Questions tagged [calculus]
For basic questions about limits, continuity, derivatives, differentiation, integrals, and their applications, mainly of one-variable functions.
132,830
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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 ...
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3
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2
answers
60
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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 $[a,b]$
0
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2
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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 ...
- 1,219
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23
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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 ...
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14
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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 ...
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1
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45
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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 ...
1
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41
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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 ...
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1
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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 ...
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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 $$\sum_{n=1}^{\infty}\dfrac{(-1)^n}{(x+n)^p}\quad,p\in\mathbb{R}.$$
- 113
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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$$, How can we exclude a specific value of x from this integration, such that this value belongs to [a,b]
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1
answer
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prove that $\int_{a}^{c} f(x) \,dx = \int_{a}^{b} f(x) \,dx + \int_{b}^{c} f(x) \,dx $
I've been working on a proof related to the additivity of Riemann integrals and would greatly appreciate insights and feedback for clarity and correctness of the proof. Because i've never seen a text, ...
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Integral involving even order Legendre polynomials
Let $a>1$. We want to evaluate the integral
\begin{equation*}
\int_{-1}^1 \frac{P_{2n}(\xi)\,d\xi}{\sqrt{a^2-\xi^2}}
\end{equation*}
Mathematica is able to evaluate special cases for various $n$, ...
- 325
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Find the range of the function $f(x)=(\sin^{-1}x)^2-(\cot^{-1}x)^2$
Find range of the function:$f(x)=(\sin^{-1}x)^2-(\cot^{-1}x)^2$
The domain of the function is $-1\leq x \leq 1$
$f(-1)=(\sin^{-1}(-1))^2-(\cot^{-1}(-1))^2=\frac{\pi^2}{4}-\frac{9\pi^2}{16}=-\frac{-5\...
- 8,692
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Finding two points when already knowing the arc length [closed]
Suppose r(t) describes a space curve C.
(a) Describe a method to find two points on C whose arc length from a given point P on C is the same distance.
(b) Use the method you described in (a) to find ...
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2
answers
36
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Taylor Expansion Subseries
If we Taylor expand an infinitely differentiable function $f: \mathbb{R} \rightarrow \mathbb{R}$ around a point $a \in \mathbb{R}$ we of course get:
$$ f(x) = f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(...
- 21
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votes
1
answer
42
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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 / ...
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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:
$$F(z)={\int_0^\infty e^{...
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1
answer
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Using Derivatives to Explore Function Behavior
This is a self-answer question taken from this youtube video.
The youtube video describes the presented problem as a no calculator problem presented in 2021, as an Olympiad qualifying
problem given in ...
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Rationale behind picking epsilon for the epsilon-delta proof for limit of 1/g(x)
So I'm trying to understand epsilon-delta proofs a lot more, but one thing that seems to be the case is that the rationale for picking various epsilons or deltas is obfuscated within the math itself. ...
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Proof $\lim _{x\to \infty }\left(a\cdot x^n\right)=\begin{cases}\infty \:&if\:\:\:a>0\\ -\infty \:&if\:\:\:a<0\end{cases}$
I must do the proof using an epsilon-delta argument:
$\lim _{x\to \infty }\left(a\cdot x^n\right)=\begin{cases}\infty \:&if\:\:\:a>0\\ -\infty \:&if\:\:\:a<0\end{cases}.$
For $a>0$: $\...
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Proving Vector Fields are Conservative
I know that a vector field is conservative if for any closed path $C$, the integral with respect to $dr =0$. Are these assumptions below correct?
To prove a vector field $F$ is conservative, all I ...
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Determine the absolute maximum and minimum of $f$ on the interval $I$
Determine the absolute maximum and minimum of $f(x)= 3x^2 + 5x − 2 $ on the interval $I=[−2, 3]$
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coupled differential equations of two bodies [closed]
can anybody please give solution for $x_1(t), x_2(t)$ form this set equations:
$$
\begin{align*}
x_1'' &= \frac{c_1}{{(x_2 - x_1)}^2} \\
x_2'' &= \frac{c_2}{{(x_2 - x_1)}^2}
\end{align*}
$$
...
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2
answers
42
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Given a decimal value, find the decimal value rounded to 4 decimal places that when multiplied with a specific number gives a whole number
Given I have two numbers, when I divide these numbers and I get a decimal number with more than 4 decimal places. How do I find the nearest decimal that when multiplied by the divisor gives me a whole ...
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Help with Variational Calculus & Leibnitz Rule
Let $f: (0,\infty)\to \mathbb{R}$ be convex and lower-semicontinuous with $f(1)=0$ and $\mu$, $\hat{\mu}$ be two probability distributions on a measurable space $\mathcal{X}$ which are absolutely ...
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integration by parts in academic paper that incorporates a discontinity
Confused about these derivations from the appendix page 15 of paper https://link.springer.com/article/10.1007/s11071-019-05117-z
bit I can't follow is in the picture here:
dot denotes differentiation ...
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answers
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If x belongs to (-pi/2, pi/2), then the number of solutions of: | sin 3x + sin x| + |sin 3x - sin x| = √3 is equal [closed]
Here's a basic image if the question. I wish to ask how do I go about solving this question.
I can't really thi aur of a legitimate approach
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4
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$f(a)=f(b)=\lambda,\frac{\mathrm d^2y}{\mathrm dx^2}<0\implies f(x)>\lambda \forall x\in(a,b)$
Let $y=f(x)$ be a smooth function in $[a,b]$ such that :
$$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)$$
I observed ...
- 2,491
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2
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Evaluate $\int_{0}^{\infty} \frac{x^2 + \sin(\pi x)}{1 + \exp(\pi x)} \,dx$
\begin{align*}
\int_{0}^{\infty} \frac{x^2 + \sin(\pi x)}{1 + \exp(\pi x)} \,dx
\end{align*}
\begin{align*}
&\int_{0}^{\infty} \frac{x^2 + \sin(\pi x)}{1 + e^{\pi x}} \,dx \\
&= \int_{0}^{\...
- 315
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votes
2
answers
61
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Putnam 2023 A1 Trigonometric Double Derivative Calculus
Question : For a positive integer $n$, let $f_n(x)=\cos (x) \cos (2 x) \cos (3 x) \cdots \cos (n x)$. Find the smallest $n$ such that $|f_n"(0)|>2023$ .
Note : Here $f_n"(x)$ denotes the double ...
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How to Derivate Divergence in Cylindrical Coordinates?
I am trying to derivate divergence in cylindrical coordinates, following is my derivation which is wrong and different from text book. I am confused why the derivation is wrong.
Denote $\nabla_x=(\...
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1
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How can I find the result of $ \sum_{n=1}^{\infty} \frac{\cos(n)}{n(n+1)} $ without computationally evaluating it
Good morning,
recently in a school near mine there was a contest for solving a infinite series that even professors couldn't, only thing we knew was the final result, approximately 0.118272222745.
The ...
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Suppose $f(x)=(\tan x)^{\frac{3}{2}}-3\tan x+(\tan x)^{\frac{1}{2}}$. Then, how can we compare the given integrals?
Let $f(x)=(\tan x)^\frac32-3\tan x+\sqrt{\tan x}$. Consider the integrals $$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$$
Then, prove that $I_1>I_3>I_2$
I ...
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Does F must be a conservative vector field?
Let $u=(x,y,z)\in \mathbb{R}^3 , r(u)=r(x,y,z)=\sqrt{x^2+y^2+z^2}$.
Denote $f: (0,\infty) \to \mathbb{R},f\in C^1$.
Let $F=f(r(u))u$ be a vector field in $\mathbb{R}^3\setminus \{{(0,0,0)}\}$.
I have ...
- 2,302
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Obtain $L^\infty$ estimate of Poisson Equation only using eigenfunction expansion
Let $\Omega\subseteq \mathbb{R}^d$ a bounded domain with sufficient smooth boundary. We consider the Poisson Equation
$$\left\{\begin{array}
--\Delta u = f, & \Omega\\
u=0, &\partial\Omega\end{...
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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 ...
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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.
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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 $F = i + 2j - \alpha*k$ ...
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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 ...
- 1,362
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1
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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 $$f(x)-f(a)- \frac{f(b)-f(a)}{b-a}(x-a)=...
- 348
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31
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Does the transformed term has meaning in Calculus Average Value?
If you calculate the average temperature of a given day, but time series data is measured as a continuous function, the initial idea is to average the temperature of each hour. Then applying the ...
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Is there a general solution for the integral that gives the force of gravity created by a 2-Dimensional body defined by a sum of cosines?
I'm trying to solve the integral which would provide the force of gravity in a 2-Dimensional universe on an object being influenced by a body with uniform density who's perimeter is defined by ...
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0
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47
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Fourier transformation of $\log(q^2)/q^4$ in $d=3$
I currently have the following Fourier transformation that I need to compute
\begin{equation}
\int \frac{d^3q}{(2\pi)^3}\frac{\log(\mathbf{q^2})}{(\mathbf{q})^4}e^{i\mathbf{q}\cdot \mathbf{r}}
\end{...
- 247
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Question about Converting Equivalency
I know for a linear programming problem that you can convert the condition maximise $c^T x$ for the objective function to $-$ minimise $-c^Tx$ and they mean the same thing. Here $x=x_1,x_2$ column ...
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2
answers
61
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Why do we solve lim in two different ways?
When solving the limit in this form
$$
\lim_{x \to \infty} \sqrt{x} \pm \sqrt{y}
$$
where $x$ and $y$ are two different expressions, I was taught that I should first find the element with the highest ...
1
vote
0
answers
43
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Applying Variational Approximation
I am trying to solve a problem involving variational approximation, where the task is to calculate a value $C$ such that
$$C > \frac{\int_{-\infty}^{\infty} |f'(y)|^2 dy}{\int_{-\infty}^{\infty} \...
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0
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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 ...
- 2,758
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0
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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 ...
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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.
