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

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1
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1answer
45 views

evaluating a sum using Cauchy condensation test

Let $$\sum\limits_{n\ge1}{\frac{(-1)^n}{n^\alpha \ln n}}$$ I want to check if the sum is converges absolutely. Hence, we need to check the convergence of $$\sum\limits_{n\ge1}{\frac{1}{n^\alpha \ln ...
1
vote
1answer
61 views

Limit of factorial how to continue

$$\lim_{n\to\infty}\left(\dfrac{(n+1)^{n+1}\cdot n!}{2^{(n+1)!-n!}}\right)=\lim_{n\to\infty}\left(\dfrac{(n+1)^{n+1}\cdot n!}{2^{n!\cdot n}}\right).$$ How to continue? the answer is $0$ ... thank you ...
0
votes
2answers
21 views

Problem about moving sides of triangle

Imagine a triangle XOY which sides lie on x-axis and y-axis with hypotenuse XY of length 5 m. Suppose the point X moves away from the (0,0) along x-axis with speed = 1 m per second. What speed the ...
1
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3answers
113 views

Evaluate $\int \frac{1}{(2x+1)\sqrt {x^2+7}}dx$

How to do this indefinite integral (anti-derivative)? $$I=\displaystyle\int \dfrac{1}{(2x+1)\sqrt {x^2+7}}dx$$ I tried doing some substitutions ($x^2+7=t^2$, $2x+1=t$, etc.) but it didn't work out.
2
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1answer
44 views

$\lim_{n \rightarrow \infty} \frac{3}{n} \sum_{k=1}^n \left(\frac{2n+3k}{n} \right)^2$

How to solve this: $$\lim_{n \rightarrow \infty} \frac{3}{n} \sum_{k=1}^n \left(\frac{2n+3k}{n} \right)^2$$ The answer is supposed to be 39. My attempt: ...
2
votes
0answers
1k views

The negative integral meaning

Whenever I take a definite integral in aim to calculate the area bound between two functions, what is the meaning of a negative result? Does it simly mean that the said area is under the the x - axis, ...
3
votes
2answers
1k views

Frenet-Serret formula proof

Prove that $$\textbf{r}''' = [s'''-\kappa^2(s')^3]\textbf{ T } + [3\kappa s's''+\kappa'(s')^2]\textbf{ N }+\kappa \hspace{1mm}\tau (s')^3\textbf{B}.$$ What is $\tau$, I can't figure that part out. ...
0
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2answers
35 views

Equivilent first order differential and initial condition?

I have another homework question that I'm struggling a bit to understand exactly what I'm asked to do. I understand what an initial condition is, but I'm not quite sure how I specify such a condition. ...
1
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0answers
33 views

Check: Find $h(x)$ when $\hat{h}(y)=\frac{1}{(1+y^2)^2}$.

Find h(x) when $\hat{h}(y)=\frac{1}{(1+y^2)^2}$. $$\hat{h}(y)=\frac{1}{(1+y^2)^2} =\frac{1}{1+y^2} \times \frac{1}{1+y^2} =\hat{\frac{1}{2}e^{-|x|}} \times \hat{\frac{1}{2}e^{-|x|}}$$ Let ...
3
votes
1answer
274 views

Volume of a solid(between two planes)?

A solid lies between planes perpendicular to the y-axis at $ y=0$ and $y=1$. The cross-sections perpendicular to the y-axis are circular disks with diameters running from the y-axis to the parabola ...
1
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1answer
94 views

Is $\hat S$ a function?

Let $S(x)=\sum_{n=-\infty}^\infty (-1)^n\chi_{(n,n+1)}(x)$. Find the Fourier transformation of $S(x)$. Is $\hat{S}$ a function? $$\hat{S}=\int_R \sum_{n=-\infty}^\infty (-1)^n\chi_{(n,n+1)}(x) ...
1
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1answer
42 views

Length of a curve?

I know how to find arc length and set up the equation in normal circumstances, but I have failed in all attempts to even set up this problem. I cannot even find a good example similar to this to get ...
2
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2answers
92 views

Shell method to find the volume of a solid?

Region bounded by $y=3x-2$, $y=\sqrt{x}$, and $x=0$ about the $y$-axis. I have been doing the washer method for all of my problems up to this one, and cannot seem to find a good resource to help guide ...
0
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4answers
83 views

Prove: For any real number a,b : |a - b| >= |a| -|b| [duplicate]

I think this question has to do with field axioms so was wondering if you can claim that |a - b| - |a| - |b| >= 0 and go from there.
5
votes
5answers
791 views

Am I allowed to apply L'Hospital's Rule inside of the natural logarithm function?

I have the following limit: $$\lim_{x\rightarrow \infty} \ln\left(\frac{2x^2+1}{x^2+1}\right)$$ If I was finding the limit of only the terms inside the natural log function, I would have the ...
1
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1answer
475 views

Parametric equation of a circle given starting point.

Find the parametric equations of a circle with radius of $5$ where you start at point $(5,0)$ at $v=0$ and you travel clockwise with a period of $3$. So, I know that I require to have a $x(v)$ and ...
1
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0answers
66 views

please help me completing this proof (Lagrange remainder for Taylor formula)

I'm trying to prove that the remainder of a $n$-th grade Taylor formula is $$R_n=\frac{f^{(n+1)}(\mu) (x-x_0)^{n+1}}{ (n+1)!}$$ where $\mu$ is a value between $x$ and the centre $x_0$. For $n=1$ it ...
1
vote
2answers
96 views

Calculating appearance of size of object at given distance

Here's the problem I'd like to solve. If I'm 1 ft away from a computer screen and a word on the screen appears a certain size, is there an equation or calculation that will tell me how big that ...
6
votes
1answer
288 views

Where to learn integration techniques?

Is there any book or any website that let you learn integration techniques? I'm not talking about the standard ones like integration by Parts Substitution (trigonometric) Partial fractions Order ...
5
votes
1answer
216 views

Hard integral, low hints… [duplicate]

$$\int_{ - \pi /2}^{\pi /2} \frac1{2007^{x} + 1}\cdot \frac {\sin^{2008}x}{\sin^{2008}x + \cos^{2008}x} \, dx .$$ This integral stuns me for a while, I just can't solve it! I tried integration by ...
5
votes
2answers
115 views

Is $\int^x \cos \frac1t$ differentiable at zero?

From Spivak's Calculus, 4th ed., exc 14-20: Let $$f(x) = \begin{cases} \cos \frac1x, & x\neq 0\\ 0, &x=0. \end{cases}$$ Is the function $\int_0^xf$ differentiable at zero? I'm having ...
1
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2answers
59 views

Differentiate $y = |x|(5 − x^2)^.5$

The curve: $$y = |x|(5 − x^2)^.5$$ is called a bullet-nose curve. Find an equation of the tangent line to this curve at the point $(2, 2)$. How would I differentiate the absolute value?
-1
votes
1answer
64 views

Nearest and farthest point from a function to another [closed]

Find the nearest and farthest point from the ellipse $ x^2 + 3y^2 =3 $ to the segment made by $ x+y = 3 $ in the first quadrant. Found in a multivariable calculus course. So I have to find the ...
1
vote
0answers
70 views

second order ODE :- solution

We have $y''-Py'-Qy = 0 $ where P,Q are $P = K_1+K_2x, Q =K_2 $. $K_1,K_2$ are constants. y' means derivative with respect to x . Please suggest a solution for y. Thanks
0
votes
1answer
81 views

Show that $\int_{\pi/4}^{\pi/2} \frac{\sin x}{x}\,dx\leq \frac{\sqrt{2}}{2}$

Show that $$\int_{\pi/4}^{\pi/2} \dfrac{\sin x}{x}\,dx\leq \dfrac{\sqrt{2}}{2}$$ Any Ideas, how to start ?!
6
votes
8answers
350 views

Evaluate: $\lim_{x \to \infty} \,\, \sqrt[3]{x^3-1} - x - 2$

Find the following limit $$\lim_{x \to \infty} \,\, \sqrt[3]{x^3-1} - x - 2$$ How do I find this limit? If I had to guess I'd say it converges to $-2$ but the usual things like L'Hôpital or clever ...
0
votes
2answers
60 views

Find the Intervals when $\cos\left(\frac{1}{2}\pi x^2\right)$ is positive.

I am looking for a general solution, not in any restricted domain.
1
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1answer
27 views

Normal line to a curve $C_1$

Find the interval for $a$ so that $(3-a)x+ay+(a^2-1)=0$ is normal to the curve $xy=4$ $(C_1)$. I approached it this way-- $C_1$ is $xy=4$. So, $\dfrac{dy}{dx}$ for $C_1$ is $\dfrac{-4}{x^2}$. ...
0
votes
1answer
25 views

Calculating optimum values of $u$ and $m$ from $\mathbb V(\bar {y_2}\prime)=\frac{S_2^2(n-u\rho^2)}{n^2-u^2\rho^2}$

I have to find optimum sample size in sampling on two occasions. Suppose that the samples are of the same size n on both occasions. In selecting the second sample, $m$ of the units in the first ...
0
votes
4answers
86 views

How to find this limit of $\tan(n\pi)/(n-1)$?

I tried to find this limit $$\lim_{n\rightarrow1}\frac{\tan(n\pi)}{n-1}$$ but I couldn't. Here's what I tried $$ \begin{align} \lim_{n\rightarrow1}\frac{\tan(n\pi)}{n-1} &= ...
1
vote
1answer
23 views

When $f(t)=ab(1-e^{-t})-c(e-e^{-t})>0$

Consider the function $f(t)=ab(1-e^{-t})-c(e-e^{-t})$, for $t>0$, where $a,b,c>0$. Can we find a sufficient condition on $a,b,c$ such that $f(t)>0$ for all $t>0$ ?
2
votes
4answers
156 views

What is the integral of x/ln(x)?

Well, I'm french so excuse me if I make some mistakes in english... I have to calculate this integral : $$ \int_{e}^{2e} \frac{x}{\ln(x)} dx $$ But I don't know how, can you help me please? Thank ...
5
votes
1answer
79 views

Maximum value problem

A function $\hspace{0.1cm}$$f:[0,1]\to[-1,1]$$\hspace{0.1cm}$ satisfying$\hspace{0.1cm}$ $|f(x)|\leq x$$\hspace{0.1cm}$ $\forall x\in[0,1]$. Then find the maximum value of: ...
1
vote
3answers
72 views

How to check the function is convex or not?

How one can check the function whether it is convex or not. I know one method by using Hessian Matrix but I think it did not fit for the following example. I think Hessian matrix method cannot be ...
3
votes
5answers
121 views

Taylor expansion of $\frac{1}{1+x^{2}}$ at $0.$

I am trying to find the Taylor expansion for the function $$f(x) = \frac{1}{1+x^{2}}$$ at $a=0.$ I have looked up the Taylor expansion and concluded that it would be sufficient to show that ...
1
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2answers
166 views

What is the probability of going bankrupt in roulette?

Imagine that the bank has the money $M_1$ and the player has the money $M_2$. The rules are the following: You win with a chance of $\frac{17}{36}$ and lose with $\frac{19}{36}$ each round. Now you ...
11
votes
2answers
459 views

Integral $\int_1^{\sqrt{2}}\frac{1}{x}\ln\left(\frac{2-2x^2+x^4}{2x-2x^2+x^3}\right)dx$

Calculate the following integral: \begin{equation} \int_1^{\sqrt{2}}\frac{1}{x}\ln\left(\frac{2-2x^2+x^4}{2x-2x^2+x^3}\right)dx \end{equation} I am having trouble to calculate the integral. I ...
4
votes
4answers
136 views

If $f(x)$ is discontinuous at $x=0$, can $\int_{-1}^1 f(x)dx$ exist.

I am interested in the reasoning. All help is appreciated
-1
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2answers
84 views

Prove that $\displaystyle\int_a^b f(x)dx = \int_a^c f(x)dx + \int_c^b f(x)dx$

Prove that $\displaystyle\int_a^b f(x)dx = \int_a^c f(x)dx + \int_c^b f(x)dx$ Note that $c$ need not belong to $(a, b)$ And $f(x)$ is a continuous function. All ideas are appreciated.
0
votes
1answer
55 views

$L^\infty(\Omega)$ space

Consider Lebesgue spaces $L^p(\Omega)$, $\Omega$ is a bounded domain. Let $f \in L^p(\Omega)$ for all $p$. Is it true that $f \in L^\infty(\Omega)$?
1
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1answer
31 views

Double integral calculation where $x=(y-1)^{2}-1$ and $y=x$. Not sure whether I should do it in terms of $y$ or $x$?

This is what it looks like: My first strategy was to separate it into two by drawing a vertical line at x=0 and calculate the first half in terms of x first, and the second half in terms of y ...
12
votes
3answers
618 views

Another integral for $\pi$

Here is a new integral for $\pi$. $$\int_{0}^{1}\sqrt{\frac{\left\{1/x\right\}}{1-\left\{1/x\right\}}}\, \frac{\mathrm{d}x}{1-x} = \pi $$ where $\left\{x\right\}$ denotes the fractional part of ...
2
votes
0answers
42 views

Find all differentialbles function [closed]

Find all differentialbles function $f:[0,\infty)\rightarrow\mathbb R$ such that: a) $f^{\prime}$ is non-decreasing; b) $x^{2}f^{\prime}(x)=f^{2}(f(x)),~\forall x\in\lbrack0,\infty)$
0
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2answers
31 views

Find power series representation of $ x/(x^{2}+9)^{2}$

I'm not sure how to do it since the entire bottom term is squared. Is there a geometric series I should use? Or differentiation?
1
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1answer
77 views

Double integration. Do I have to integrate $(1+x^{3})^{1/2}$? (which I thought was impossible)

Or is there a theorem(or method) that will allow me to avoid that?
3
votes
2answers
89 views

Composition of functions is constant in $\mathbb{R^2}$.

Let $\hspace{0.05cm}f:\mathbb{R^2}\to\mathbb{R^2}$ $\hspace{0.05cm}$ and $\hspace{0.05cm}$$g:\mathbb{R^2}\to\mathbb{R^2}$ $\hspace{0.05cm}$ be such that$\hspace{0.05cm}$ $g\circ f$$\hspace{0.05cm}$ ...
0
votes
2answers
80 views

Prove for a monotone function, if $x_0$ is interior on interval $I$ then the one-sided limits exists

The proof goes like this: Lets suppose $f$ is nondecreasing (for nonincreasing we'll observe the function $-f(x)$). Let $x_0$ an interior point in the interval $I$, and $\left\{x_k\right\}$ an ...
5
votes
3answers
197 views

Examples of “difficult” integrals with are easier to solve with a series?

Yesterday someone posted an extremely elegant solution to a seemingly bizarre series where the integral: $$\int_{0}^{1} x^{m}\ dx = \frac{1}{m + 1}$$ was utilized. Oftentimes one will also ...
1
vote
5answers
244 views

Solve $e^x-1= 2x$ with numerical or analytical methods. [closed]

Find the value of x for which \begin{equation} e^x-1= 2x \end{equation} Some numerical or analytical methods are appreciated, thanks. Using graphic inspection, we have that We can observe that ...
1
vote
1answer
60 views

Convergence of a subsequence .

If every subsequence of $x_n$ has a further subsequence which converges , is it true that the sequence is convergent? NOTE : This is not a duplicate ofthis . In this problem it is not given that the ...