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

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3
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30 views

Finding the length of an elliptical spiral

Okay, i had a very strange thought, it was "Is it possible to find the length of an elliptical spiral whose major and minor axes were decreasing?" Like for example lets say that $$ \frac{a}{b} = n ...
3
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49 views

If $f$ is differentiable at $x_0$, then $\lim_{h\rightarrow 0}(f(x_0 + ah) - f(x_0))/h = af'(x_0)$

If $f$ is differentiable at $x_0$ and $a\in \mathbb R$, show that $ \lim_{h\rightarrow 0} {f(x_0 + ah) - f(x_0) \over h} = af'(x_0)$ EDIT Suppose $u = ah $ then $ h = a/u$. So now we have $$ ...
3
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25 views

Analysing an integral with its inverse integral [Image, and good explanation inside]

I have a function $f(x) = \dfrac{x}{\sqrt{1+x^2}}$ This means $f^{-1}(x) = (\pm)\dfrac{x}{1-x^2}$, where the negative solution is ignored for this problem. If I want to find a relation between the ...
3
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31 views

Is there any diffeomorphism from A to B that $f(A)=B$?

I've been thinking about this problem. $A = \{(x,y) \in \mathbb{R}^2 : x \ge 0 \wedge y \ge 0\}$ $B = \mathbb{R}^2 \setminus \{(x,y) \in \mathbb{R}^2 : x>0 \wedge y>0 \}$ Is there any ...
3
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26 views

To limit an integral by a function

Being $$u(x,t)=\frac{1}{\sqrt{4 \pi t}} \int_{-\infty}^{\infty} e^{\frac{-(x-y)^2}{4t}} u(y,0) dy$$ suppose $0\leq u(x,0)$, $||u(-,0)||_1 < \infty$, and $u(x,0)$ nonzero. Prove that there is a ...
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51 views

Closed-form of $\int_{0}^{\infty} \frac{{\text{Li}}_2^3(-x)}{x^3}\,dx$

Is there a possibility to find a closed-form for $$\int_{0}^{\infty} \frac{{\text{Li}}_2^3(-x)}{x^3}\,dx$$ We have $$I=\int_0^1\frac{Li_2^3(-x)+x^4Li_2^3(-\frac{1}{x})}{x^3}\,dx$$ After repeatedly ...
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24 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 ...
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46 views

Implicit differentiation gives different results

I want to find an expression for the derivative of the implicit function below: $$\arctan(xy)=\frac{\pi}{4}e^{x-y}, \qquad \text{at point } (1,1)$$ I've tried to derive this using both Maple and ...
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58 views

Can this summation be expressed differently?

Lets say I have a sum that states the following $$ \sum_{j=0}^{k-c} {k-c \choose j}\ln(a)^{k-c-j} \frac{d^j}{dx^j}[(x)_c] $$ where $(x)_c$ is the falling factorial such that $$ (x)_c = ...
3
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110 views

Evaluate $ \int_0^3 \frac{x^3}{1-x^4}\, dx. $

Evaluate $$ \int_0^3 \frac{x^3}{1-x^4}\, dx. $$ I evaluated the integral and got $\left[\dfrac{-\ln(1-x^4)}{4}\right]_0^3$ which ended up diverging. Any help is appreciated!
3
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43 views

Why can't we generalize straight line equations for all curves?

Apologies, but I'm confused. Let $y = f(x)$ be curve such that at a point $(a,b)$ lying on it, the slope of the tangent kissing that point is $f'(x)$ Now, the equation of the tangent passing through ...
3
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32 views

Evaluating the sum of a series to a power and finding the limit

I wish to find the limit of the following series: $$ \lim \limits_{n \to \infty} a_n := (1 + 2^2 + ... + n^n)^{(\frac{1}{n^2})} $$ I'm currently doing basic real analysis although I've never come ...
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50 views

Tangent Line for Polar Curve R=$3-3sin\theta $

i start by changing polar coords into x and y and then find the derivatives to get the slope. $$x=(3-3\sin\theta)\cos\theta $$ $$x=3\cos\theta -3\cos\theta \sin\theta $$ and took $x'=(-3\sin\theta ...
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37 views

How to find the Maclaurin series for the integral of $e^{x^2}$?

I am trying to find the Maclaurin series for the integral of $e^{x^2}$? What I done so far is that the Maclaurin series for $e^{x^2}$ is $$e^{x^2}=\sum_{n=0}^{\infty}\frac{x^{2n}}{n!}$$ So would ...
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55 views

How to evaluate the following integrals

$$\int\limits_0^{\frac{\pi }{2}} {{x^2}{{\ln }^2}\left( {\sin x} \right)\ln \left( {\cos x} \right)dx} ,\int\limits_0^{\frac{\pi }{2}} {x\ln \left( {\sin x} \right){{\ln }^2}\left( {\cos x} \right)dx} ...
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49 views

Is the exponential function the one this problem is hinting at?

Suppose that $f$ is holomorphic on all of $\mathbb{C}$ and that $$\lim_{n\rightarrow \infty} \left(\frac{\partial}{\partial z}\right)^nf(z)$$ exists, uniformly on compact sets, and that this limit ...
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64 views

Find the length of the arc of the curve

$$x = \frac{y^5}{5}+\frac{1}{12y^3}\quad\text{for } 2\leq y \leq4$$ I have the following so far: $$x' = y^4-\frac{1}{4y^4}$$ $$1 + (x')^2 = 1+y^8-\frac12+\frac{1}{16y^8} = ...
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29 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 ...
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80 views

I have one month preparation. Please suggest some books.

I have one month for my GRE subjective Mathematics test. I am from India. I have learnt $75\%$ of the syllabus in my UG and high school mentioned in the ETS. I am starting today, will I be able to ...
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36 views

How to solve integrals of the form $\int u^{-\alpha} e^{-\beta u} du$?

I have to simplify an integral of the form $\int u^{-\alpha} e^{-\beta u} du$, where $\alpha, \beta \in \mathbb{R}^{++}$. Is it a standard integral, or a family that subsumes gamma integrals? Is there ...
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32 views

Given a point $A$, describe those points to which a catenary cannot be drawn from $A$.

Background An elementary problem in the calculus of variations shows that among all curves joining two points $A$, and $B$ in the first quadrant, the one which generates the surface of minimum area ...
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86 views

How can I show that the integral equals zero?

Problem: Show that $$\int_{0}^{\pi / 2} \ln\left(\tan x - \sqrt{2 \tan x} + 1\right)\,\mathrm{d}x = 0 $$ I'd like to use, if possible, only single-variable Calculus methods, and it does not include ...
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34 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 ...
3
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114 views

Squeeze Theorem: Finding the limit of a trig function

I'm stuck on finding the limit of a complex fraction/trig function. Could someone please assist, or point out where I'm going wrong? Determine $$\lim\limits_{x \to 0} ...
3
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40 views

Maximize or find an upper bound of the function $kx^{k-1}\exp(-\mu(x^k-x))$

I was programming some random variable simulation using the acceptance-rejection method and I encounter with the Weibull$(k,\lambda)$ distribution. This random variable is posible to simulate with ...
3
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53 views

Integration indefinite integral of multiple functions

I need help integrating $$\frac{x}{1-\exp(-x^2/a^2)}\exp((x-u)^2/2s^2)$$ wrt $x$, where $a$ and $u$ are constants
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98 views

Is this proof correct? Divergence of $\int_{1}^{\infty} \left| \frac{\sin x}{x} \right| \, \mathrm{d}x $

Problem: Show that $$ \int_{1}^{\infty} \left| \frac{\sin x}{x} \right| \,\mathrm{d}x $$ diverges. I know that there are many questions in which this problem is solved, but I want to know if my ...
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0answers
85 views

Dealing with absolute values after trigonometric substitution in $\int \frac{\sqrt{1+x^2}}{x} \text{ d}x$.

I was doing this integral and wondered if the signum function would be a viable method for approaching such an integral. I can't seem to find any other way to help integrate the $|\sec \theta|$ term ...
3
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53 views

Are there some functions that cannot be optimized using calculus?

I've been working on a project to maximize a functions output using a genetic algorithm. However, from the limited calculus I know I thought there were methods to find the maximum of a mathematical ...
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50 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 ...
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115 views

Help on the Integration of $\int_0^{\infty} e^{-bx}\sin ax^2 \, \mathrm{d}x$.

I have had the misfortune of coming across the following integral, for real $b$ and $a > 0$: $$\int\limits_{0}^{\infty} e^{-bx} \sin\left(ax^{2}\right) \, \mathrm{d}x.\tag{1}$$ Naturally, I ...
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146 views

Any comments on Lax's “Calculus with Applications, 2e”

There's a new calculus book titled Calculus with Applications by Peter Lax (2nd edition of an old one). I really liked his linear algebra and functional analysis books, and I was wondering if this ...
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68 views

Integral substitution paradox

Assume $f \in L^+(\mathbb{R})$ and $x>0$. Consider the integral $$ \int_0^\infty \frac{f\left(\frac{x}{y}\right)}{y} \: dy. $$ I am trying to make the substitution $u=x/y.$ I seem to get $$ ...
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43 views

Gaps between roots of trigonometric polynomials?

Given a polynomial in $e^{\mathrm{i}k t}$ of the form $$ p(t) = \sum_{-n\leq k\leq n} c_k e^{\mathrm{i}k t} $$ with $\bar c_{-k} = c_k$, is there a good way of characterising how close its roots can ...
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52 views

Where to post a Calculus review guide?

I created a PDF document (using LaTeX) in which I wrote relevant review materials and Calculus problems for Calculus 1, 2, and 3. Is there an appropriate forum where I could try to post this to ...
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30 views

Convergence of sum of antiderivative and derivative

This question is inspired by this question: Solutions for $ \frac{dy}{dx}=y $?. It makes me wonder if there are any function where the sum of all antiderivative and derivative converges. The ...
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72 views

Ordinary differential equation­

$$\dfrac{dy}{dx}-\dfrac{\tan y}{1+x}=(1+x)e^x\sin y$$ I tried $\sin y=t$ but failed. It seems to immune to methods I know of or I am just unable to make the right substitution... Wolfram alpha ...
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192 views

Question on Moment of inertia about center of mass of a smooth plane curve.

This question is in continuation to this and motivated to answer this question. If I have an answer to it, then I can prove a special case of this question. Consider two smooth plane curves $C \equiv ...
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75 views

Closed-form of $\sum\limits_{n=1}^{\infty}\dfrac{x^n}{\sqrt{n\pi}}$

What is the closed-form of the following series $$\sum\limits_{n=1}^{\infty}\frac{x^n}{\sqrt{n\pi}}\ ?$$ From a different approach I got the answer as $(1-x)^{-1/2}$, by using the fact that ...
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38 views

What does $|d x|$ denote in $\int_{\gamma} |f| |dx|$?

What does $|d x|$ denote in $\int_{\gamma} |f| |dx|$? I'm not sure how to interpret this notation. Is it $\int_0^1 |f(\gamma(t))| |\gamma'(t)| dt$? In the context where I see it that would give the ...
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110 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 ...
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0answers
51 views

Does this integral variable change makes sense to you?

I was Reading a book about calculus when I've found this part about variable substitution in integrals: Consider $f$ defined in na interval $I$. Suppose that $x =\phi(u)$ is inversible, and its ...
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41 views

Help proving(well, disproving) the convergence of $\sin^3(x)$

So I'm stuck on a question where it's asking for the power series and radius of convergence of $\sin^3(x)$ I've done the power series ok, but my problem is that when I apply the ratio test it's ...
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120 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 ...
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65 views

How to practice applied mathematics calculation skill

As a natural science student in university, you may encounter so many problems that might require a deep understanding in integrating skills and series calculation. But as many of the college students ...
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48 views

Maximize an integral

I have the following integral to maximize but I don't know what to do with this $f(b,y)$ in the first integrand: $$ F=\intop_{a}^{b}[f(b,y)+\intop_{a}^{b}f(x,y)dx]dy $$ a and b are constants. Do I ...
3
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27 views

Intersection of functions and the ratio of curvature and slope

I have two functions $f: \mathbb{R_+} \rightarrow \mathbb{R}$ and $g: \mathbb{R_+} \rightarrow \mathbb{R}$, both strictly increasing, strictly concave, and twice differentiable. For all $x \in [a,b]$, ...
3
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76 views

Proving $\sum^\infty_{n=1}a_n$ converges absolutely iff each each sub series converges

We have a series $\displaystyle\sum^\infty_{n=1}a_n$ and a sub series $\displaystyle\sum^\infty_{k=1}a_{n_k}$ where $n_k\in\mathbb N$. Prove that $\displaystyle\sum^\infty_{n=1}a_n$ converges ...
3
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0answers
40 views

Integration over time by having derivation

Assume we want to find the following integration: \begin{equation}\int_{t=0}^{\infty} p(t)dt\end{equation} where $p(0)=p$ and also $$\frac{dp(t)}{dt}=-p(t)(1-p(t))\mu$$. Is there any easy way to ...
3
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0answers
119 views

Differentiation under integral sign

There is this integral that I used a lot in my research: $$\int_{-\infty}^{\infty}\exp\left(-b^{2}\left(x-c\right)^{2}\right)\mathrm{erf}\left(a\left(x-d\right)\right)\,\mathrm{d}x = ...