Questions tagged [calculus]

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

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293 views

The $d$ in Leibniz's Notation [duplicate]

Possible Duplicate: Usage of dx in Integrals I have read some tutorials about Leibniz's notation, and I am still wondering about the $d$ beside variables, and when you can de-attach the $x$ from $...
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2answers
149 views

$\lim_{n\to \infty }\sqrt[n]{a_{n}}< 1$, $a_{_{n}}\geq 0$ for every $n \in \mathbb{N}$- prove that $a_{n}$ is convergent

It is very similar to this question that I posted not a while ago, But I'm still having a hard time to transalte or use the solution that was given. Now ,the sequence $a_{_{n}}$ applies these ...
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3answers
606 views

A question on differentiability of the inverse of strictly monotonically increasing functions

I'd like to know if the following statement is true ? If $f : (0,1) \to \mathbb{R}$ is a strictly monotonically increasing function and $f$ is differentiable at some $x \in (0,1)$ then $f^{-1}(y)$ is ...
2
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2answers
240 views

$\lim_{n\to \infty }\frac{a_{n+1}}{a_{n}}< 1$, $a_{_{n}}> 0$- does $a_{_{n}}$ converge?

I'd like your help with this: The sequence $a_{_{n}}$ applies these condition: $a_{_{n}}> 0$ for every $n \in \mathbb{N}$ $\lim_{n\to \infty }\frac{a_{n+1}}{a_{n}}< 1$. I need to prove that $...
7
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2answers
260 views

Evaluating the limit $\lim \limits_{x \to \infty} \frac{x^x}{(x+1)^{x+1}}$

How do you evaluate the limit $$\lim_{x \to \infty} \frac{x^x}{(x+1)^{x+1}}?$$
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0answers
556 views

Maximizing two codependant profit equations for Bertrand Model Oligopolies

For this problem I was given the Fixed Cost, Marginal cost, and demand curves for two firms (x and y). So far, from this information I derived the profit (π) function for each firm. π(x)=-0.01x^2+70x+...
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1answer
85 views

Two variable linear function

It's been a while since I've been in any math classes. I have a linear function $(T-c)m = B$. I have many samples of $T$ and $B$, but no idea what $m$ and $c$ are. How do I solve this? My first ...
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1answer
107 views

Root of a multi-variable derivative

How I got to the problem: Let $f(x,y)=\frac{1}{\sqrt{(x-a_1)^2+(y-a_2)^2}}+\frac{1}{\sqrt{(x-b_1)^2+(y-b_2)^2}}$, where $a_1,a_2,b_1,b_2 \in \mathbb{R}, a=(a_1,a_2)\neq (b_1,b_2)=b$ are fixed and $x,...
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2answers
3k views

Olympiad calculus problem

This problem is from a qualifying round in a Colombian math Olympiad, I thought some time about it but didn't make any progress. It is as follows. Given a continuous function $f : [0,1] \to \mathbb{R}$...
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1answer
508 views

Prove that $\int_0^x f^3 \le \left(\int_0^x f\right)^2$

This problem comes from Calculus by Spivak, namely in Chapter 14- "The Fundamental Theorem of Calculus". Suppose that $f$ is a differentiable function with $f(0)=0$ and $0<f'\le1$. Prove ...
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1answer
518 views

Where's the trick in this partial fractions question?

Here is a partial fractions question from page 287, q22 of Schaum's Calculus 5e: $$ \int{ \frac{x^6 + 7\, x^5 + 15\, x^4 + 23\, x^2 + 25\, x - 3}{{\left(x^2 + 1\right)}^2\, {\left(x^2 + x + 2\right)}^...
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2answers
2k views

Proving and deriving a Gamma function

I'm having a hard time trying to prove this Gamma function and trying to derive the duplication formula: a.) Prove that $$\frac{\Gamma (p)\Gamma (p)}{\Gamma (2p)} = 2\int_0^{1/2}x^{p-1}(1-x)^{p-1}\...
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2answers
140 views

Sequence of functions (convergence)

Let be $f(x)=\frac{2x}{1+x} $ function and $ x_0 > 0 $. With the help of this, form the $x_{n+1}=f(x_n)$ sequence. Is $x_n$ convergent and if yes what is the limit? Thank you very much in advance!
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3answers
786 views

How can I sum the infinite series $\frac{1}{5} - \frac{1\cdot4}{5\cdot10} + \frac{1\cdot4\cdot7}{5\cdot10\cdot15} - \cdots\qquad$

How can I find the sum of the infinite series $$\frac{1}{5} - \frac{1\cdot4}{5\cdot10} + \frac{1\cdot4\cdot7}{5\cdot10\cdot15} - \cdots\qquad ?$$ My attempt at a solution - I saw that I could ...
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3answers
320 views

modified gamma integral

I have the following integral $$\int_0^{+\infty} t^{z-1} e^{-t} \frac1{(kt + 1)^s}\mathrm dt$$ where $k>0, s > 0$. How would you suggest to solve it? Without $\frac1{(kt + 1)^s}$ it would be ...
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4answers
2k views

Mean Value Theorem for a Multivariate Function $\mathbb{R}^2 \to \mathbb{R}$

I am reviewing masters exams and can't recall the multivariable calculus one needs to prove that this is true. A reference would suffice. Thank you! Suppose $x_1,x_2,x_3 \in \mathbb{R}^2$ are three ...
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1answer
1k views

Fourier transform of Schrödinger kernel: how to compute it?

Let $$K_t(x)=\frac{1}{(4 \pi i t)^{\frac{n}{2}}}e^{i \frac{\lvert x \rvert^2}{4t}}\quad x \in \mathbb{R}^n,\ t \in \mathbb{R},\ t\ne 0.$$ Clearly this is not a $L^1$ or $L^2$ function with respect ...
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1answer
2k views

Maximum Likelihood Estimator for Multivariate Bernoulli

I am working on deriving Naive Bayes for document classification. Each document is represented by a binary vector $x^i$ where $i=1,..,N$ for N documents. In this vector a cell is set to 1 if that ...
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3answers
8k views

Does the series $\sum\frac{\ln n}{n^{2}}$ Converge?

Does the series $$\sum_{n=2}^{\infty}\frac{\ln n}{n^2}$$ converge? I'm searching for a solution that does not use the Integral test, Stirling, L’Hôpital or functions theorems. I tried ratio test, ...
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1answer
199 views

Finding a derivative

I'm working through a book on relativity so this may end up being a physics question but I'm pretty sure that my problem is mathematical so I'm asking here. In deriving the "special Lorentz ...
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8answers
3k views

Simpler way to compute a definite integral without resorting to partial fractions?

I found the method of partial fractions very laborious to solve this definite integral : $$\int_0^\infty \frac{\sqrt[3]{x}}{1 + x^2}\,dx$$ Is there a simpler way to do this ?
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1answer
239 views

Integrating ray casting equation

In some ray casting algorithm I want to integrate the following integral: $\int_{0}^{Z} \frac{c^x}{\|(\vec{o}+x \vec{d})-\vec{l}\|^3} dx$ $Z$ is a constant until which I want to integrate $\vec{o}$ ...
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4answers
215 views

Prove that $a_{n+1}=3\frac{a_{n}+1}{a_{n}+3} , a_{1} =a> 0$ is convergent

The sequence $a_{n}$ is given in this recursive form: $$a_{n+1}=3\frac{a_{n}+1}{a_{n}+3} , a_{1} =a> 0$$ How does one show that it is convergent? I tried to prove that it is monotone ...
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2answers
118 views

Behavior of $2^{1−1/x}−x^{−1/x}−1$

I'm studying the function $f(x):=2^{1−1/x}−x^{−1/x}−1$ and want to show that there is a $n>0$ such that $f(x)>0 \, \; \forall x>n$. Do you have any suggestion? I tried to show that this ...
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1answer
217 views

Integration Order Reversal

I have a question regarding integration order reversal in a stochastic integral. This is a homework problem of the form "Show this is true". My problem is 1) my results are not exactly the same as the ...
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2answers
327 views

Integrating $\int_0^\infty \frac{1}{x^2 + 2x + 2} \mathrm{d} x$

I've been trying to integrate this: $$\int_0^\infty \frac{1}{x^2 + 2x + 2} \mathrm{d} x .$$ Unfortunately I haven't found a way so far. I've been trying to factor the denominator in order to end up ...
2
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2answers
238 views

Maximization of Two Areas — Calculus 1

"A 10-meter length of wire is available for making a circle and a square. How should the wire be distributed between the two shapes to maximize the sum of the enclosed areas?" Here's what I have: $$...
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1answer
2k views

What is the sum of the series $\sum\limits _{k=1}^\infty \frac{1}{k^2}$? [duplicate]

Possible Duplicate: Different methods to compute $\sum_{n=1}^\infty \frac{1}{n^2}$. What is $\lim \limits_{n\to\infty} \sum\limits_{k=1}^n \frac{1}{k^2}$ as an exact value?
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1answer
543 views

Resulting Solid Questions Help?

Q: A ball of radius 17 has a round hole of radius 8 drilled through its center. Find the volume of the resulting solid. A: 4500pi I am mostly confused about the question...If we have a bead-like ...
3
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1answer
271 views

Use a change of variables to evaluate a double integral

Use the change of variables $$x = u \quad y = \frac{v}{u}$$ to evaluate the double integral $$\iint \frac{x}{1+x^2y^2} \, \mathrm{dA}$$ I would like some direction as to how to solve this. Thank ...
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2answers
1k views

Finding the limit of $\frac{Q(n)}{P(n)}$ where $Q,P$ are polynomials

Suppose that $$Q(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{1}x+a_{0} $$and $$P(x)=b_{m}x^{m}+b_{m-1}x^{m-1}+\cdots+b_{1}x+b_{0}.$$ How do I find $$\lim_{x\rightarrow\infty}\frac{Q(x)}{P(x)}$$ and what ...
2
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1answer
273 views

Extended Lambert W function?

I need to solve something like this $$\exp(-a x) = \frac1{x^b}$$ where $a$ and $b$ are positive real values. Do other results exist when $b > 1$ or do I have to rely on numerical inspection?
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1answer
222 views

Show that $\limsup \frac1{x_n}\cdot \limsup x_n\geq 1$

Given a sequence $x_{n}$ and initial data that $0< a\leq x_{n}\leq b< \infty $, for $a,b\in \mathbb{R}$. I need to show that: $$\limsup \frac{1}{x_{n}}\cdot \limsup x_{n}\geq 1.$$ I think ...
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2answers
2k views

How to prove $\limsup (x_{n}+y_{n})=\lim x_{n}+\limsup y_{n}$?

Given a convergent sequence $x_{n}$ and bounded sequence $y_{n}$ I need to prove that $\limsup (x_{n}+y_{n})=\lim x_{n}+\limsup y_{n}$, when $n$ tends to $\infty$. I chose $z_{n}=x_{n}+y_{n}$, we ...
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4answers
161 views

A few things about about derivatives and integrals

I'd like to know the reason behind some notations used when handling derivatives and integrals. For example, why does $x' = 1 \frac{d}{dx}$ and not simply $1$? Related to this, why is integral of ...
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3answers
11k views

Relation between root of a function and its derivative

I am given the following function $f : \mathbb R \mapsto \mathbb R, f(x) = x^4 - 4x + p\ \ \ $ and am asked to find $p$ such that $f$ has two identical real roots. The proposed solution is to get ...
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1answer
179 views

What does the notation $A_a^n$ mean?

Given a set of matrices $$M = \left\{\begin{bmatrix}1&a\\0&3\end{bmatrix} \mid a \in \mathbb R\right\},$$ what does the notation $A_a^n, n \in \mathbb N$ mean?
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3answers
115 views

Tiny question about implicit differentiation

The following differentiated implicitly with respect to $\theta$: $3x = \tan \, \theta $ The book says $3 dx = \sec^2 \theta \, d \theta$ One could start the calculation like this (I think): $ \...
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6answers
4k views

Prove $\sin(\pi/2)=1$ using Taylor series

Prove $\sin(\pi/2)=1$ using the Taylor series definition of $\sin x$, $$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots$$ It seems rather messy to substitute in $\pi/2$ for $x$. So we have $$\sin(\...
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1answer
187 views

Finding derivative in an implicit function

I searched and couldn't find anything specific to my question, so I'll ask it here. I'm asked to find the indicated derivative: $${\operatorname{d}y\over\operatorname{d}x} \sin(xy^2)-x^2 = x+5$$ ...
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1answer
97 views

Lipschitz contradiction

Assume that $\phi:\mathbb{R}^n\rightarrow \mathbb{R}^n$ is a smooth vector field, and assume that we can find vectors $y_k,x_k$ ($k$ positive integer) such that $(\phi(x_k)-\phi(y_k),x_k-y_k)\geq k \...
1
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1answer
88 views

an estimate for derivative

let $F$ a closed convex subset of $\mathbb{R}^n$, let $x,y\in F$ and assume that for any $s\in[0,1]$ we have $f(s):=\mid sx+(1-s)y-z\mid\geq \mid y-z\mid$ why is it true that $\frac{\partial}{\...
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3answers
212 views

For continuous and bounded $f(x,y)$ prove $g(x)=\sup_y f(x,y)$ is continuous too

Suppose $f(x,y)$ is continuous and bounded. Please prove $g(x)=\sup_y f(x,y)$ is continuous too.
3
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1answer
344 views

References for the basic theory of surfaces of revolution, cylinders and cones

I'm looking for references to books were the following types of problems about finding the equation defining a surface of revolution, a cylinder or a cone are treated. These are problems that are ...
1
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1answer
206 views

Solution for $x = -c_1 e ^ x + c_2 e ^{-x}$

How can I solve this equation, $$x = -c_1 e ^ x + c_2e ^{-x}, \;\;\; 0 < c_1, c_2 < 1$$ We can use $t = e^x$ which will result in, $$t \ln(t) + c_1 t ^ 2 - c_2 = 0, \;\;\; 0 < c_1, c_2 < 1$...
7
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3answers
863 views

Is this a justified expression for the integral of the floor function?

Mathematica seems to agree with me in general with saying that $\displaystyle\int \lfloor x \rfloor dx = \frac{\lfloor x\rfloor (\lfloor x\rfloor-1)}{2}+\lfloor x\rfloor \{ x \}+C = \frac{\lfloor x\...
11
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2answers
6k views

Integral of floor function: $\int \,\left\lfloor\frac{1}{x}\right\rfloor\, dx$

How would you go about solving integral of a floor? The particular problem I have is: $$\int \,\left\lfloor\frac{1}{x}\right\rfloor\, dx$$
3
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1answer
104 views

Trigonometric re-write I don't understand

$$\int \sin^3{x}\,\cos^5{x}\,dx = \int \sin{x}\,(\cos^5{x}-\cos^7{x})\,dx$$ My ignorance amuses me hehe. Even if I multiply it out I still don't get it.
1
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1answer
113 views

How can this be re-written with the following identity?

Can this: $$\frac{\cos x}{4 + \sin^2 x}$$ Be re-written using the fact that: $$\cot(t) = \frac{\cos (t)}{\sin (t)} = \frac{1}{\tan (t)}$$ I'm not good with algebra, but I'm getting there. I'm ...
18
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3answers
13k views

How to integrate $\int\frac{1}{\sqrt{1+x^3}}\mathrm dx$?

In a course, my teacher told us that the following integral is convergent and used the comparison test to prove it; my question is how to find the antiderivative in closed form? It seems to exist; if, ...