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Questions tagged [calculus]

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

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Integrate the product of a heaviside step and the absolute value?

I have a rather tricky integral here: $$\underbrace{\int_0^R r_0\Theta(R-r_0)|r-r_0|dr_0}_{(1)} - \underbrace{\int_0^R r_0\Theta(R-r_0)|r+r_0|dr_0}_{(2)} \ \ \ \ \cases{0\le r < \infty \\ R=1}$$ ...
Researcher R's user avatar
1 vote
1 answer
52 views

Using Graph of $\frac{1}{x}$ to Find $\delta$

Hello, This is a question I was assigned for homework and I am struggling to find a correct answer. The question states "Use the graph of $f(x)=\frac{1}{x}$ below to find a number δ such that $|f(...
Squishy698's user avatar
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0 answers
9 views

Solving challenging 4D integrals arising from triangle-triangle gravitational interaction

I am trying to find a closed form for two related integrals, coming from a physics problem partially solved here, about attractive forces between two triangles : $$\begin{align} {\bf F}_1 &= -G ...
user1420303's user avatar
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2 answers
72 views

I can't calculate this limit

I am struggling to calculate what $h(x)$ tends to in $$\lim_{x\to\infty}\left[\cos\left(\dfrac{1}{x}\right)\right]^{h(x)},$$ where $$h(x)=\dfrac{x^4+x^2-1}{2x+1}\sin\left(\dfrac{1}{x}\right)$$ This is ...
sofischh's user avatar
1 vote
0 answers
12 views

How to calculate the functional derivative of this composite function?

I think I am not unfamiliar with the functional derivative, while recently I encounter a paper which gives the functional derivative expression like this $$W=\ln[\sum_{N=0}^{\infty}\frac{1}{N!h^{3N}}e^...
Xeh Deng's user avatar
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0 answers
23 views

Partial Derivative of a Scalar Function

I am working on deriving expressions related to a scalar function $\theta$ defined as $\theta = \cos^{-1} \left( \frac{v_{io}^T}{\|v_{io}\|} \frac{x_{oi}} {\|x_{oi}\|} \right)$, where $v_{io}$ and $x_{...
Thinesh's user avatar
  • 13
3 votes
0 answers
25 views

Determining the significance of a curve's factors

Given the equation $x^2+x+1$ you could easily determine that $x^2$ will have the greatest overall impact on the curve--then $x$ and finally $1$. And this holds true for any coefficients present as the ...
SlavaCat's user avatar
0 votes
1 answer
24 views

Bump function with integral $1$ and value $1$ at zero

How can i contruct a smooth bump function $F$ on $\Bbb{R}^n$ such that $F(0)=1$ and with integral $1$? I have tried to manipulate the function $f(x)=e^{-\frac{1}{x^2}}$ if $x>0$ and $f(x)=0$ if $x \...
Marios Gretsas's user avatar
1 vote
0 answers
33 views

If $f$ is discontinuous but continuous for each component,is there a continuous function $g$ that makes $f \circ g$ discontinuous for some component?

For example $$f(x,y)= \begin{cases} \frac{xy}{x^2+y^2} & (x,y)\neq (0,0)\\ 0 &(x,y)=(0,0)\end{cases}$$ $f$ is discontinuous but continuous for each component, In polar coordinates(Equivalent ...
9sky's user avatar
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1 vote
0 answers
30 views

Is there such a thing as the "Second Passage Time"?

I am learning about the First Passage Times (https://en.wikipedia.org/wiki/Hitting_time) of Stochastic Processes - how to derive the Probability Distribution for the time required for a Stochastic ...
konofoso's user avatar
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-1 votes
1 answer
31 views

Recursive piecewise integral formula

I have the recursive formula for the integral $1/(x^2+a^2)^n$, which is, in fact, the one that Ng Chung Tak provides in this link. My problem is that when finding a specific integral, for the case $n=...
Emerson Villafuerte's user avatar
3 votes
2 answers
208 views

Why can the Binomial Distribution be Approximated by a Normal Distribtuion?

As a practice problem, I am trying to prove the relationship between the Normal Distribution and the Binomial Distribution. I have seen several proofs of this before (e.g. Justifying the Normal Approx ...
konofoso's user avatar
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29 views

Concavity where second derivative is undefined at a point but negative everywhere else

I want to say that $y=-|x|^{1.5}$ must be strictly concave since $y'$ is continuous and $y''=-.75/\sqrt{|x|}<0$ everywhere except at $x=0$ where it's undefined. (I'm working in real numbers only.) ...
Rookie's user avatar
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-2 votes
2 answers
136 views

What is the sign of $I_n = \int _{0}^{1}\frac{x^{2n+1}}{x^{2}+1}dx$ [closed]

I was given an exercice to calculate $I_0$ and then $I_0 + I_1$ and then deduce $I_1$, and then asks the sign of $I_n$, can someone help? I tried deductive reasoning but I don't know how to complete ...
Nassim Ouali's user avatar
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1 answer
61 views

Issue in numerical integration of $\frac{1}{4\pi^{3/2}i}\int_Ce^{-\frac{z^3}{12}-tz}z^{-1/2}dz$

I am trying to numerically integrate the integral representation of $\operatorname{Ai}^2(x)$. The representation is $$\operatorname{Ai}^2(t)=\frac{1}{4\pi^{3/2}i}\int_Ce^{-\frac{z^3}{12}-tz}z^{-1/2}dz....
random0620's user avatar
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Is there a process by which we can simplify the limit definition of the derivative for any differentiable function into something usable?

When finding the derivative of a specific function using the limit definition (i. e. $f’(x) = \frac{f(x + h) - f(x)}{h}$, generally I’ve used a lot of tricks. With polynomials, the $h$s cancel out ...
APersonThatExists's user avatar
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Arrangements of fixed k-polyplets in a $n\times n$ matrix

Recently, I asked a question about the number of arrangements of $k$ elements inside a $n\times n$ matrix with certain restrictions. The one I´m actually interested in for this question is in its 2. ...
Cardstdani's user avatar
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1 answer
57 views

Find a sequence function for combinatorial sequences

I´m trying to find a sequence function for the following sequence: $0, 1, 85, 419, 973, 1747, 2741, 3955, 5389, 7043, 8917, 11011, 13325, 15859, 18613, 21587, 24781, 28195$ The first term is generated ...
Cardstdani's user avatar
0 votes
0 answers
35 views

Decompose a function of multiple variables to product of functions of single variable

For simplicity, we consider a two variable function $f = f(x,y): \mathbb{R}^2 \to \mathbb{R}$. As stated in this post, in general, we can't write $$f(x,y) = g(x) h(y)$$ as a product of single variable ...
nyan's user avatar
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0 answers
29 views

Related to double integration

In my research work, I got the following expression: $\int_0^{\infty} \int_0^{Z_{lim}} 1-\exp(-\lambda*\frac{(\gamma_{th}([A+\beta^2y^2](P_u/jj1)+1/jj1)-\beta^2yz(\alpha_u-\gamma_{th}\alpha_t))}{\...
Heretolearn's user avatar
3 votes
2 answers
60 views

Why this formula cannot be used here? [duplicate]

My book says Why can't I apply it to $$g(x)=\int_{0}^{x}e^{x-t}dt$$ Which gives $g'(x)=e^{x-x}$ but when we integrate properly we get $g'(x)=e^x$ What I thought $g(x)=\int_{u(x)}^{v(x)}f(t,x)dt$ As ...
Anshu Gupta's user avatar
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0 answers
21 views

Independent Variable Technique in Derivative of a function with respect to another function

This question differentiates $y=x$ with respect to $x^2$ by introducing a variable $u$, and answers say his method is valid. On the other hand, the same technique when used here has been downvoted, ...
Starlight's user avatar
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4 votes
0 answers
52 views

If $\lim_{x\to a}f(x)=\infty$ and $\lim_{x\to a}g(x)=c$, where $c$ is a real number, prove the following

If $\lim_{x\to a}f(x)=\infty$ and $\lim_{x\to a}g(x)=c$, where $c$ is a real number, prove the following: $\lim_{x\to a}[f(x)+g(x)]=\infty$ $\lim_{x\to a}[f(x)g(x)]=\infty $ if $c>0$ $\lim_{x\to a}...
EpicFaceInc100's user avatar
4 votes
0 answers
71 views

Validity of Python-derived solution for contour integral $\oint f(z)f(z-\overline{z})~dz$

$\newcommand{\on}[1]{\operatorname{#1}}$ $$ \mbox{Consider the function:}\quad \on{f}\left(z\right) = \frac{{\rm e}^{tz}}{\left(1 + z^{2}\right)^{3}}\, \left(\sqrt{t} - t\right)\ \ni\ t,z \in \mathbb{...
MASTER DHRUV's user avatar
-5 votes
0 answers
63 views

Differential Equations riddle: $f=(f’)(f’’)(f’’’)(f’’’’)\dots$ [closed]

$$f=(f’)(f’’)(f’’’)(f’’’’)\dots$$ I found this question someone posted in a group chat, and no one has solved it yet
Brian Li's user avatar
2 votes
3 answers
142 views

Why $\lim_{x \rightarrow \infty } \frac {P(x)}{Q(e^x)} = 0$ for polynomials $P(x)$ and $Q(x)$?

I am trying to reason why for any $2$ polynomials $P(x)$ and $Q(x)$ defined over the reals, $\lim_{x \rightarrow \infty } \frac {P(x)}{Q(e^x)} = 0$. This assertion was made in this answer which I am ...
Princess Mia's user avatar
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1 vote
0 answers
57 views

Stokes Theorem on standard simplices

I was studying the de Rham homomorphism, and more specifically Stokes theorem for chains, and I'm wondering whether one could avoid talking about manifolds with corners and just give a definition of ...
Paz's user avatar
  • 21
0 votes
0 answers
23 views

Integral of Poisson Kernel

This doubt comes from Dupaigne's book named stable solutions of elliptic partial differential equations. The Poisson Kernel is \begin{equation} P(x,y)=\frac{\partial G(x,y)}{\partial n_{y}}=\frac{1-|x|...
Richard's user avatar
  • 89
2 votes
0 answers
27 views

Attempt to derive Taylor expansion using discrete time steps

I was playing around trying to check if the nth order Taylor approximation about 0 can be explained and obtained by using a physical reasoning argument like below for the case $n = 2$ (that is given $...
Alejandro's user avatar
  • 191
0 votes
1 answer
53 views

Prove that a functional equation has at least one root

Let $f$ be a strictly increasing and continuous function that satisfies the equality $$f^3(x)+βf^2(x)+γf(x)=x^3-2x^2+6x-1, \forall x\in\mathbb{R} $$ with $β,γ\in\mathbb{R}$ s.t $β^2<3γ$ Prove that ...
Antony Theo.'s user avatar
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1 vote
2 answers
72 views

About solution to homogeneous ODE $u'' + u = 0$

It is known that the solution to $u'' + u = f(x)$ with $u(0) = u'(0) = 0$ is $$u(x) = \int_0^x\sin(x - \xi)f(\xi)d\xi,$$ where $\sin(x-\xi)$ is a solution to $$\frac{d}{dx}R + R = 0, \ \ \ x > \xi,$...
user57's user avatar
  • 796
0 votes
2 answers
51 views

Find solution of the IVP $ y' = y+ \frac12 |\sin(y^2)|,\,\, x>0,\,\, y(0) = -1$

Consider the following initial value problem $$ y' = y+ \frac12 |\sin(y^2)|,\,\,\,\,\,\, x>0,\,\,\, y(0) = -1$$ Which of the following statements are true? 1.) there exists an $\alpha \in (0,\infty)...
Ark's user avatar
  • 135
0 votes
1 answer
64 views

Why is this bound correct?

I recently encountered the following statement: Let $S:=\big\{(x,y,z)\in\mathbb{R}^3 \mid x>0, y>0, z>0, x+y+z<7\big\}$. Let $f : S \to \mathbb{R}$ such that: $$f(x,y,z) = \ln x+2\ln y +3 \...
Anon's user avatar
  • 1,791
0 votes
1 answer
37 views

Solving a combined limit with an $1^{\infty}$ form nested inside a 0×∞ form

I came across this limit problem: $\lim _{x \rightarrow \infty}\left\{\left(\frac{x+1}{x-1}\right)^x-e^2\right\} \cdot x^2$ Plugging this into desmos, one can see that the limit approaches $\frac{2 e^...
Afsheen's user avatar
  • 45
0 votes
0 answers
33 views

composite derivative and variable change

I'm asking for help with this calculation that I've been racking my brain over for two days! We have a variable change: $$p=e^{-\gamma t}\hat p$$ $$dp = d \hat p e^{-\gamma t}$$ Now the pdf becomes $$\...
Ged's user avatar
  • 11
2 votes
0 answers
45 views

Alternate proof to the Extreme Value Theorem

I'm following Spivak's Calculus and was revisiting some of my notes when I think I found a much more straightforward proof for the Extreme Value Theorem, compared to the one given in the book. I was ...
Aryaan's user avatar
  • 271
1 vote
1 answer
42 views

How to write this in a rigorous way? [duplicate]

I was studying for a test and encountered the following problem in a textbook: Suppose $f(x)$ is continuous, with $f >0$ for all $x$ and $\lim_{x \to \infty} f(x) = \lim_{x \to -\infty}f(x)= 0$ ...
Artur Stolf's user avatar
-2 votes
0 answers
92 views

Evaluating $\int_{-\pi/2}^{\pi/2} \frac{\sin^2(nx)}{\sin^2(x)(1+e^x)}\,dx $ [closed]

Try this amazing integral....This will be either too much pain or too much fun $$ \int_{-\pi/2}^{\pi/2}\frac{\sin^{2}\left(nx\right)}{\sin^{2}\left(x\right)\left(1 + {\rm e}^{x}\right)}\,{\rm d}x $$
X_xBABAIx_X's user avatar
4 votes
3 answers
121 views

Find minimum value of $f(x)= x^{1.5} + x^{-1.5} -4(x + x^{-1})$

Problem: Find minimum value of $f(x)= x^{1.5} + x^{-1.5} -4(x + x^{-1})$ My first attempt involved letting $\lambda = \sqrt{x}+\frac{1}{\sqrt{x}}$ followed by successive squaring and cubing and then ...
Vansh Chandak's user avatar
0 votes
0 answers
37 views

Solving integral involving minimum $\int_{L^n} \bigwedge_{j=1}^n \Big( \sum_{k=1}^j x_k \Big) \ dx$.

Consider $L = [s,t]$ with $s<t$ and write $a \wedge b := \text{min}(a,b)$. I am trying to solve the following integral. $$ \int_{L^n} \bigwedge_{j=1}^n \Big( \sum_{k=1}^j x_k \Big) \ dx $$ My ...
justAGuy's user avatar
4 votes
2 answers
169 views

usage of Leibniz notation for things like $\frac{d^2y}{dt^2}$ and $\frac{dy'}{dy}$

I've read the other posts on this site about whether you can treat $\frac{dy}{dt}$ as a fraction. There are a lot of conflicting opinions, but many seem to be saying that treating it as a fraction ...
Ishaan Jain's user avatar
0 votes
2 answers
99 views

Does $\frac{1}{(1+x)\cdots(1+x^n)}$ converges uniformly for $x\geq 0$?

$\newcommand{\on}[1]{\operatorname{#1}}$ Does $\displaystyle \frac{1}{\left(1 + x\right)\cdots\left(1 + x^{n}\right)}$ converges uniformly for $x\geq 0$ ?. $$ \mbox{Let}\ \on{f}_{n}\left(x\right) = \...
xldd's user avatar
  • 3,593
0 votes
0 answers
23 views

Why do you need the arc length element in surface area integrals? [duplicate]

From ChatGPT: "When you revolve a curve $y=f(x)$ around the x-axis, you are generating a surface composed of infinitesimally thin strips that are themselves small frustums (truncated cones). Each ...
Ruchir Kavulli's user avatar
1 vote
1 answer
119 views

Is $\frac{{dy}^2}{{dx}^2} = ({\frac{dy}{dx}})^2$?

Is $\frac{{dy}^2}{{dx}^2} = ({\frac{dy}{dx}})^2$? Problem: If $x = t \cos(t)$ and $y = t + \sin(t)$ then find $\frac{d^2 x}{dy^2}$ at $t = \frac{\pi}{2}$. My attempt: $\frac{d^2 x}{dy^2}=\frac{d^2 x}{...
rohit1729's user avatar
-4 votes
1 answer
92 views

I really need verification and validation of my view of integrals! [closed]

Questions: A runner is running at a rate of $3 \frac{m}{s}$. How long has he traveled in $4$ seconds? A runner is running at a rate of $3x \frac{m}{s}$. Thoughts: For question one, given that the ...
Idiamine's user avatar
0 votes
0 answers
10 views

Boundary terms of integration by part depends on the order to exchange the differetial operator

I have a very simple question while learning calculus. Suppose $\Omega\subset\mathbb{R}^2$ is a smooth domain. $f,g\in C^\infty(\Omega)$. We consider the integration by part here: $$\begin{aligned} \...
Holden Lyu's user avatar
2 votes
3 answers
80 views

How can I simplify $\prod_\limits{n=1\atop n\ne m}^{a}\frac{nx - 1}{n - m}:\ ? $

I ran into the following product while doing a Lagrange Polynomial Interpolation for a math puzzle I created for myself and am struggling to simplify it further: $$ \prod_{\Large n = 1\atop \Large n \...
Dylan Levine's user avatar
  • 1,728
-1 votes
0 answers
12 views

The trailing cone of a guided long range torpedo is to be conical with slant edge s cm. [closed]

The trailing cone of a guided long range torpedo is to be conical with slant edge s cm. Radius is r and height is h. The cone is hollow and must contain the maximum possible value of fuel. Find the ...
antara's user avatar
  • 1
1 vote
2 answers
65 views

Two similar formulas for ellipse circumference?

Does $$ \int_{0}^{2\pi}\sqrt{a^2\cos^2x+b^2\sin^2x}dx= \int_{0}^{2\pi}\sqrt{a^2\sin^2x+b^2\cos^2x}dx $$ The right expression is the ellipse circumference.
boaz's user avatar
  • 4,811
2 votes
2 answers
91 views
+100

'deducing' a bound using the first order taylor series. How to make it more precise?

So, I just saw a ‘proof’ that the generalized birthday problem has a median of C*sqrt(n). Though the probability in question is interesting, this question is more about calculus and maybe asymptotics ...
josinalvo's user avatar
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