Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [calculus]

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

0
votes
0answers
5 views

Inequality of cauchy

I found the following inequality in a book, dubbed "Cauchy's inequality for subunitary exponent": $$ sb^{s-1}(b-a) < b^s - a^s < sa^{s-1}(b-a)\text{ } (0<a<b, s \in (0,1))$$ I tried ...
1
vote
0answers
13 views

Differential expression in Fourier's paper “Theory of Heat”

The Following is from Fourier's paper in which he says: It is easy now to generalize this result and to recognize that it exists in every case of the varied movement of heat expressed by the equation ...
0
votes
0answers
9 views

Surface integral with flat triangle surface

Let $G=2x\hat{i}+xy\hat{j}+z\hat{k}$ and let S be the flat triangular surface with vertices $(0,0,0),(0,0,0),(1,1,0),(2,2,2)$. Assume $S$ is oriented towards the positive $x$-axis. Find $\iint GdS,$ (...
2
votes
2answers
18 views

How can I simplify the derivative further to match the correct answer?

I've been stuck on trying to simplify my expression in order to match the correct answer but I can't seem to get the correct solution. Guidance towards the proper steps would be greatly appreciated! ...
0
votes
0answers
6 views

How to integrate$\int_0^{t} xe^{i(ax^2)}J_0(bx)\,\mathrm dx$?

Find $$\int_0^{t} xe^{i(ax^2)}J_0(bx)\,\mathrm dx$$ Given $a>0,\ b>0, t>0$. Previously I asked a more difficult integration problem. It seems that it is too difficult to find a solution. ...
0
votes
0answers
10 views

Given a positive sequence $\{a_n\}$ where $a_{n+1}=a_n+\frac{n}{a_n}$, can one find an asymptotic expansion of $a_n-n$?

Given a positive sequence $\{a_n\}$ such that $$a_{n+1}=a_n+\frac{n}{a_n}$$ Can one find an symptotic expannsion of $a_n-n$? I want one which has the term $O(1/n)$, or a stronger one.
1
vote
0answers
31 views

Verification of Limit Proof for $\lim_{x \to 9} \sqrt{x-5} = 2$

I am trying to solve this question $\lim_{x \to 9} \sqrt{x-5} = 2$. If you could tell me if I am going about this the right way I'd appreciate it. Given $\epsilon > 0$, Let $\delta =min(9,\epsilon)...
0
votes
0answers
12 views

Change of variable in double integrals.

Problem Express the double integral $\iint_S f(x,y) \,dx \,dy$ as an iterated integral in polar coordinates,where $S$={$ (x,y)|x^2 \leq y,-1 \leq x \leq 1$}. Doubt I have difficulty finding ...
0
votes
1answer
41 views

Proving that $f(x) = \ln x$ is continuous at $a = 3$.

Proving that $f(x) = ln x$ is continuous at $a = 3$. The book wrote this solution: But when I started by $|f(x) - f(3)| < \varepsilon$ to find the appropriate $\delta$, I arrived at $ 3e^{-\...
0
votes
0answers
9 views

Conditions to tell that the total flux in a vector field across a closed surface is zero

When is the total flux across a closed surface zero. I am trying to find a set of values for an equation to prove that it has a total flux of 0 across a closed surface. I am not given any specific ...
0
votes
1answer
14 views

Use relation to determine limit of function

I was preparing for a calculus test tomorrow on limits and continuity and I came upon this problem while I was practicing. I have no idea how to attempt it, so any help would be appreciated. The only ...
1
vote
0answers
13 views

What is the explanation of Big-O notation in the following

Please, can someone let me know how Big-O notation appears in part (b) of the Eq. below. Please note that $f_\mathcal{R}(r) = 2\pi\lambda_f re^{-\pi.\lambda_fr^2}$ This question is part of my reading ...
1
vote
1answer
38 views

Find a continuous function that is always less than or equal to its own integral

I am having difficulty coming up with examples of functions $f$ defines on $[0,1]$ such that $$f(t) \leq \int_{0}^{t} f(s) ds$$. I see that the is required to be that $f(0)=0$, and that the ...
1
vote
1answer
16 views

why does the variable bound (i.e. the bound that changes) must always be upper bound

I'm currently learning integral calculus, in particularly, integral function with the lower bound been the variable and upper bound is the constant. Below is a question and solution from khan ...
0
votes
1answer
18 views

Jacobian of vec of a matrix into vec of its inverse

Suppose that $X$ is invertible and $n\times n$ and the transformation: $$ \varphi:\operatorname{vec}(X)\mapsto\operatorname{vec}(X^{-1}). $$ For example, with $n=2$ and $X=\begin{pmatrix}a & b \\ ...
0
votes
1answer
48 views

Integral of $\sin^5(x)\cos(x)$

I'm trying to solve the following integral: $$\int\sin^5(x)\cos(x)$$ I assumed I would do u-substitution where: $$u = \sin(x)$$ $$du = \cos(x) dx$$ Which would then cancel out the $\cos(x)$ And ...
0
votes
0answers
34 views

How does following follows Fundamental Theorem of Calculus?

In the attached figure, I cannot understand how we got (b). Also, how does Big-O notation appears here? Please note that $f_{\mathcal{R}}(r) = 2\pi\lambda_f re^{-\pi \lambda_f r^2}$ where $\lambda_f$...
0
votes
1answer
18 views

How to get 4x4 matrix from integrating column vector and its transpose?

This is from Finite Element Methods. Part of potential energy approach. Can someone show me how to come to this symetric matrix: $$ [K]=EI/L^3\begin{bmatrix} 12&6L&-12&6L\\ &4L^2&-...
0
votes
0answers
16 views

Roots of function through integral

$$\int_{-2}^{-1} (ax^2 -5)dx = 0$$ and $$5 + \int_{1}^2 (bx+c)dx = 0,$$, then: a) $ax^2 - bx + c = 0 $ has at least one root $(1,2)$ b) $ax^2 - bx + c = 0$ has at least one root in $(-2,-1)$...
0
votes
3answers
39 views

About dy/dx in application

We know that the integral of dy/dx is y+c. however, I've encountered a problem modeling the growth of population, which can be expressed as dP/dt=kP . where P is the population and t is time ...
0
votes
3answers
36 views

can you please see if my solution is correct for the folowing equation

The equation is $ z^7+z^6+z^5+z^4+z^3+z^2+z=0 $ I tried to solve it that way $z(z^6+z^5+z^4+z^3+z^2+z+z+1)=0 $ then No root was found algebraically so $z=0$ !!!! Am I right here? Special ...
2
votes
1answer
14 views

Understanding functional composition example in Untyped $\lambda$-Calculus

The following equalities are shown as an application of $\alpha, \beta, \eta$-conversion found within Untyped $\lambda$-Calculus. I am unable to reconcile the first equality. Is the example missing a $...
0
votes
1answer
9 views

Why can we substitute any arbitrary variable in for this expression of the CDF?

I'm trying to follow this example of a variable transformation, which defines the PDF and CDF as follows: PROBLEM: The above definition is used in solving this problem: Question: Now, where I get ...
0
votes
1answer
16 views

Rotating a figure around x-axis whose area is given

Let's say I have a 2d-figure above the x-axis and I'm already given it's area (so I don't have to integrate). All of this figure touches the x-axis, so there's no hole in the middle when rotating How ...
-2
votes
1answer
21 views

Can we integrate exactly this problem.

$$\int \frac{\cos x}{2x}dx$$ when $\cos x$ is a finite differential family and $x^{-1}$ is a infinite differential family then how we can solve this.
2
votes
2answers
80 views

Compute $\lim \limits_{n \to \infty}\left(\frac {\sqrt[n]a + \sqrt[n]b}2\right)^{n} ~~~ (a, b>0)$

$$\lim_{n \to \infty}\left(\frac {\sqrt[n]a + \sqrt[n]b}2\right)^{n} ~~~ (a, b>0)$$ I extended its domain and applied L'Hopital's rule to get the answer $\sqrt{ab}$. However, is it possible to ...
0
votes
0answers
17 views

Related rates of triangle

A radar gun is positioned 50 feet from the edge of a straight road. At a certain moment, the straight-line distance between the gun and an approaching car is 130 feet, and the distance is decreasing ...
0
votes
1answer
19 views

PDF of sums of independent random variables confusion

Suppose that $X$ and $Y$ are independent continuous RVs with PDFs $f_X$ and $f_Y$ respectively. I want to find the PDF of $Z = X + Y$. The CDF of $X + Y$ is $$F_{X+Y}(z) = P(X + Y \leq z)$$ $$=\...
-1
votes
4answers
53 views

solve this equation:$A=(2-\sqrt{3})^4+(2+\sqrt{3})^4$ [on hold]

Hi please help me solving this puzzle. Calculate and solve this equstion using relations between roots of quadratic equations. $$A=(2-\sqrt{3})^4+(2+\sqrt{3})^4$$ Use this relations: $$ ax^2+bx+c= x^2-...
-1
votes
2answers
32 views

find a quadratic equation that : $\alpha^2+\beta^2-\alpha\beta=7$ and $(\alpha+2)(\beta+2)=20$ [on hold]

Hi please help me solving this puzzle $\alpha$ $\beta$ are roots of an quadratic equation. I want to create an equation using this rules about Its roots: $$\alpha^2+\beta^2-\alpha\beta=7$$ $$(\alpha+2)...
-3
votes
0answers
40 views

solve this equation in complex no. and write the answer in polar and rectangular form [on hold]

solve this equaation in complex no. and write the answer in polar and rectangular form $$z^7+z^6+z^5+z^4+z^3+z^2+z=0$$ !! any one please can solve this >> thanks
0
votes
2answers
24 views

how would you show that $\lim_{n\rightarrow \infty} \sum_{i=1}^{i=n} 2\pi f(x_i *)(\sqrt{1+f'(x_i*)^2}-1) (x_i-x_{i-1})$ does not necessarily equal 0

how would you rigorously show that $\lim_{n\rightarrow \infty} \sum_{i=1}^{i=n} 2\pi f(x_i *)(\sqrt{1+f'(x_i*)^2}-1) (x_i-x_{i-1})$ does not necessarily equal 0
0
votes
0answers
42 views

Evaluating $\int_0^1\frac{\ln(x)}{1-x/2}\int_0^{x/2}\frac{\ln(1-t)}{t}\,dt\,dx$

According to this post $$Q=\int_0^1\frac{\ln(x)}{1-x/2}\int_0^{x/2}\frac{\ln(1-t)}{t}\,dt\,dx=\frac{1}{8}\zeta \left( 4 \right) - \frac{1}{2}\zeta \left( 2 \right){\ln ^2}(2) + \frac{1}{{12}}{\ln ^4}(...
1
vote
1answer
37 views

Evaluate $\oint_L (x-y)dx+xydy$

Evaluate $$\oint_L (x-y)dx+xydy$$ where $ L\in \{x=a\cdot \cos ^3t,y=a \cdot \sin ^3t\}, 0\leq t \leq \Pi$ and $\{x=a\cdot \cos t,y=a\cdot \sin t\}, \Pi\leq t \leq 2\Pi $ So I try $$=\int_0^\Pi ((a\...
1
vote
0answers
31 views

Is there an exact solution to the minimum distance from a $\tan(x)$ curve to the origin?

Of course, there is the trivial answer that the minimum distance is 0, since tan passes through the origin, but what I'm interested in is the solution when $x < -\frac{\pi}{2}$, ie. this point here:...
0
votes
2answers
42 views

Can we take a derivative with respect to $y$?

For example, suppose we have the function $f(x)= y$ . Can we take the derivative of the function with respect to $y$? Or maybe we can't because of the definition of derivative?
0
votes
2answers
51 views

Proof inequality $\frac{\sqrt{\pi}}{2}\sqrt{1-e^{-a^2}} < \int_0^a e^{-x^2}dx < \frac{\sqrt{\pi}}{2}\sqrt{1-e^{-2a^2}}$

I'm asked to prove the inequality $$\frac{\sqrt{\pi}}{2}\sqrt{1-e^{-a^2}} < \int_0^a e^{-x^2}dx < \frac{\sqrt{\pi}}{2}\sqrt{1-e^{-2a^2}}$$ After playing around for a while I was able to find (...
1
vote
0answers
29 views

How to integrate $\int_0^{t} xe^{i(a x^4-bx^2)}J_0(cx)\,dx$?

Find $$\int_0^{t} xe^{i(a x^4-bx^2)}J_0(cx)\,\mathrm dx$$ Given $a>0,\ b>0, c>0, t>0$. I decompose it into two parts: $$\int_0^{t} x\cos{(a x^4-bx^2)}J_0(cx)\,\mathrm dx,$$ $$\int_0^{t}...
0
votes
1answer
27 views

Condition of the method of Lagrange Multiplier?

Based on my Calculus textbook, the method of Lagrange multipliers is stated as follow: Suppose that $f(x,y,z)$ and $g(x,y,z)$ are differentiable and $\nabla g \ne \mathbf 0$ when $g(x,y,z) = 0$. To ...
0
votes
1answer
41 views

Find $p$ such that $f(x) = e^{−\frac1{|x|}}$ is differentiable at $0$

The function $f$ is given by $$ f(x) = \begin{cases} e^{−\frac{1}{|x|}} & \text{if } x \ne 0\\ p & \text{if } x = 0 \end{cases}. $$ I have to find out which value of $p$ I need, so that the ...
0
votes
0answers
10 views

Let $f:I\mapsto \mathbb{R}$, find the Taylor's polynomial with Peano Remainder of order $3$ centered at $0$. Use $f(x)=\sqrt{x+1}$

Let $f:I\mapsto \mathbb{R}$, find the Taylor's polynomial with Peano Remainder of order $3$ centered at $0$. Use $f(x)=\sqrt{x+1}$ I'm stuck, and i didn't find any material on peano remainder besides ...
1
vote
1answer
65 views

Show that $f$ is differentiable at $x=0$

Let $f,g: I \mapsto \mathbb{R}$ defined on an interval such that $0 \in I$. Suppose that $(f.g):I\mapsto \mathbb{R}$ and $ \frac{f}{g}: I\mapsto \mathbb{R} $ are differentiable at $x=0$ with $g(0)\...
0
votes
0answers
28 views

Calculate surface integral given $G = x^2\,\mathbf i + xy\,\mathbf j + z\,\mathbf k$

Let $G = x^2\,\mathbf i + xy\,\mathbf j + z\,\mathbf k$ and let $S$ be the flat triangular surface with vertices $(0, 0, 0)$, $(1, 1, 0)$, $(2, 2, 2)$. Assume $S$ is oriented towards the postive $x$-...
1
vote
2answers
55 views

Integrate $ \int \frac{x^4 +1}{x^6 - 1}dx $

Integrate $$ \int \frac{x^4 +1}{x^6 - 1}\, \mathrm dx$$ I have tried using partial fractions but to no use. Thanks for help.
-1
votes
1answer
32 views

How to solve the sum of this Infinite series? [on hold]

I'm a 10-th grade student and I haven't learn calculus yet. When doing a programming problem, I suddenly come up with a mathematical model, however, it contains an infinite series which I haven't ...
0
votes
2answers
28 views

Bivariate functions - minimum, maximum, saddle point [on hold]

In the following, $k$ is a real number. Define a bivariate function $f_k$ by $$f_k(x, y) = kx^3 + x^2 + 2y^2 − 4x − 4y.$$ The addition of $k$ confuses me. Normally I would derive with respect to both ...
0
votes
1answer
41 views

Find $\lim_{n\to\infty}\frac{(a-1)^{-n+1}}{a^{-n-1}}$ [on hold]

Hello could you please help me solve this limit? $$\lim_{n\to\infty}\frac{(a-1)^{-n+1}}{a^{-n-1}},\quad a\ge1.$$
0
votes
2answers
59 views

better way to explain $\lim\limits_{x\to 0} e^ \frac {−1}{|x|} $

$$\lim\limits_{x\to 0} e^\frac{−1}{|x|}$$ I know that if $x$ goes to $0$, that $\frac{-1}{|x|}$ goes to $-\infty$ and thus, the limit goes to $0$. But, is there a more mathematical way to explain ...
0
votes
0answers
22 views

Volume under the surface parameterized by $\Phi(t,θ)=\left(\;\frac{\cos\theta}{\cosh t}, \frac{\sin\theta}{\cosh t}, t−\tanh t\;\right)$ [duplicate]

Let $C$ be the parametrised surface given by $$\Phi(t,θ)=\left(\;\frac{\cos\theta}{\cosh t}, \frac{\sin\theta}{\cosh t}, t−\tanh t\;\right)$$ for $0\leq t$ and $0\leq θ< 2\pi$. Let $V$ be the ...
0
votes
2answers
36 views

Proving $\sum_{n=1}^\infty \frac{\cos \pi n}{2^n}$ converges by the comparison test [on hold]

$$\sum_{n=1}^\infty \frac{\cos \pi n}{2^n}$$ How can I show whether or not the following series converges using the comparison test?