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Questions tagged [calculus]

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

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0answers
8 views

Gamma function and limit of integral

For $n \to \infty$, prove $\int_{2n}^{\infty} t^n e^{-t}\ dt = o(\Gamma(n+1))$. I have no clue to resolve this. I started like this : $\Gamma(n+1) =\int_{0}^{\infty} t^n e^{-t} dt$ and $\Gamma(n+1)=...
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8 views

Vector and Scalar potential conditions

Given that A is a 3-by-3 matrix, with constants a11, a12, a13, ..., a31, a32, a33. And vector F is A*x, where x is a vector. What are the general conditions on vector A for vector F to have a 1) ...
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0answers
9 views

How to justify this limit (Gamma function)?

For all $x, t \in \mathbb R_*^+$ and $n \ge 2$ such that $t < n$. And from this equation: $\left(1-\frac t n \right)^n \times \exp\left( n \int_0^{\frac{t}{n}} \left( \frac t n - \sigma \right) \...
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1answer
22 views

Show that a function is negative over its domain

I would like to demonstrate that the following function is negative \begin{equation} -\frac{t}{4\sqrt{x}^3}\bigg[1-\bigg(1+\sqrt{x}\bigg)^{\frac{1}{t-1}}\bigg]+\bigg(\frac{1}{1-t}\bigg)\bigg(\frac{t}{...
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1answer
26 views

Singularity of holomorphic function

Let $f: \mathbb C \rightarrow \mathbb C$ be a holomorphic function in an open set around some $c \in \mathbb C$, but excluding $c$. Moreover, assume that the Laurent series for $f$ around $c$ ...
3
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3answers
42 views

Find $\lim_{n \to \infty} \prod_{k=1}^{n} \frac{(k+1)^2}{k(k+2)}$

I have to find the following limit: $$\lim\limits_{n \to \infty} {\displaystyle \prod_{k=1}^{n} \dfrac{(k+1)^2}{k(k+2)}}$$ This is what I tried: $$\lim\limits_{n \to \infty} {\displaystyle \prod_{k=...
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1answer
15 views

Volume bounded by rotating a curve

Consider the region bounded by the curves $y=\sqrt{4−x^2}$, $y=2$, and $x=2$. This region is rotated about the $y-$axis. What is the volume of the generated solid?
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1answer
15 views

Gradient vector parallel to postition vector

Let $f:\mathbb{R}^3\to \mathbb{R}$ be a differentiable fuction such that $\nabla f(x,y,z)$ is parallel to $(x,y,z)$ for all $(x,y,z)\in \mathbb{R}^3$. Show that, $f(a,0,0)=f(-a,0,0)$ for all $a\in \...
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34 views

Prove that the second derivative has a zero

my question is: Can I use Rolle's theorem to prove that the second derivative has a zero? Consider a Real function of Real variable defined by $f(x)=(x+1)\cdot e^{x^2}$. Prove that the second ...
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49 views

Show that $f(x) = \arctan(x)$ is uniformly continuous in R, directly from the definition

I need a proof like this, just for $\arctan x$ (without using derivate) $ϵ>0$ and $x,y∈R$ $|f(x)−f(y)|<\epsilon$ ⟹ $|\arctan x−\arctan y|<\epsilon$ ?? $|\arctan x−\arctan y|$=$|\arctan \...
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0answers
19 views

proving that a function is continuous using delta and epsilon

Prove that $x^3$ is continuous at $x=-2$ How can we prove that the limit is continuous using epslion delta proof$?$
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1answer
12 views

x-axis graph question. very short question

I have the following equation $(x+a)y=0$ where x and y are non-negative variables. and a is strictly positive. So, the graph go this equation is x-axis graph? Is this right? Regardless of value ...
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20 views

$Ψ(x, y) = (e^x cos y, e^x sin y) =: (u(x, y), v(x, y))$ range and solution check

Define $Ψ:ℝ^2→ℝ^2$ by $Ψ(x, y) = (e^x cos y, e^x sin y) =: (u(x, y), v(x, y))$ (a) What are the images, under $Ψ$, of lines parallel to the coordinate axes? (b) What is the range of $Ψ$? Describe ...
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1answer
5 views

derivative of position vector is perpendicular to position vector

$\Large \vec{r}(s) = x(s)~\hat{\text{i}} + y(s)~\hat{\text{j}} + z(s)~\hat{\text{k}}$ $\Large \frac{\vec{r}(s)}{ds} = \frac{dx(s)}{ds}~\hat{\text{i}} + \frac{dy(s)}{ds}~\hat{\text{j}} + \frac{dz(s)}{...
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3answers
30 views

sum of continuous functions

Is there any possible way to prove that $\sum_{i=1}^{N}f(x_i)$ is smaller or equal to $0$ if we know that $f$ is smooth monotonously growing function with $f(0)=0$ and $\sum_{i=1}^{N}x_i=0$?
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2answers
20 views

Multivariable limit with min and max

1) I'm trying to show that $$ \lim_{(x,y) \rightarrow (0, 0)} \frac{\max(x,y)}{\min(x,y)} = 1. $$ I'm not totally sure the limit exists, but I couldn't find a counterexample so far. I've tried the ...
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1answer
12 views

List the critical numbers of the following function in increasing order. Enter N in any blank that you don't need to use

List the critical numbers of the following function in increasing order. $$f(\theta)=14\cos(\theta)+7\sin^2(\theta),-\pi \leq \theta \leq \pi$$
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2answers
41 views

Solve integral $\int_\sqrt{x}^0 \sqrt{1+\sec^2 p}\ \mathrm dp $

How would I go about solving the following integral: $$T(x) = \int_\sqrt{x}^0 \sqrt{1+\sec^2 p}\ \mathrm dp $$ I went ahead and started by flipping the bounds, but am confused on next steps, this ...
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2answers
26 views

Proving $ f(x)=x \sin{(\frac{1}{x})}-\cos{(\frac{1}{x})} $ has no limit at x=0

How to prove that $ f(x)=x \sin{(\frac{1}{x})}-\cos{(\frac{1}{x})} $ has no limit in x=0? One-sided limits in x=0 do not exist, according to WolframAlpha.
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20 views

Find $a_0,a_1$, and $a_2$ from the fact that Simpson’s rule is exact for $f(x) = x^n$ when $n = 1,2$, and $3$.

I found the answer on Slader . I just don't understand the concept behind it and need someone to explain it in a little for detail.
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0answers
17 views

algebraic vectors question [on hold]

a=(m,n) If (10 - m,m + 6n)=a how do i find m,n and |a|
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8 views

Reverse Lipschitz condition and continuity implies bijection

I am trying to prove the following: Let $f:\mathbb{R}\to\mathbb{R}$ be continuous function with $\mid f(x)-f(y)\mid\geqq c\mid x-y\mid$ for all $x,y\in\mathbb{R}$, where $c>0$ does not depend on ...
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1answer
20 views

Find $a$ and $b$ such that the improper integral converges

Find $a$ and $b$ so that the integral $$\int_1^\infty \frac{x}{(x-1)^a(1+x^2)^b}\, dx$$ converges. I know that $\int_1^\infty \frac{1}{x^p}$ converges if $p>1$ and I've tried integration by parts ...
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0answers
25 views

Prove that the equations have exactly one real solution $(x,y,z)$ with $x,y,z \geq 0$.

Let $a,b,c$ be constants and $a,b,c$ are positive real numbers. Prove that the equations $$2x+y+z=\sqrt{c^2+z^2}+\sqrt{c^2+y^2}\\ x+2y+z=\sqrt{b^2+x^2}+\sqrt{b^2+z^2}\\ x+y+2z=\sqrt{a^2+x^2}+\sqrt{...
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1answer
43 views

Does this double limit of $\cos^{2n}(m! \pi x)$ exist?

I've figured out that the repeated limit exists: $$ \lim_{m \to \infty} \lim_{n \to \infty} \cos^{2n}(m! \pi x) = \begin{cases} 1,&x\text{ is rational}\\ 0,&x\text{ is irrational}\end{cases} $...
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2answers
29 views

finding coordinates with derivatives

Find the coordinates of the point on the curve $x^2+xy+y^2=7$ where the slope is the same as at the point $(2,1)$. I’ve already found $y’$ to be $\frac{(-2x+y)}{(x+2y)}$ and the slope at the point ...
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1answer
18 views

Lagrange multiplier volume maximatisation

Using method of lagrange multiplier show that among all rectangular parallelepiped inscribed in a given sphere cube has the maximum volume
2
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2answers
60 views

Setting up the volume $\iiint_{?}^{?}dV$

Let $$S = \{ (x,y,z) | x=a+b, y = b+c, z = -b, ac-b^2\ge0, c\ge0, a\ge0\}$$ and $$x^2+y^2+z^2\le 1$$ Compute the volume of S. My work: $$V=\iiint_R 1 dV = \int\limits_{-1}^{1}\int\limits_{-\sqrt{(1-...
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0answers
37 views

Mechanics question : Tension

I think there are 10 unknowns. $$A_y,A_x,B_y,B_x,C_x,C_y,D_x,D_y,E_x,T$$ And unable to obtain 10 equations so uanble to solve this question. The answer is 138 lb according to the book for the ...
3
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0answers
75 views

Challenging Sum: compute $\sum_{n=1}^\infty\frac{H_n}{2n+1}\left(\zeta(3)-H_n^{(3)}\right)$

How to prove $$\sum_{n=1}^\infty\frac{H_n}{2n+1}\left(\zeta(3)-H_n^{(3)}\right)=\frac74\zeta(2)\zeta(3)-\frac{279}{16}\zeta(5)+\frac43\ln^3(2)\zeta(2)-7\ln^2(2)\zeta(3)\\+\frac{53}4\ln(2)\zeta(4)-\...
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0answers
14 views

What's the connection between Legendre polynomials, Associated Legendre polynomials, and Sphereical Harmonic?

Associated Legendre Polynomials and Legendre Polynomials are strongly connected. From what I read, Legendre polynomials is a special case of Associated Legendre polynomials where $m=0$. Thus the ...
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0answers
17 views

How to find direction in which Directional Derivative is zero? [on hold]

Please help me with this question ***Find the direction in which the directional derivative of f(x, y) = (x^2 - y^2)/xy at (1, 1) is zero. **
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2answers
39 views

Stability without Lyapunov methods

I've been having some problems trying to solve a problem which appears in the book im following, so any help would be really appreciated. Defn. A fixed point $x_{0}$ is asymptotically stable if it is ...
0
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3answers
22 views

Find $f(x)$ given $f'$' and two initial values

I was given that $f''(x) = 6x + \frac{1}{x^2}$ , $f(1)=0$ and $f(2) = 6$. I have gotten down to $x^3 + ln(|{x}|)+Cx + D$ and I am confused on how to find the values for $Cx$ and $D$ .
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1answer
37 views

Improper integrals(Limit comparison test)

Here is the question that someone gave to me. For the $f(x) = {x-3 \over x^2 \sqrt{\vert x-2\vert \vert x-3 \vert}}$ , Determine $\int_{2}^\infty$$ f(x)dx$ is converge or not So my colleague ...
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1answer
25 views

How to turn a limit to a definite integral

I am stuck as to where to go with this problem. What I know now is that this may be a Reimann Sum problem but I am not 100% sure. I also know that the integral is lower bound by 0 and upper bound by 4....
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2answers
23 views

Applying the Intermediate Value Theorem on periodic functions

I have just recently covered the Intermediate Value Theorem, and I wanted to practice solving problems involving this theorem. However, I encountered a problem that I am not exactly sure how to tackle ...
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0answers
32 views

intersections of decreasing sequences of convex sets [on hold]

Let $\{C_k\}_k$ denote a decreasing sequences of convex sets, where $C_1 \supseteq C_{2} \supseteq\cdots $, its intersection $C = \bigcap_k C_k \neq \emptyset$. For any vector $x\in\mathbb{R}^n$, ...
3
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4answers
48 views

Find $\lim\limits_{n \to \infty} \sqrt[3]{n^3+2n^2+1}-\sqrt[3]{n^3-1}$.

I have to find the limit: $$\lim\limits_{n \to \infty} \sqrt[3]{n^3+2n^2+1}-\sqrt[3]{n^3-1}$$ I tried multiplying with the conjugate of the formula: $$(a-b)(a^2+ab+b^2)=a^3-b^3$$ So I got: $$\lim\...
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2answers
26 views

How do I find a function $f(x,y)$ such that $\nabla f = \langle y,-x\rangle$?

I've been stuck on this problem for a while and I am starting to think that it's not possible. Could someone please point me in the right direction? Thanks!
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2answers
28 views

Use the intersection of the following sets to prove the integral

if $0 \in \{ f(0),g(0)\} \cap \{f(e),g(e)\}$ prove that$\int_0^e f(x)g'(x)= -\int_0^e g(x)f'(x)$. I am lost on this one this looks very similar the theory of integration by parts
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2answers
26 views

Attempt of a question dealing with the Intermediate Value Theorem

I'm new to proofs, and I recently got introduced to the Intermediate Value Theorem and decided to practice some questions. So, I was wondering if I got the following question right and if I'm even on ...
1
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1answer
19 views

Derivative of a generic polynomial function

Let $P:M_n(\mathbb{C})\to\mathbb{C}$ be a polynomial function, $A=(A_{ij}),E_{ji}\in M_n(\mathbb{C})$,where $E_{ji}$ is defined as matrix filled with zeros except $1$ in j-th row and i-th column and $...
0
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2answers
32 views

Question involving the application of Intermediate Value Theorem

I'm new to mathematical proofs, and I have just covered the Intermediate Value Theorem. I have tried to practice my understanding of the theorem, but I have encountered a question that I'm not sure ...
1
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2answers
48 views

Use the $\epsilon$ - $\delta$ definition to show that $\lim_{x\to \sqrt2} \frac{1}{2}(\frac{2}{x}+x) = \sqrt2$

Use the epsilon-delta definition to show that $\lim_{x\to \sqrt2} \frac{1}{2}\left(\frac{2}{x}+x\right) = \sqrt2$. I have been shown the following approach to solve this: Let first $\epsilon > ...
1
vote
0answers
47 views

Converting an Integral to a Sum

I have an integral over $t$ where I choose $t$ takes discrete values $t=0,1,2,...$ and would like to write the integral as a sum using the same function. I realize that this is not necessarily ...
0
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2answers
31 views

Calculation of limit (two sides) [on hold]

I don't have a clue, how to solve this limit from each side (+-). It is +inf*0 from left side and -inf*0 from right side. Can ...
4
votes
2answers
60 views

Evaluate $\sum_{n=0}^{\infty} \frac{z^{kn+p}}{(kn+p)!}$

$$\sum_{n=0}^{\infty} \frac{z^{kn+p}}{(kn+p)!}$$ I have found out that $$\sum_{n=0}^{\infty} \frac{z^{kn}}{(kn)!} = \frac{1}{k} {}\sum_{i=0}^{k-1} e^{z\varepsilon_i}$$ where $\varepsilon_i$ is the ...
3
votes
2answers
57 views

What is the supremum of the set?

What is the supremum of the set consisting the following real numbers: • 0.200 . . . • 0.2500 . . . • 0.25200 . . . • 0.252500 . . . Would it just be 0.252500?
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4answers
72 views

Solve the ODE $y''-x^{-1}y'+x^{-2}y=0$.

Consider ODE $y''-x^{-1}y'+x^{-2}y=0$. I have found the values of the constant $n$ for that $y(x) = x ( \ln x)^n $ satisfies the ODE as $0$ or $1$. How do you calculate the general solution I'm ...