Questions tagged [calculus]
For basic questions about limits, continuity, derivatives, differentiation, integrals, and their applications, mainly of one-variable functions.
134,424
questions
4
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Does this series violate the integral test condition for convergence/divergence?
My professor is asking me to determine if the following series is convergent or divergent using the integral test specifically.
$$
\sum^{\infty}_{n=1}n^2e^{-n}
$$
However, this seems to violate the ...
1
vote
1
answer
43
views
Is there a nice closed form of the following function: $f(k) = \lim_{n \rightarrow \infty}\prod^{kn}_{i = 0}\frac{n-2i+1}{n-2i}$
I am tring to find the closed from of the following function:
$$f(k) = \lim_{n \rightarrow \infty}\prod^{kn}_{i = 0}\frac{n-2i+3}{n-2i+2},$$
where $k \in [0,\frac{1}{2}]$
If the numerator is $n-2i+4$ ...
-2
votes
0
answers
32
views
"Check Stokes' theorem in the plane where $f(x, y) = y^2\mathbf{i} + x^2\mathbf{j}$ and the region formed is bounded by the circle ($x^2+y^2=4$)."
Resolution
"The question is a problem from Leithold's calculus book. I didn't understand the ($x = 5 \cos(t)$). Shouldn't it be ($x = 2 \cos(t)$)?
Resolution
The theorem in the book is this. From ...
0
votes
1
answer
33
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What is the significance of defining the partial derivative as a one-sided limit or a two-sided limit?
Let $f:\mathbb{R}^n\to\mathbb{R}$ be a function. We can define the partial derivative of f with respect to its first argument using either a two-sided limit ($h\to 0$)
$$\lim_{h\to 0} \frac{f(x_1+h,...
0
votes
0
answers
18
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Critical points and Points of Extremum
Wikipedia says that:
By Fermat's theorem, all local maxima and minima of a continuous function occur at critical points
But I think that all local maxima and minima, irrespective of whether the ...
-1
votes
0
answers
23
views
How to calculate the ranking of metrices [closed]
In the image I have list of metrices against two accounts. I need to create a mechanism to calculate the ranking of metrices. The condition is that some metrices are good on lower side (like attrition)...
0
votes
4
answers
79
views
What is the maximum and minimum value of the following function?
I came across a question in the book Calculus for the Practical Man. The question stated to find the maximum or minimum value of the following function:
$$f(x) = \frac{1}{4}\cos^2x - \sin 2x$$
I tried ...
-1
votes
0
answers
21
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Essential Range of a measurable function [closed]
im trying to come up with a solution to the next problem, but i dont know how to handle it. Im relative new to $L^p$ spaces and i would appreciate any help. Thanks in advance!
"Given ($\Omega,\...
4
votes
2
answers
63
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Showing divergence of $\sum\limits_{k=1}^{\infty} \log\left(1+\frac{(-1)^{k+1}}{k^\alpha}\right)$ where $0<\alpha<\frac{1}{2}$
I am trying to prove that $$\sum\limits_{k=1}^{\infty} \log\left(1+\frac{(-1)^{k+1}}{k^\alpha}\right)$$ diverges, for any $0<\alpha<\frac{1}{2}$.
I tried showing this by taking the Taylor ...
0
votes
0
answers
35
views
Evaluating the limit of a function involving exponential and hypergeometric functions as it approaches infinity.
I am trying to evaluate the limit of
$$
e^{-BY} \left[
B \,e^{2Y} {}_2F_1 \left( 1,1-\frac{B}{2}; 2-\frac{B}{2}; -e^{-2Y} \right)
-(B-2)\, {}_2F_1 \left( 1,-\frac{B}{2}; 1-\frac{B}{2}; -e^{-2Y} \...
5
votes
4
answers
196
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Can I avoid differentiation to find the remainder when $x^{73}-2x^{15}+3x-1$ is divided by $(x-1)^2$
The problem states:
Find the remainder when $x^{73}-2x^{15}+3x-1$ is divided by $(x-1)^2$.
So, what I did is,
Assume, the remainder to be linear, i.e $r(x) = ax+b$
By Eucild''s,
$x^{73}-2x^{15}+3x-1 ...
-2
votes
0
answers
23
views
Derivative of Gamma and partial sums [closed]
I need to prove that
$$
S_N(x)=\sum_{j=1}^N{\frac{1}{j-x}}=\Psi(N+1-x)-\Psi(1-x)
$$
Any help is welcome, thanks.
0
votes
1
answer
28
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finding work required to pump water out of a tank
The tank has the shape of a horizontal cylinder with radius $r$ and length $l$. The water exists through a small opening on the top right side of the cylinder. It seems that there are two ways to ...
-1
votes
0
answers
37
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$y''−3y' +2y = 4t+ e^(2t)$ where $y(0)= 1$ and $y'(0) = −1$. [closed]
solve the following ODE through laplace transform and then apply partial fraction then use inverse laplce transform.
1
vote
1
answer
38
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Let 𝑓:(𝑎,𝑏)→ℝ be a differentiable function. Show that if 𝑓(𝑐)=sup{𝑓(𝑥)∣𝑥∈(𝑎,𝑏)} for some 𝑐∈(𝑎,𝑏) , then 𝑓′(𝑐)=0 .
Here is what I have so far:
suppose f'(c) > 0
then lim (f(x) - f(c)) / (x - c) = f'(c) > 0
there exists some delta > 0 s.t. if abs(x-c) < delta then lim (f(x) - f(c)) / (x - c) > 0
take ...
0
votes
0
answers
43
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In what sense are differentials from calculus covectors in differential geometry? Why would there be a covector present in integration?
To preface, I am currently following eigenchris' series on tensor calculus in an effort to learn GR. I believe I understand the sense in which differential operators representing derivatives are ...
3
votes
1
answer
79
views
+50
The positive Laplacian is indeed the negative Laplacian
I know this question sounds like a joke. And it probably is:).
I found it kind of annoying, but also interesting, to call $-\Delta=-\sum_{j=1}^n\partial^2_{jj}$ "the positive Laplacian" as ...
0
votes
0
answers
31
views
When is a particle speeding up/down based on position graph
Question: Based on this position graph of a particle, how am I supposed to know when it is speeding up or slowing down?
I know that with a velocity graph, when velocity and acceleration have the same ...
0
votes
2
answers
88
views
Using triple integration for the area of a triangle
For practice I am trying to use the standard definition of volume for a set $X \subset \mathbb{R}^n$ given by:
$$vol(X) := \int_X dx_1 \ \ldots \ dx_n$$
to compute the area of the triangle spanned by ...
0
votes
0
answers
36
views
What is the derivative of x with respect to 2x? [closed]
What is $$\frac{d}{d(2x)}(x)$$
Is it $\frac{1}{2}$ since setting $y=2x$, and rewriting in terms of y
$$\frac{d}{dy}(\frac{1}{2}y) = \frac{1}{2}$$
or is it not defined?
2
votes
0
answers
19
views
Negative $r$ in polar coordinate while integrating [duplicate]
The question asks $\iint_R (3x+4y^2)\; dA$ where $R$ is the region in the upper half plane bounded by the circles $x^2+y^2=1$ and $x^2+y^2=4$
$$\int_0^\pi \int_1^2 (3r\cos \theta + 3r^2 \sin^2 \theta) ...
2
votes
3
answers
82
views
Why are these primitives containing $\arcsin x$ equal up to a constant?
While trying to solve $\displaystyle\int\sqrt{14x-x^2}\;dx$, I obtained three different primitives in three different ways:
Method 1: completing the square
$c(x)=\dfrac{1}{2}\left[49\arcsin\left(\...
6
votes
2
answers
125
views
Seeking clarification: Convergence of the series $\sum_{n=1}^{\infty} \frac{1}{e^{n^2}}$
I was asked to show whether the following series converges:
$$\sum_{n=1}^{\infty}\frac{1}{{e^{n^2}}}$$
Here's my first attempt which I thought was pretty straightforward:
$$e^{n^{2}}=(e^{n})^{n}\ge e^{...
4
votes
2
answers
117
views
Prove that $\lim_ {x\to \infty}\min_{m,n \in \mathbb{Z}}|x-\sqrt{m^2+2n^2}|=0$ [closed]
For $x\gt0$, let $f(x)$ be the minimum value of $|x-\sqrt{m^2+2n^2}|$ over all integers $m, n$. Then Prove that $$\lim_ {x\to \infty} f(x)= \; 0$$
I am completely stuck with this question. If I take $\...
0
votes
0
answers
34
views
Is this right for converting an infinite sum into an integral with the limit $[0; \infty]$?
I've done the following to get from an infinite sum to an integral, the question is now, did i do everything properly and didn't made any mistakes, because i coulnd't find anything like that when ...
0
votes
0
answers
37
views
Washer method around X-axis vs other axis
I had a question I was hoping someone could answer. I'm working on this homework problem, and when graphed, the volume looks like it would be the same revolving around the x-axis or on the axis of $y=...
0
votes
1
answer
19
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Analysis of Metric Properties in an Infinite Set with Discrete Metric
I am not sure if my solution to the following problem is correct
Let x be an infinite set. for x $\in$ X and y $\in$ X we define:
\begin{equation}
d(x,y) = \left\{
\begin{array}{ll}
1 & x \...
0
votes
1
answer
55
views
Understanding the limit of a function
I've always heard that if the value of a function approaches a given limit as the value of the argument approaches a specific value, then the limit exists. But here is an example that has me confused;
...
0
votes
0
answers
18
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How can one compute rounding preservant integrable functions?
Background & Context : The background of the question is an engineering problem.
I want to efficiently represent a set of integers as rounded real valued functions and quickly be able to calculate ...
0
votes
0
answers
29
views
Can billiards fill any figure?
For any 2D closed shape, is there always a point and angle for which the billiard movement of a ball fully completes the interior of the shape? In other words, does the path eventually converge to the ...
1
vote
1
answer
81
views
$f(x)$ is continuous on $[a, b]$ and assumes one single critical point at $x = c$ that is a local minimum. Prove that $f(c)$ is the absolute minimum.
I was working on a textbook problem yesterday when I found myself needing to prove a statement to make my main argument more rigorous. The statement seems relatively simple at first, yet I ended up ...
-3
votes
1
answer
85
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Calculation of $\lim_{n \to \infty}\frac{ 1^n + 2 ^n + \dots + n^n}{n^n}$ in two different ways giving different answers [closed]
I calculated $ L= \lim_{n \to \infty} \frac {1^n + 2 ^n + \cdots + n^n }{n^n}$ in two different ways getting two different answers.
$$L
= \lim_{n \to \infty}\left[1 + (\frac {n-1}{n})^n + (\frac {n-...
1
vote
1
answer
83
views
Transformation of a random variable vs. joint transformation of several random variables
Let $X \sim f_X(x)$ and $Y = g(X)$. If $g(X)$ is a differentiable, monotonic function with inverse such that $X = g^{-1}(Y)$ then the PDF of $Y$ can be described:
$$
f_Y(y) = f_X(g^{-1}(y)) \bigg| \...
1
vote
2
answers
114
views
An integral question which i have never encountered before
I = $\int _{0}^{3}\left( 1+x^{2}\right) d[ x]$
$a)12$
$b)17$
$c)15$
$d)19$
where $[x]$ is the greatest integer less than or equal to $x$,
shouldn't this integral be $0$ since $d[x]$ is 0 for all $x$ ...
3
votes
1
answer
29
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Inconsistent Function Monotonicity from hand and Mathematica image
$g(x)=\frac{\phi(x)}{1-\Phi(x)}$, where $\phi(x)$ and $\Phi(x)$ are p.d.f and c.d.f of standard normal distribution respectively.
$g'(x)=\frac{\phi'(x)(1-\Phi(x))+\phi^2(x)}{(1-\Phi(x))^2}=\frac{\phi(...
1
vote
0
answers
49
views
Residue theorem integral - calculating with trigonometric functions
I am trying to solve the following integral:
$$I = \int_0^{2\pi} \frac{d\phi}{a + 2 b \cos(\phi) + 2 c \sin(\phi)}$$
We can assume that the denominator is always strictly positive ($a \gg b, c$). ...
1
vote
0
answers
49
views
Difficult Vectors Problem (Calculus & Vectors 12)
Find parametric equations of a line that intersects line 1 and line 2 at right angles.
Line 1: $[x,y,z] = [4,8,-1] + t[2,3,-4]$ and
Line 2: $[x,y,z] = [7,2,-1] + k[-6,1,2]$.
I've tried solving this ...
0
votes
1
answer
37
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Quantifiers uniform continuity
According to this answer:
https://math.stackexchange.com/a/2582334/1098426
We know $\forall x \ \exists y \ \forall z$ differs from $\forall x \forall z\ \exists y$ insofar as $y$ depends on $x$ ...
0
votes
0
answers
25
views
Maximizing a function for $k \in [-\pi, \pi]$
I want to find the maximum of $|g(\kappa)|$ defined as follows
For some $A>0$, $\alpha\in (0,\infty)$, and $\kappa \in [-\kappa^{*},\kappa^{*}]\subset [-\pi,\pi]$, define $g(\kappa )$
\begin{align}...
1
vote
0
answers
58
views
Show that there exists $L \in \mathbb{R}_+^*$ such that for all $x \in [a,b]$ $|g(x) - c| \leq L|x - c|^2$
I encountered an exercise from the admission exam for the Moroccan preparatory class Al-Zahrawi where I faced a small problem with question 4. I was able to answer the question, but the statement ...
0
votes
4
answers
81
views
What is wrong about this seemingly simple false proof? [duplicate]
It came accros my mind when doing an exercise in calculus. Consider the following inequalities:
$1\leq 2\leq 5$
$1\leq 3\leq 5$
Adding the equaions up is totally fine. But subtracting them, we get:
$0\...
4
votes
3
answers
117
views
How to approach an Hyperbolic Integral that doesn't appear to be solvable in closed form.
I'm interested in tackling the following integral:
$$\int_{-\ln (2+\sqrt 5)}^{\ln (2+\sqrt 5)} \sqrt{4+\sinh^2(x)} dx$$
While I've attempted various techniques, it appears challenging to find a closed-...
1
vote
2
answers
194
views
Compute $\int \frac{\sin(x)\cos(x)}{x^2+1}dx$
Compute $$\int \frac{\sin(x)\cos(x)}{x^2+1}dx$$
Attempt
$u=\sin(x)\Rightarrow du=\cos(x)dx$
$$\int\frac{u}{x^2+1}du$$
But, now I have both $u$ and $x$ in the function. How do I resolve?
Second attempt:...
3
votes
2
answers
397
views
Are there two contradictory definitions of the limit of a function at a point?
I have found two different definitions of the limit of a function f at a point a in very popular calculus/analysis texts.
In one, the function f is asked to be defined on an open interval containing ...
-3
votes
0
answers
27
views
Shouldn't it be like that when Lim ∆x→0 Tan Ψ=tan ∅ [closed]
see the highlighted portion and please try to clear my doubt.
$$\tan \psi = \lim_{\Delta x \to 0} \tan \theta$$
-1
votes
0
answers
47
views
Using double integrals to find the center of mass [closed]
Let $f(x,y)$ be the density of mass in the region;
$$R:\begin{cases}
0<x<1\\
0<y<\sqrt{(1-x^2)}\\
\end{cases}$$
Find the center of mass and the moments of inertia $I_x$, $...
1
vote
1
answer
40
views
Expectation of X^2 using complement of CDF
For a nonnegative random variable
$$E[X]=\int_0^\infty P(X >x)dx$$
This video on a question from IIT JAM 2023 extends it to
$$E[X^2]=\int_0^\infty 2x P(X >x)dx$$
How was the result was arrived ...
0
votes
3
answers
83
views
Confusion on finding if a limit exists or not.
I have started the introduction of calculus in high school, and I am confused about this problem.
Define $f(x) = -x$ for $x<0$. Then find out if $\lim_{x \to 0} f(x)$ exists or not. If it does, ...
-1
votes
0
answers
26
views
That's right ? $\operatorname{Re}(\sqrt{-z}\delta)>0~$
Helle, everyone. I want to check this question and find it in an article.
so that
$$s(r_{0},\theta_{0})~:=\left\{z\in\mathbb{C}\setminus\{0\}:|z|\geqslant r_{0}~~\text{ et}~~|\arg\left(z\right)\bigr|...
1
vote
2
answers
119
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Generalised integral $\int_{0}^{\infty} \int_{0}^{y} \sqrt{x^2 + y^2} e^{-x^2 - y^2} dx dy$
I am having trouble understanding how to compute this integral $$\int_{0}^{\infty} \int_{0}^{y} \sqrt{x^2 + y^2} e^{-x^2 - y^2} \, dx \, dy$$
My idea is to consider the whole plane xy plane instead of ...