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

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Math Subject GRE 1268 Question 55

If a and b are positive numbers, what is the value of $\int_0^\infty \frac{e^{ax}-e^{bx}}{(1+e^{ax})(1+e^{bx})}dx$. A: $0$ B: $1$ C: $a-b$ D: $(a-b)\log(2)$ E: $\frac{a-b}{ab}\log(2)$ I really ...
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1answer
11 views

Question on continuity and differentiability of min() and max() functions.

Question: $f(x)=x^2-2|x|$. Test the continuity of $g(x)$ in the interval $[-2,3]$ if $g(x)$ is defined as: attempt: $f(x)$ is defined as: But i am finding it difficult to understand $g(x)$. ...
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40 views

What does it mean for a function to “preserve the limits of sequences”?

I've translated the English Wikipedia page "Limit of a sequence". What does the following statement mean? In fact, any real-valued function f is continuous if and only if it preserves the limits ...
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1answer
29 views

Calculus stay to Real Analysis as $x$ stay to Functional Analysis

Hi guys i had a look to book which treat the subject of Calculus (of course...) Analysis and Functional Analysis. Is that correct to state that Calculus is more focused on "computing" while ...
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0answers
33 views

Problem with a step involving a type of Riemann integration

I am reading this text, , and I find it unclear how the ratio of the considered rectangle's area to its length tends to become the derivative of a function $S$ as the lenght of the considered ...
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2answers
21 views

Factoring out quotient of an expression.

I am following the MIT Open Courseware on Single Variable Calculus and in the first lesson when taking the derivative of a simple function I found myself confused because of my knowledge gap in ...
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10 views

How to determine the minimum number of basis functions thats linear superposition best reproduces a set of curves?

How to determine the minimum number of basis functions that's linear superposition best reproduces a set of arbitrary curves?
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1answer
79 views

Convergence of $\sum \frac{2n+1}{(n^2+n)^n}$

I have to choose the right option: The series $$\sum_{n\geq 1} \frac{2n+1}{(n^2+n)^n}$$ a. Converges to 1. b. Converges to a number >1. Using ...
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24 views

different combinations of numbers [on hold]

Can anyone help me pick 1,000 combinations of six numbers from two separate pools of numbers - five different numbers from 1 to 75 and one number from 1 to 15?
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0answers
49 views

An integral that I cannot simplify.

Good day, esteemed students of mathematics! I have been trying to prove that the convolution of $2q$ Gaussian probability distributions is another $q$ Gaussian probability distribution with the same ...
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1answer
21 views

An inequality involving supremum and integral

Let $g$ be a positive function defined on $(0,\infty)$. Is the following inequality always true ? $$ \sup_{r<t<\infty}g(t)\leq C\int_{r}^{\infty}g(t)\frac{dt}{t}, $$ where positive constant $C$ ...
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2answers
50 views

Trouble with finding the limit of this sequence

Well I was trying to find the limit of - $$ \lim_{x\rightarrow \infty } \lim_{n\rightarrow \infty} \sum_{r=1}^{n} \frac{\left [ r^2(\sin x)^x \right ]}{n^3}$$ obviously $$ ...
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2answers
482 views

How to recognise intuitively which functions grow faster asymptotically?

There are some cases where it is not so simple to decide which function grows faster asymptotically. For example, in the following cases, why (intuitively) $g(n)$ should grow faster than $f(n)$, or ...
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3answers
58 views

Prove that series $ \sum^{+\infty}_{n=0}a_n(x-x_0)^n $ and $ \sum^{+\infty}_{n=0}(n+1)a_{n+1}(x-x_0)^n $ have the same radius of convergence.

I want to prove that these two power series $$ \sum^{+\infty}_{n=0}a_n(x-x_0)^n $$ and $$ \sum^{+\infty}_{n=0}(n+1)a_{n+1}(x-x_0)^n $$ have the same radius of convergence. What I've done so far is: ...
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28 views

derivative of exponential function proof- [on hold]

I will post link because stackE won't accept size or type of file. This is part of proof for derivative of exponentinal function. 1.part http://postimg.org/image/iz97da581/ How do they get bottom ...
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0answers
70 views

Integrating functions with $x^3$

After learning the integration of various functions with $x^2$ involved, I was given the following integration, as a challenge: $$\sqrt{1+x^3}$$ I tried various methods - too long to even try and ...
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2answers
49 views

What are some unfamiliar and/or special tricks used to evaluate limits?

What are some neat tricks used to evaluate limits that might be otherwise a problem to deal with? I'm not asking for methods akin to L'Hopital's rule.. which is often used. My question is geared ...
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1answer
37 views

Finding the Zeros

A word problem gives this cost equation and asks to find the x where the average cost is minimized, to do so, I need to solve for average cost and derive it and then set it to zero and solve; $c(x) = ...
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1answer
68 views

Old calculus books?

this is really a question about math and not books. I am mainly wondering if reading really old calculus books is still beneficial for undrgraduate students today. I was told that the material covered ...
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2answers
17 views

Div$f$ is invariant under an orthogonal change of coordinates

Let $f: \mathbb{R^n} \to \mathbb{R^n}$ and $Df$ exists. I need to show that div$f$ is invariant under an orthogonal change of coordinates. Let $T:\mathbb{R^n} \to \mathbb{R^n}$ be an orthogonal ...
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117 views

$\lim x_n^{x_n}=4$ prove that $\lim x_n=2$ [duplicate]

Let $(x_n)$ be a sequence of real numbers, such that: $\lim x_n^{x_n}=4$, prove that $\lim x_n=2$ I'm not sure if my proof is right. I assumed that $\lim x_n $ isn't 2 and using Cauchy's criterion: ...
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25 views

Please Help Me - Is this conclusion true?

In a sliding mode control, we have : $ s = \dot e + \Lambda e $ we know that e and $ \dot e $ are independent variables. Now in order to find control effort from the system dynamics, we rewrite the ...
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1answer
14 views

Flatten grid coordinates to find display order.

I have a grid measuring (3 * 3) for example (though there may be more rows, there will always be 3 columns only), and i need to find the display order of a particular item, given it's x, y position. ...
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3answers
59 views

Consider the function f(x)=sin(x) in the interval x=[π/4,7π/4]. The number and location(s) of the local minima of this function are?

This is MCQ of a competitive exam(GATE), Answer is (d) given by GATE , and from other sources ,explanation is (b) somewhere and (d) somewhere , I am going with (b) as minimum at $270$, I have drawn ...
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4answers
283 views

Basic of Partial Differential Equation

I pretty new to calculus and I am trying to understand the following transformations: $2uu_{t} = \frac{\partial }{\partial t}u^{2} $ $2u_{t}u_{tt} = \frac{\partial }{\partial t}u_{t}^{2} $ $2uu_{xx} ...
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1answer
49 views

Prove or disprove that a series is convergent

I was given the following task which I struggle with. Prove the following statement, or disprove it by giving a counter example: if $\sum_{n=1}^\infty a_n$ is convergent then $\sum_{n=1}^\infty ...
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9 views

limit of homogeneous polynomials as the order approaches infinity?

Let $f_n(x,y)=\sum a_ix^iy^{n-i}$ be a sequence of homogeneous polynomials with fixed coefficients $a_i$, is it possible to make $f_n$ converge as $n\rightarrow\infty$ after possible renormalization? ...
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2answers
91 views

Name of $|x|^p+|y|^p\le (|x|+|y|)^p$ ($p\ge 1$)?

I checked these What is the difference between square of sum and sum of square? Prove $(|x| + |y|)^p \le |x|^p + |y|^p$ for $x,y \in \mathbb R$ and $p \in (0,1]$. It is easy to see $p$-th power ...
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1answer
26 views

Understanding second step in derivation of joint distribution

This is a follow up to an earlier question Understanding step in derivation of joint distribution. In a derivation I am trying to understand, there is the following argument: $$ n! \int \prod_{i=1}^n ...
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2answers
32 views

Evaluating the bounds for a triple integral

I've working on the problem: Evaluate $\iiint_Q$ $1/(x^2 + y^2 + z^2)$, where Q is the solid region ABOVE the xy-plane (and we must do this in spherical coordinates). What I've done thus far is ...
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3answers
30 views

A balloon rises at a certain rate (in body), What is velocity of balloon after 40 seconds?

A balloon rises vertically from the ground so that its height after $t$ seconds is $h(t) =\frac12t^2+\frac12t$ feet where $t$ is between $0$ and $60$. What is the velocity of the balloon after $40$ ...
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1answer
62 views

How to find unkown height of triangle without hyptenuse

I been trying to solve this question and have tried to solve it for many days, but do not know how, any help would be much oblidged. A cable company owns the roads marked with the dotted lines in ...
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2answers
32 views

Limit with fraction in numerator

I need to find the limit for $$ \lim_{x\,\rightarrow\,0} \frac {\frac{1}{x+3} - \frac{1}{3}}{x} $$ It's supposed to be $-1/9$. I've tried changing it around multiple ways, and get the $9$ but never ...
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32 views

Can someone suggest some reference for a particular kind of infinite series

Can someone suggest some reference for the properties of the series $\lim_{n \to \infty} \sum_{i = 0}^{n} f(n, i)$ For example, when can we apply approximations to $f(n, i)$ that only works when $n ...
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19 views

Will this method always find the maximum of a positive-definite function?

Definitions: A real-valued, continuously differentiable function $f$ is positive definite on a neighborhood of the origin, $D$, if $f(0) = 0$ and $f(x) > 0$ for every non-zero $x \in D$. ...
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21 views

How do you calculate an area enclosed by four tangents by using the integration method?

For example, make it $y=3x-6$, $y=3x-15.48$, $y=-0.25x+1.25$, and $y=-0.25x-1.06$. It's been taken by finding the tangent line of a curve $y=(x-2)(x-3)(x-5)$.
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1answer
20 views

A question about fixed-point iteration sequence of a two times continuously differentiable function

I am stuck at this problem: Let $g:[a,b]\to[a,b]$ be a 2 times continuously differentiable function that satisfy: for all $x\in [a,b]$, $g''(x)\neq 0$ And let $s$ be an arbitrary fixed-point of ...
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2answers
62 views

how to prove that $ \lim x_n^{y_n}=\lim x_n^{\lim y_n}$?

Let $(x_n)$ and $(y_n)$ be two sequences of real numbers, such that: $\lim x_n = a > 0$ and $\lim y_n = b \in \mathbb R$ I need to prove that: $$\lim \left( x_n ^{y_n} \right) = a^b$$ I tried ...
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4answers
62 views

evaluate $\lim_{n\to\infty}\sum_{r=1}^{n-1}\frac{e^{r/n}}{n}$

Each term in the equation given in title tends to zero. $e^{\frac{r}{n}}$ tends to 1 and the denominator tends to infinity. Also, even the greatest numerator in the summation $e^{\frac{n-1}{n}}$ ...
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2answers
53 views

GRE Math Subject Practice Test Question $53$

f and g are function of a real variable such that $g(x) = \int_0^x f(y)(y-x)dy$ for all $x$. If g is three times continuously differentiable, what is the greatest integer n for which f must be $n$ ...
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simplifying complex expression

Hi I am trying to simplify the following expression:$$ \left|\frac{1}{a+ib}\left(\frac{J_1(c x)}{J_1(c b)}-x\right)\right|^2,\quad a,b,x\in \mathbb{R}, \ c\in \mathbb{C} $$ Is there a simple way of ...
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52 views

Integral $\int z^2\Re(J_1(z))dz$

$$ \int x^2 \, \Re\left[{J_1(a x)}\right]dx=\frac{1}{a^3}\int z^2 \Re\left[\frac{z}{2}\sum_{k\geq 0} \frac{(-1)^k}{k!\Gamma(k+2)} \left(\frac{z}{2}\right)^{2k}\right]dz $$ where $a\in \mathbb{C}$ and ...
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104 views

Could it possibly have a nice closed form? $\int _0^1\int _0^1\frac{x y}{(x+1) (y+1) \log (x y)}\ dx \ dy$

Using multiple integrals it's not hard to show that the present integral reduces to some integral over squared digamma functions, but then things become harder. How would you tackle the problem? ...
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1answer
23 views

Understanding step in derivation of joint distribution

In a derivation I am trying to understand, there is the following argument: \begin{align} &=\int n!\prod_{i=1}^n f_X(x_i)\mathbb{I}_{x_1\le x_2\le\ldots\le ...
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1answer
25 views

How to calculate the partition function of a given distribution?

As noted in A FULL BAYESIAN APPROACH FOR INVERSE PROBLEMS, let $ y = Ax + n$, where $y$ is a $m$ dimensional signal and $n$ is white Gaussian noise with precision $\beta$, so we have: $$ y|x, \beta ...
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2answers
25 views

Show that any 2D vectors can be expressed in the form…

(a) Show that any 2D vector can be expressed in the form $s \begin{pmatrix} 3 \\ -1 \end{pmatrix} + t \begin{pmatrix} 2 \\ 7 \end{pmatrix},$ where $s$ and $t$ are real numbers. (b) Let $u$ and $v$ be ...
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1answer
47 views

Condition on p for convergence of $\sum{\frac{1}{n(\log(n))^p}}$

For what values of $p$ is the series $\sum{\frac{1}{n(\log(n))^p}}$ divergent and for what values it is convergent?
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3answers
75 views

Series convergence $x+\frac{2^2x^2}{2!}+\frac{3^3x^3}{3!}+\frac{4^4x^4}{4!}+\cdots$ [on hold]

Choose the right option. The series $x+\dfrac{2^2x^2}{2!}+\dfrac{3^3x^3}{3!}+\dfrac{4^4x^4}{4!}+\cdots$ is convergent if a. $0<x<1/e$ b. $x>1/e$ c. $2/e<x<3/e$ d. $3/e<x<4/e$ ...
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2answers
86 views

Problem 7 IMC 2015 - Integral and Limit

I'm trying to solve problem 7 from the IMC 2015, Blagoevgrad, Bulgaria (Day 2, July 30). Here is the problem Compute $$\large\lim_{A\to\infty}\frac{1}{A}\int_1^A A^\frac{1}{x}\,\mathrm dx$$ ...
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2answers
37 views

solve this linear equation

Using linear differential equation, solve the following equation $( y \log (x)-2) y \textrm{d} x =x \textrm{d}y$. Source: "higher engineering mathematics by grewal"