\sqrt[4]{1+x^4}} dx$ by parametrizing the curve $y^4-x^4=1$?

I found this integral in a handbook of integrals: $$\int \frac{1}{\sqrt[4]{1+x^4}} dx$$ I already have evaluated this integral by trigonometric substitutions and my answer agrees with that of the ...
4
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0answers
85 views

How to construct a diffeomorphism with $p_k \mapsto q_k$?

How to prove the following property? I cannot do anything. Let $M$ be a connected paracompact smooth manifold of dimension $m\geq 2$. Let $(p_k), (q_k)_{k\in \mathbb{N}}$ be sequences on $M$ which ...
4
votes
0answers
221 views

existence of solution of volterra integral equation of the first kind

$$ \int_0^t k(s,t)f(s)ds=g(t) $$ To know the existence and uniquness of solution of volterra integral equation(VIE) of the first kind, we differentiate it and convert to the VIE of the second kind. ...
4
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0answers
262 views

Is Courant's Introduction to Calculus and Analysis still up-to-date?

I just found this marvelous book and I think that it's the best book in this category, but I'm worried that it is not up-to-date. I've heard that Hardy's A Course of Pure Mathematics has some switched ...
4
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0answers
434 views

Proving u-substitution the hard way — use only definition of integration with partitions

I am trying to prove integration by substitution, i.e. for $f:[c,d] \rightarrow \mathbb{R}$ continuous and $\phi: [c,d] \rightarrow [a,b]$ continuous on $[c,d]$ and $\mathscr{C}^1$ on $(c,d)$. Then ...
4
votes
0answers
104 views

Convection-diffusion-reaction problem

I seek to solve to the system $$ \frac{\partial \phi_{a}}{\partial t} = D_{a} \frac{\partial^{2} \phi_{a}}{\partial x^{2}} - v_{a} \frac{\partial \phi_{a}}{\partial x} + \mathfrak{K}_{b}\phi_{b} ...
4
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0answers
98 views

Can Lagrange multipliers be used to give a good bound on the number of critical points?

I will explain my problem by illustrating a simple case. Easy question: Let $f(x,y)$ be a "generic" polynomial in two variables, of total degree $\le D$. What's a good upper bound for how many ...
4
votes
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154 views

The second derivative as a limit

It is well-known that if $f$ is twice differentiable at $a$, then $$ f''(a) = \lim_{h\to 0} \frac{f(a+2h)-2f(a+h) + f(a)}{h^2}. $$ See e.g. this question or this question. On the other hand, the ...
4
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147 views

Product of Elements in SU(2)

Let $$ V := \frac{x_4+i\vec{x}\cdot{\vec{\sigma}}}{\left|x\right|}$$ where $\left(x_1,x_2,x_3,x_4\right)\in\mathbb{R}^4$, $|x|$ is the Euclidean norm, and $\sigma^j$ are the Pauli matrices. Let ...
4
votes
0answers
162 views

Integral of product spherical harmonics

I'm trying to calculate this integral: $$\int_0 ^\pi\int_0^{2\pi} [\sin\theta(\cos\theta \sin \phi + \sin\theta \cos(2\phi))]^2 \sin\theta d\phi d\theta $$ We could compute this integral ...
4
votes
0answers
188 views

Integral $ \int_0^\infty \frac{x^n\ln x}{(x^2+\alpha^2)^2(e^x-1)}dx$

Hey I am trying to integrate $$ \int_0^\infty \frac{x^n\ln x}{(x^2+\alpha^2)^2(e^x-1)}dx,\quad \alpha,n \in \mathbb{R}^{0+}. $$ This integral is old. I am also looking for literature on these ...
4
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62 views

Calculation of $\lim_{x\rightarrow 0}\frac{\sin (\pi\cos^2 x)}{x^2}$

Calculation of $\displaystyle \lim_{x\rightarrow 0}\frac{\sin (\pi\cos^2 x)}{x^2}$ $\bf{My\; Try::}$ Given $\displaystyle \lim_{x\rightarrow 0}\frac{\sin (\pi\cos^2 x)}{x^2} = \lim_{x\rightarrow ...
4
votes
0answers
112 views

Ugly-nice double series

I'm trying to evaluate the following ugly double sum (presented in raw notation as used in my calculations): $\sum _{m=1}^{\infty } \sum _{n=1}^{\infty } \frac{4 m \cos \left(\frac{2 \pi m ...
4
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0answers
118 views

Was the Weierstrass function constructed or discovered?

Reading Halmos' I want to be a mathematician, he mentions a continuous function without a tangent. Naturally, I was curious to see how such a function could possibly exist, and I imagined it to be ...
4
votes
0answers
111 views

Symbolic math engines barf on this ostensibly tractable integral.

$$\frac14 \int_{-M\pi}^{N\pi - s} \cos(tu/M) \cos((t+s)u/M)(1-\cos(t/M))(1-\cos((t+s)/N))\space \mathrm d t$$ with integer $u$. Alpha runs out of time. Maxima gives a tremendous result that can ...
4
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0answers
161 views

$\int e^{-x} \log \log x dx$ - another special integral?

I came across this integral in some old notes. After several unsuccessful attempts I ran it in WA and got an interesting result: the antiderivative (closed form) doesn't exist, but the bounded ...
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0answers
186 views

Measuring how change in input variables contributes to output in non linear equation.

How do we measure how a variable contributes to an output as its value increases, and how it relates to other input variables? Let's say we're playing a video game, where you can buy items to augment ...
4
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0answers
242 views

Evaluating $\int\frac{\sin x}{\sin (x-a)\cdot \sin (x-b)}\,\mathrm dx$

$$\int\frac{\sin x}{\sin (x-a)\cdot \sin (x-b)}\,\mathrm dx$$ My Try:: \begin{align} &\displaystyle \frac{1}{\sin (b-a)}\int\frac{\sin \{(x-a)-(x-b)\}}{\sin (x-a)\cdot \sin (x-b)}\cdot\sin ...
4
votes
0answers
216 views

Difficult Sum: Any Tips?

I'm trying to integrate a very difficult expression, and I've arrived at a particular step involving this sum, which I want to turn into a closed expression so that I can raise it to a power, re-sum ...
4
votes
0answers
54 views

integration of simple function

So I have this question I saw this week which I do not understand the answer for it. Let $a,b \in \mathbb{R} a <b \ $ and let $f_1:[a,b] \rightarrow \mathbb{R}$ and I know that $\int_a^x ...
4
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0answers
2k views

Continuity on open and closed intervals

I will be taking Calculus I soon, and I just want to make sure I understand some concepts correctly. So far, reading my book for Calculus I, I've encountered the definition of continuity as being ...
4
votes
0answers
112 views

Extracting Taylor coefficients of a quotient

I was wondering if anybody has come across functions of the form $$\Phi_n(z):=\frac{f(z)^{n+1}}{zf'(z)-f(z)}\quad (n\geq 1).$$ Here, $$f(z)=\sum_{k=0}^{\infty} a_kz^k$$ is holomorphic on the open unit ...
4
votes
0answers
139 views

Differential calculus on Banach space

I'm revising for my upcoming test, and this problem dated back some years ago. I've been working on this problem for almost a day, but I don't even know how to start it correctly. Problem Given the ...
4
votes
0answers
48 views

Laplacian of a generalized Whitehead potential

I'm trying to calculate the Laplacian $\triangle\Phi_W^{IJ}$ of the following "generalized Whitehead potential": $$ \Phi_W^{IJ}(\vec{x}) = ...