Questions tagged [calculus]

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

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Finding constants making function continuous

Determine the values of $h$ and $k$ so that $$F(x)=\begin{cases} \frac{kx^2-kx+x+10}{x^2-4}, &x\neq2,\\ 3x+h, &x=2 \end{cases}$$ is continuous on $[0,4]$. I extracted k as Common factor and ...
Math's user avatar
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Integration over an open interval.

Is there any difference between the following integrations: $$\int_a^b f(x) dx$$ where $(a,b]$ And $$\int_a^b f(x) dx$$ where $[a,b]$
Elmarconi Abdo's user avatar
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Convergence or divergence of $\sum_{n=2}^{\infty}\frac{\ln \frac{n+1}{n}}{\ln \frac{n-1}{n}}$

I need to study convergence or divergence of this series $\sum_{n=2}^{\infty}\frac{\ln \frac{n+1}{n}}{\ln \frac{n-1}{n}}$. All the terms are negative hence $\frac{n-1}{n}<1$ which implies that $\ln ...
weymar andres's user avatar
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Proof of Area Under Polar Curves

The books use the area of sectors to derive the formula for the area under the polar curves. I couldn't understand how when the number of sectors reach infinity, the area of the region created by each ...
WBI's user avatar
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Taylor Expanding a higher power of a function

I have a (sufficiently smooth) function $f$ that is volume-preserving (Jacobian has determinant $1$) and invertible. Given two integers $\ell, k \in \{0, 1, \ldots, T\}$ with $\ell > k$ it seems ...
Euler_Salter's user avatar
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Trigonometric identity involving i

$$\sum_{k=-N}^N e^{ik θ} = \frac{\sin[(N+1/2) θ]}{\sin(θ/2)}$$ I cannot figure out how to prove this trigonometric identity. I considered using the formula of a geometric sum, but that didn’t give me ...
Aditya Patnaik's user avatar
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Modulus of x raised to a

For what values of $a \in R^+ $ the function $$f(x) = |x|^a$$ is differentiable at $x=0$. According to definition of differentiablity $$\lim_{x\to0} \frac{f(x)-f(0)}{x-0}$$ should exist & equal ...
DEEPAK KUMAR's user avatar
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1 answer
25 views

Optimization Problem: Triangle and Circle Problem

A wire b units long is cut into two pieces. One piece is bent into an equilateral triangle and the other is bent into a circle. If the sum of the areas enclosed by each part is a​ minimum, what is the ...
spice_math_guy's user avatar
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Find the interval convergence of a strange series of functions

Find the interval convergence (also absolute convergence) of the series of functions $$\sum_{n=1}^{\infty}\dfrac{(-1)^n}{(x+n)^p}\quad,p\in\mathbb{R}.$$
Harry's user avatar
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Problem with evaluating a line integral

I'm trying to find the following integral using Green's theorem: $$\oint _C y^3 \, dx -x^3 \, dy$$ where $C$ is a circle with radius of 2. Applying the theorem yields: $$\iint _R -3x^2 - 3y^2 \, dx\,...
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For an integration of a closed interval , how can we exclude a certain point from this integration?

For the integration over a closed interval [a,b]: $$\int_a^b f(x) dx$$, How can we exclude a specific value of x from this integration, such that this value belongs to [a,b]
Elmarconi Abdo's user avatar
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1 answer
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prove that $\int_{a}^{c} f(x) \,dx = \int_{a}^{b} f(x) \,dx + \int_{b}^{c} f(x) \,dx $

I've been working on a proof related to the additivity of Riemann integrals and would greatly appreciate insights and feedback for clarity and correctness of the proof. Because i've never seen a text, ...
rllynotgoodwithmath's user avatar
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Integral involving even order Legendre polynomials

Let $a>1$. We want to evaluate the integral \begin{equation*} \int_{-1}^1 \frac{P_{2n}(\xi)\,d\xi}{\sqrt{a^2-\xi^2}} \end{equation*} Mathematica is able to evaluate special cases for various $n$, ...
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Find the range of the function $f(x)=(\sin^{-1}x)^2-(\cot^{-1}x)^2$

Find range of the function:$f(x)=(\sin^{-1}x)^2-(\cot^{-1}x)^2$ The domain of the function is $-1\leq x \leq 1$ $f(-1)=(\sin^{-1}(-1))^2-(\cot^{-1}(-1))^2=\frac{\pi^2}{4}-\frac{9\pi^2}{16}=-\frac{-5\...
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Finding two points when already knowing the arc length [closed]

Suppose r(t) describes a space curve C. (a) Describe a method to find two points on C whose arc length from a given point P on C is the same distance. (b) Use the method you described in (a) to find ...
Amungus's user avatar
2 votes
2 answers
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Taylor Expansion Subseries

If we Taylor expand an infinitely differentiable function $f: \mathbb{R} \rightarrow \mathbb{R}$ around a point $a \in \mathbb{R}$ we of course get: $$ f(x) = f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(...
shea's user avatar
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1 answer
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Finding flux of a vector field through a hemisphere

This question is given in a (publicly shared) past exam at my university: Let S be the upper hemisphere of $x^2 + y^2 + z^2 = 4$ with normal vector pointing toward the origin, and $\vec F = z \vec x / ...
aces12590's user avatar
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Is the function $F(z)=\int_0^\infty e^{-zt}\,\Bbb dt$ for $z$ such that $\operatorname{Re}z>0$ holomorphic?

Is the function $F(z)=\int_0^\infty e^{-zt}\,\Bbb dt$ for $z$ such that $\operatorname{Re}z>0$ holomorphic? I wanted to prove that it is analytic and find its power series: $$F(z)={\int_0^\infty e^{...
romperextremeabuser's user avatar
1 vote
1 answer
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Using Derivatives to Explore Function Behavior

This is a self-answer question taken from this youtube video. The youtube video describes the presented problem as a no calculator problem presented in 2021, as an Olympiad qualifying problem given in ...
user2661923's user avatar
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Rationale behind picking epsilon for the epsilon-delta proof for limit of 1/g(x)

So I'm trying to understand epsilon-delta proofs a lot more, but one thing that seems to be the case is that the rationale for picking various epsilons or deltas is obfuscated within the math itself. ...
ekorel's user avatar
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Proof $\lim _{x\to \infty }\left(a\cdot x^n\right)=\begin{cases}\infty \:&if\:\:\:a>0\\ -\infty \:&if\:\:\:a<0\end{cases}$

I must do the proof using an epsilon-delta argument: $\lim _{x\to \infty }\left(a\cdot x^n\right)=\begin{cases}\infty \:&if\:\:\:a>0\\ -\infty \:&if\:\:\:a<0\end{cases}.$ For $a>0$: $\...
SergeyMartin's user avatar
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Proving Vector Fields are Conservative

I know that a vector field is conservative if for any closed path $C$, the integral with respect to $dr =0$. Are these assumptions below correct? To prove a vector field $F$ is conservative, all I ...
Dam's user avatar
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Determine the absolute maximum and minimum of $f$ on the interval $I$

Determine the absolute maximum and minimum of $f(x)= 3x^2 + 5x − 2 $ on the interval $I=[−2, 3]$
Alessandro Terminiello's user avatar
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coupled differential equations of two bodies [closed]

can anybody please give solution for $x_1(t), x_2(t)$ form this set equations: $$ \begin{align*} x_1'' &= \frac{c_1}{{(x_2 - x_1)}^2} \\ x_2'' &= \frac{c_2}{{(x_2 - x_1)}^2} \end{align*} $$ ...
Vedant Kalkotwar's user avatar
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Given a decimal value, find the decimal value rounded to 4 decimal places that when multiplied with a specific number gives a whole number

Given I have two numbers, when I divide these numbers and I get a decimal number with more than 4 decimal places. How do I find the nearest decimal that when multiplied by the divisor gives me a whole ...
moleculezz's user avatar
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Help with Variational Calculus & Leibnitz Rule

Let $f: (0,\infty)\to \mathbb{R}$ be convex and lower-semicontinuous with $f(1)=0$ and $\mu$, $\hat{\mu}$ be two probability distributions on a measurable space $\mathcal{X}$ which are absolutely ...
pablopez's user avatar
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integration by parts in academic paper that incorporates a discontinity

Confused about these derivations from the appendix page 15 of paper https://link.springer.com/article/10.1007/s11071-019-05117-z bit I can't follow is in the picture here: dot denotes differentiation ...
AlexLovesToto's user avatar
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If x belongs to (-pi/2, pi/2), then the number of solutions of: | sin 3x + sin x| + |sin 3x - sin x| = √3 is equal [closed]

Here's a basic image if the question. I wish to ask how do I go about solving this question. I can't really thi aur of a legitimate approach
Shubham Sagar Singh's user avatar
1 vote
4 answers
79 views

$f(a)=f(b)=\lambda,\frac{\mathrm d^2y}{\mathrm dx^2}<0\implies f(x)>\lambda \forall x\in(a,b)$

Let $y=f(x)$ be a smooth function in $[a,b]$ such that : $$f(a)=f(b)=\lambda,\frac{\mathrm d^2y}{\mathrm dx^2}<0$$ then, we need to prove that : $$f(x)>\lambda \forall x\in(a,b)$$ I observed ...
An_Elephant's user avatar
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Evaluate $\int_{0}^{\infty} \frac{x^2 + \sin(\pi x)}{1 + \exp(\pi x)} \,dx$

\begin{align*} \int_{0}^{\infty} \frac{x^2 + \sin(\pi x)}{1 + \exp(\pi x)} \,dx \end{align*} \begin{align*} &\int_{0}^{\infty} \frac{x^2 + \sin(\pi x)}{1 + e^{\pi x}} \,dx \\ &= \int_{0}^{\...
James's user avatar
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3 votes
2 answers
61 views

Putnam 2023 A1 Trigonometric Double Derivative Calculus

Question : For a positive integer $n$, let $f_n(x)=\cos (x) \cos (2 x) \cos (3 x) \cdots \cos (n x)$. Find the smallest $n$ such that $|f_n"(0)|>2023$ . Note : Here $f_n"(x)$ denotes the double ...
S.Ragnork1729's user avatar
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2 answers
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How to Derivate Divergence in Cylindrical Coordinates?

I am trying to derivate divergence in cylindrical coordinates, following is my derivation which is wrong and different from text book. I am confused why the derivation is wrong. Denote $\nabla_x=(\...
Yamato's user avatar
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1 answer
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How can I find the result of $ \sum_{n=1}^{\infty} \frac{\cos(n)}{n(n+1)} $ without computationally evaluating it

Good morning, recently in a school near mine there was a contest for solving a infinite series that even professors couldn't, only thing we knew was the final result, approximately 0.118272222745. The ...
JoshuaKasa's user avatar
4 votes
1 answer
85 views

Suppose $f(x)=(\tan x)^{\frac{3}{2}}-3\tan x+(\tan x)^{\frac{1}{2}}$. Then, how can we compare the given integrals?

Let $f(x)=(\tan x)^\frac32-3\tan x+\sqrt{\tan x}$. Consider the integrals $$I_1=\int_0^1f(x)dx$$ $$I_2=\int_{0.3}^{1.3}f(x)dx$$ $$I_3=\int_{0.5}^{1.5}f(x)dx$$ Then, prove that $I_1>I_3>I_2$ I ...
Eraser head's user avatar
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Does F must be a conservative vector field?

Let $u=(x,y,z)\in \mathbb{R}^3 , r(u)=r(x,y,z)=\sqrt{x^2+y^2+z^2}$. Denote $f: (0,\infty) \to \mathbb{R},f\in C^1$. Let $F=f(r(u))u$ be a vector field in $\mathbb{R}^3\setminus \{{(0,0,0)}\}$. I have ...
Algo's user avatar
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2 votes
0 answers
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Obtain $L^\infty$ estimate of Poisson Equation only using eigenfunction expansion

Let $\Omega\subseteq \mathbb{R}^d$ a bounded domain with sufficient smooth boundary. We consider the Poisson Equation $$\left\{\begin{array} --\Delta u = f, & \Omega\\ u=0, &\partial\Omega\end{...
MikeMichael_maths's user avatar
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1 answer
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Problem Help in Studying for Final.

I am reviewing past material and these were two problems that I was stuck on. I want to make sure that I understand them for my final. Problem 1: An isosceles triangle has two sides of length 10. The ...
MSM's user avatar
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-1 votes
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How to parameterize a vector field in terms of $u$ and $v$ given three components.

I am given the following vector field $$F(x,y,z) = xyz \mathbf i + xy \mathbf j + x^2yz \mathbf k.$$ I need to parameterize the vector field $r(u,v)$ to use it with Stoke’s theorem.
Weejaw's user avatar
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-2 votes
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How do you solve this?

Let C be the intersection of the cylinder $x ^ 2 + y ^ 2 = 1$ with the plane $x + 2y + 2z = 3$ counterclockwise when viewed from the origin. Let F be a vector field with curl $F = i + 2j - \alpha*k$ ...
Surya's user avatar
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6 votes
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155 views

Can $\ln$ be written as a ratio of polynomials?

Is it possible that $\ln(x)=\frac{p(x)}{q(x)}$ for all $x>0,$ where $p$ and $q$ are polynomials with real coefficients? I think the answer is no. Suppose two such polynomials did exist. Take the ...
aqualubix's user avatar
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1 vote
1 answer
69 views

Prove that $\exists c \in [a,b] \: \text{s.c}\: f(x)-f(a)- \frac{f(b)-f(a)}{b-a}(x-a)=\frac{(x-a)(x-b)}{2}f''(c)$

Given function $f$ differentiable on $[a,b]$, and has derivative $f''(x)$ on $[a,b]$, prove that for all $x\in[a,b]$, we have at least one $c\in[a,b]$ such that $$f(x)-f(a)- \frac{f(b)-f(a)}{b-a}(x-a)=...
区なしま's user avatar
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Does the transformed term has meaning in Calculus Average Value?

If you calculate the average temperature of a given day, but time series data is measured as a continuous function, the initial idea is to average the temperature of each hour. Then applying the ...
John HHU's user avatar
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0 answers
21 views

Is there a general solution for the integral that gives the force of gravity created by a 2-Dimensional body defined by a sum of cosines?

I'm trying to solve the integral which would provide the force of gravity in a 2-Dimensional universe on an object being influenced by a body with uniform density who's perimeter is defined by ...
ImIsaacKane's user avatar
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0 answers
47 views

Fourier transformation of $\log(q^2)/q^4$ in $d=3$

I currently have the following Fourier transformation that I need to compute \begin{equation} \int \frac{d^3q}{(2\pi)^3}\frac{\log(\mathbf{q^2})}{(\mathbf{q})^4}e^{i\mathbf{q}\cdot \mathbf{r}} \end{...
MathZilla's user avatar
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Question about Converting Equivalency

I know for a linear programming problem that you can convert the condition maximise $c^T x$ for the objective function to $-$ minimise $-c^Tx$ and they mean the same thing. Here $x=x_1,x_2$ column ...
Dam's user avatar
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1 vote
2 answers
61 views

Why do we solve lim in two different ways?

When solving the limit in this form $$ \lim_{x \to \infty} \sqrt{x} \pm \sqrt{y} $$ where $x$ and $y$ are two different expressions, I was taught that I should first find the element with the highest ...
Yusuf Bouzekri's user avatar
1 vote
0 answers
43 views

Applying Variational Approximation

I am trying to solve a problem involving variational approximation, where the task is to calculate a value $C$ such that $$C > \frac{\int_{-\infty}^{\infty} |f'(y)|^2 dy}{\int_{-\infty}^{\infty} \...
Newbie's user avatar
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When to include the boundary points in the convexity analysis?

I was wondering about this: suppose I have a function $f: D \to \mathbb{R}$; suppose $(a, b) \subset D$ ($D$ can either be bounded or unbounded), and say $f$ is convex in $(a, b)$. What is the ...
Numb3rs's user avatar
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1 vote
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Integration of rational functions. Integration techniques.

By the fundamental theorem of algebra a real nonconstant polynomial $Q$ has factorisation into real prime factors $$Q=g_1^{k_1}g_2^{k_2}\cdots g_l^{k_l}$$ The prime factors $g_j$, all distinct, are ...
emil agazade's user avatar
-4 votes
0 answers
30 views

What do I need to learn for AP BC Calculus exam? [closed]

Please name all of the subjects that I need to learn in order to take this as a 6th grader.
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