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

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1answer
13 views

Simple Question about Derivative property

Suppose $f:[-1,1] \to \mathbb{R}$ is twice differentiable and $f(-1) = f(1) = 0$ and $f(0) = 1$. Prove that there exists $x_0 \in (-1,1)$ with $f''(x_0) = -2$. I tried establishing this with ...
1
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1answer
14 views

Area Bounded by a Curve, 2 Lines and the $x$-axis - Solution at Odds with Textbook

I was solving a problem today, and it appears my approach is at serious odds with the provided solution. The Problem Find the area between $y = x^3 - 3x^2$, the $x$-axis and the lines $x = 2$, $x = ...
0
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2answers
41 views

If $x^2+y^2=14x+6y+6$, find the maximum value of $3x+4y$

Since $(x,y)$ is constrained to be in a circle, $3x+4y=73$. (The lines $3x+4y=c$ and the circle should touch each other and should be as far away as possible from origin. From this, $c$ can be ...
2
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0answers
25 views

Area interpretation of integrals for the floor function [duplicate]

Use the area interpretation of integrals to prove that; $$\int_{0}^n \lfloor t \rfloor dt = \frac{(\lfloor n\rfloor-1)\lfloor n\rfloor}{2}$$ for all integers n $\ge$ 0, and hence find a formula for ...
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1answer
50 views

I'm trying to find $\lim\limits_{x\to 0^+} \dfrac{x^2\sin(1/x)}{\sin x}$, I get $\dfrac{\infty}{0}$, what is that?

If $ \frac{\infty}{0} $is not an indeterminate form (like $ \infty \times 0, 1^\infty, \infty-\infty, \infty^0, 0^0 $) then what is it? I know $ \frac{0}{\infty}$, for example, is zero...so what is ...
0
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1answer
33 views

Integrating Floor Functions

I am having difficulties understanding and solving the following problem. So far I think I understand the first line. I know what the graph of floor[t] looks like and I understand they're asking to ...
0
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1answer
26 views

Is this differential equation exact?

The definition of exact differential equation reads that: "A differential equation of the form $M(x, y)dx+N(x, y)dy=0$ is said to be exact if its left hand member is the exact differential of some ...
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0answers
51 views

Weird problem with L'Hopitals: $\lim\limits_{x\to 0^-} x^3e^{1/x}$

$$\lim\limits_{x\to 0^-} x^3e^{1/x}$$ If I plug in zero, I get $0\cdot\infty$. So, this, I thought was a hint to try to rewrite the problem to try to get $\frac{\infty}{\infty}$ or $\frac{0}{0} so ...
2
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0answers
29 views

Limit of Stirling approximation theorem

This limit I tried to solve it by using Stirling approximation and here is the limit and my tried $\lim_{0\to\infty }\frac{ \sqrt[n]{\dbinom{n}{1} \dbinom{n}{2} \dbinom{n}{3} \dbinom{n}{4} \cdots ...
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1answer
38 views

“Not” indeterminate form problems

"...are not indeterminate forms. Find the following by inspection:" $\displaystyle\lim_{ x\to \pi/2} (\cos x)^{\tan x}$ and $\displaystyle\lim _{x\to \pi/2} [ (2/\pi-2x) + \tan x ]$ These are ...
2
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2answers
21 views

Do roots lead to two antiderivatives that differ in their non-constant terms?

Consider the following example: $f'(x)= x^{-3/2}$ and $f(4)=2 $ $f'(x)= x^{-3/2}\Rightarrow \frac {x^{(-3/2) + 1}} {-1/2} \Rightarrow$ $\frac{-2}{\sqrt x} +C =f(x)$ This is where the problem ...
0
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0answers
37 views

Some questions about computing integrals?

I am trying to compute some integrals in the paper. My question is about the example on page 119 after Lemma 5.2.8 on page 118. Why $$\int_0^T e^{-\alpha_2(\pi)} = \int_0^T ds ...
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1answer
12 views

Model equation for customer numbers.

I am trying to formulate an equation to determine the number of customers signed up to a subscription as a function of time. It's been 10 years since I studied maths, so bear with me! The assumptions ...
2
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4answers
45 views

Integrate $\tan^2(\frac{\pi}{12} \cdot y)$

Integrate $\tan^2(\frac{\pi}{12} \cdot y)$ Wolfram gives the answer: $$\frac{12 \tan(\frac{\pi y}{12})}{\pi} -y + \text{constant}$$ But why is the value of $\tan^2$ not getting differentiated? ...
5
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1answer
27 views

Find $\sum_{k=0}^{\infty}(1-1/n)^{2k}\frac{e^{-n\theta}(n\theta)^{k}}{k!}$ (the variance of $(1-1/n)^{X_1+\cdots+X_n}$)

Given a random sample $X_1,\ldots,X_n$ from Poisson distribution with an unknown parameter $\theta>0$.$T:=(1-1/n)^{X_1+\cdots+X_n}$. Find $\operatorname{var}(T)$. My work: I find $T$ is a UMVUE ...
2
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1answer
37 views

Showing that $f_{xy} = f_{yx}$ for the following function.

Show that for the function $$f(x,y) = 9x^3y+2y^3+10x^2y^2+9$$ satisfies the equality $$f_{yx} = f_{xy}$$ by computing the partial derivatives. I know that $f_y= 9x^3+6y^2$ because we exclude all ...
-1
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1answer
31 views

Why does the second derivative method allow us to classify stationary points?

I'm learning how to find the nature of a stationary point using the second derivative method. I understood the first derivative method where you pick two close points near the target point and find ...
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2answers
33 views

Evaluating $\int_{0}^{\frac{\pi}{2}} (\sin (x) +\cos (x))^4 - (\sin(x) - \cos (x))^4 dx$

I would like to evaluate $$\int_{0}^{\frac{\pi}{2}} (\sin (x) +\cos (x))^4 - (\sin(x) - \cos (x))^4 dx.$$ One approach I used was to split the integral into two pieces and evaluating each piece ...
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0answers
27 views

Interpreting notation for a differential equation

I'm looking at the following differential equation (see Appendix B of the following paper): $$ \partial_t\mu[V_E]=-\mu[V_E] + C_{EE}[x\nu_E+(1-x)v_{ext}]-C_{EI}J_{EI}\nu_I, $$ where $\mu[V_E]$ is ...
2
votes
2answers
59 views

Why is this curve not closed?

Consider the curve $$\gamma (t) =\left( \cos(t^3+t), \sin(t^3+t) \right) $$ I am asked to show that a reparamaterization of a closed curve is not necessarily closed. The book provides this as a ...
1
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1answer
28 views

Integral of cosine over a triangle

I need to integrate $\pi \cos (\pi x)$ over a triangle T with vertices $(0,0)$, $(1,1)$ and $(2,0)$. Me reasoning is: $$\pi \int_T \cos (\pi x) dxdy =\pi\int_0^1 \int_0^{y=x}\cos (\pi x) ...
0
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3answers
31 views

Convergent Series and Proofs

I am trying to get some clarity as to what exactly this is asking me? Is this series convergent or divergent and prove: $$s_n=\begin{cases} \frac{1}{n} & n\,\text{odd} \\ 0 & n\,\text{even} ...
-1
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1answer
57 views

Is this limit of an infinte product infinity, undefined or something else

I am wondering what is the following limit is: $$\lim_{a \to \infty} \prod^{\infty}_{i=a} i$$ It might be infinity because it is infinity for all real $a$. But my intuition is telling me that the ...
0
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3answers
81 views

Help solving $5^{-4} \left(\frac{2x+5}{8}\right)^2 = 49$

I am trying to solve a simple formula but I get lost in the orders of operation. The formula is: $5^{ -4}{ \left( \frac { 2x+5 }{ 8 } \right) }^{ 2 } =49$ The easy parts: $5^{-4} = 0.0016$ ...
8
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3answers
159 views

Stirling approximation note

During my study to Stirling approximation I find this formula $n! \approx \sqrt{2\pi n} n^{n}e^{-n} $ but we know that $ 0! =1 $ And in this formula if we replace every $ n $ with $ 0$ we will ...
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2answers
37 views

Evaluating a basic integral of the exponent.

Upon reading some mathematical literature, I have encountered the following computation: $x\in X$, a Banach space, $\alpha=\text{Re }(z)$ for $z\in\mathbb{C}$ and $\omega$ is the growth bound. ...
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3answers
36 views

The value of a series .

Is it possible to find the value of $$\sum _{t=0}^{n} \binom{n}{t}(q-1)^t$$ where $q$ is a positive integer. How about the following series: $$\sum _{t=0}^{n} t\binom{n}{t}(q-1)^t$$
5
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1answer
94 views

My proof ɛ-δ Definition of a limit only disproves the limit… What have I done wrong?

Whilst attempting to prove that $$\lim_{x\to8} \sqrt[3]{(3x+3)} = 3 $$ I came up with the following as my proof: $$\lim_{x\to8} \sqrt[3]{(3x+3)} = 3 ⇔ ...
0
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1answer
21 views

Existence of the partial derivatives ${\delta^{2}f}\over {\delta x \delta y}$ and ${\delta f}\over {\delta x}$

The question is can the partial double derivative ${\delta^{2}f}\over {\delta x \delta y}$ exist without the derivative ${\delta f}\over {\delta x}$ existing? I don't know , I am ...
7
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2answers
106 views

Calculating in closed form $\int_0^{\pi/2} \arctan\left(\sin ^3(x)\right) \, dx \ ?$

It's not hard to see that for powers like $1,2$, we have a nice closed form. What can be said about the cubic version, that is $$\int_0^{\pi/2} \arctan\left(\sin ^3(x)\right) \, dx \ ?$$ What are ...
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0answers
29 views

$n$th derivative of $\sin(nx)$? [on hold]

Don't know how to solve this one please answer in a detailed manner. Thanks in advance... Answer is given in a recursive form. I have thought of using the expansion of $\sin(nx)$ and then ...
-1
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1answer
64 views

Find the $nth$ derivative of $y=\sin(x^2)$ [on hold]

Please help me and its not a homework.I have tried it a hell lot of many times and even asked my seniors but none had solved it ...So plzzz help me know step by step how to solve it..thanks in advance ...
0
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0answers
11 views

To show that $L(f) = \sup \{L(P;f) : P \in P^*\}$.

Let $a>0$ and let $J = [-a , a]$. Let $f: J \to \Bbb R$ be bounded and $P^*$ be the set of all partitions $P$ of J that contain $0$ and are symmetric. Show that $L(f) = \sup \{L(P;f) : P \in ...
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0answers
32 views

Finding critical points of $f(x) = \frac{\ln(x^2-1)}{x^2-x}$

In sketching the function below, I'm having a hard time finding the critical points of $f'(x)$, mostly because of factoring. The function is $$f(x) = \frac{\ln(x^2-1)}{x^2-x}$$ and it's derivative ...
2
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3answers
59 views

Proving a function is bounded

Here's my question, and a suggestion for a solution. Please let me know if I'm wrong. Prove that the function $$f(x)=\frac{\ln(x+1)}{x}$$ is bounded in $(0,\infty)$ Solution: Using L'Hospital ...
2
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3answers
192 views

Derivative of integral in interval

Let $$F(x)=\int_{2}^{x^3}\frac{dt}{\ln t}$$ and $x$ is in $(2,3)$. Find $F'(x)$. Can somebody give me idea how to do this? Thank you
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3answers
649 views

How to calculate the limit that seems very complex..

Someone gives me a limit about trigonometric function and combinatorial numbers. $I=\displaystyle ...
1
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1answer
42 views

Limit of finite Sum [duplicate]

$$\lim_{n\to\infty}\left(\frac{1}{n}\sum_{i=0}^{n} \sqrt{\frac{i}{n}}\right)$$Any tips how to solve this problem, because clearly I have no idea how to approach it.
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0answers
35 views

String problem with two equal particles

Two equal particles are connected by a string one point of which is fixed and the particles are describing circles of radii $a$ and $b$ about this point with the same angular velocity so that the ...
0
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1answer
29 views

Calculate the flux through a surface S and my approach using Divergence theorem

Since my previous, introductory question Calculate the flux through a surface S from a field described by vectors about this example raised even more questions that I had initially - I was advised to ...
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0answers
10 views

On the properties of extended integrals

Suppose that {$A_1,A_2,...$} is a countable collection of subset of $\mathbb{R}^n$ such that $A_i\cap A_j=\emptyset$, for $i\ne j$. Put $A=\bigcup \left \{ A_i \right \}$. Let $A$ be open. I will use ...
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1answer
21 views

Closed set and derivative

$A= \{(x, y, z) ∈ R^3 : 0 ≤ x < 1, 0 ≤ y < 1, 0 ≤ z < 1 \}.$ Is this closed or open set? My opinion is open.Also I don't know how to solve this problem: Does $(xy)^\frac{1}{3}$ have ...
2
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1answer
59 views

Why it's wrong to use L'Hôpital method twice?

We have, $$ f(x)=\left\{\begin{matrix} \frac{g(x)}{x},&if\ x\not=0\\ 0,&if\ x=0 \end{matrix}\right. $$ given $g(0) = {g}'(0) = 0$, ${g}''(0) = 3$, try to solve ${f}'(0)$ $$ ...
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1answer
22 views

Convergent infinite sum [duplicate]

For which p the sum is convergent? $$ \sum_{n=2}^{\infty}\frac{1}{(\sqrt{n+1}+\sqrt{n})^p}\ln\frac{n-1}{n+1} $$ Clearly I have no idea how to solve problems of this type,tipsI think will help me
0
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1answer
21 views

Convergence of a Sequence Involving arctan - Is my solution correct?

Here's my question: Let $(a_{n})$ be a sequence where $(a_{1}) > 0$, defined as: $$(a_{n+1})=\arctan*(a_{n})$$ for all $n$. Prove that $a_{n}$ has a limit $L$ and calculate it. Solution: ...
2
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2answers
67 views

Can anybody help me in solving this integration?

i've tried all the existing methods to solve it but failed. if anybody can suggest any way to solve this that would be a great help to me. ...
0
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3answers
48 views

¿What mistakes can be made when differentiating power series (e.g $\sin x$ power series)?

I know that the derivative of $\sin x$ is $\cos x$, but I don't know what is wrong with the following: $$\sin x = \sum_{n=0}^{\infty}(-1)^n\dfrac{x^{2n+1}}{(2n+1)!}.$$ Now if I want to find its ...
0
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0answers
17 views

Partitions of unity from Spivak's Calculus on manifolds

This question has (in part) already been answered. Let $A_n=[1-1/2^n,1-1/2^{n+1}]$. Suppose that $f:(0,1)\to \mathbb{R}$ satisfies $\int_{A_n}f=(-1)^n/n$ and $f=0$ for $x \notin$ any $A_n$. Show that ...
0
votes
1answer
25 views

Derivative and length of a curve

Let $\alpha:[a;b]\to \mathbb{R}^2$ is a vector function with 1 argument $\alpha$ different from$(0,0)$.Find the derivative of : $$ \phi(t)=\ln(\|\alpha(t)\|)+\langle b,\alpha(t)\rangle$$ where ...
-3
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0answers
26 views

Equally convergent function [on hold]

Is the function $f_{n}=\frac{nx}{n+x+1}$ equally convergent in [0;1].