# Tagged Questions

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

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### evaluating limit of two functions

Let $P(n) = a^{P(n-1)}-1$ such that for all $n = 2 ,3 ,$ and so on. And let $P(1) = a^x -1$ where a belongs to all real positive numbers, then we have to evaluate $\lim\limits_{x\to0} P(n)/ x$ ...
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### $\mathcal{T_B}$ is the intersection of all topologies containing $\mathcal{B}$

Let $\mathcal{B}$ be a basis on a set X, and let $\mathcal{T_B}$ be the topology it generates. Show that $\mathcal{T_B} =\bigcap \{ \mathcal{T} \subseteq P(X) \mathcal{T}$ is a topology on X and ...
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### Where can I find the demonstration of $\sum_{j=0}^J \Delta x f(x_j) \to \int_0^1 f(x) dx$?

Let $x \in \left[0, 1\right]$. Define a partition $x_j = j \Delta x$ with $j=1,\ldots, J$ and $J\Delta x = 1$. Then, $\Delta x = \frac{1}{J}$. I can't find the demonstration of ...
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### How to find directional derivative of f at given p? [closed]

Question: Compute the differential of $$(f \circ g )(a,b)=\left(\cos\left(\frac{2 ab}{\pi}\right),\sin\left(\frac{2 ab}{\pi}\right),\cos\left(\frac{2 ab}{\pi}\right)\right)$$ Could anybody tell ...
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### Conditional extremes, solving $xa+yb < (x^p+y^p)^{\frac{1}{p}}(x^q+y^q)^{\frac{1}{q}}$ if.

Conditional extremes, solving $$xa+yb \leq (x^p+y^p)^{\frac{1}{p}}(x^q+y^q)^{\frac{1}{q}}$$ using lagrange multipliers.. If $\frac{1}{q}+\frac{1}{p}=1$ and $p,q>1$. This reminds me of Holders ...
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### Definite integration of an algebraic expression

Evaluate $$\int_{0}^{1}\frac{1-x}{1+x}\frac{dx}{\sqrt{x+x^2+x^3}}$$ I think none of the properties of definite integral will be useful here so I think I will have to integrate. But I am unable to ...
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### Domain of Integral $\int_{5}^{x} \frac {dt}{(1-t^2)}$

A function reads $$F(x) = \int_{5}^{x} \frac {dt}{(1-t^2)}$$ Barrons says that the domain of F must be that $x >1$. But why can't $x$ be less than $1$ as well? As long as $x$ does not equal ...
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### Where am i going wrong in solving this equation?

Fing the least value of $a$ for which $f(x)$ is increasing, where $$f(x)=2e^x-ae^{-x}+(2a+1)x-3$$ What i tried for increasing $f'(x)\ge 0, \forall x\in \mathbb R$. So ...
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### Need help integrating $\frac{1}{160}\log \left(5x-25\right)\left(\left(y-146\right)^2+\left(x-7\right)^2\right)=10$

The project is quite simple that I am making much more difficult because it is fun to make things difficult. Create a single function for a water tower design with a narrow part and reservoir at least ...
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### $f \in C^2(\mathbb R)$ , $(f(x))^2 \le 1$ ; $(f'(x))^2+(f''(x))^2 \le 1$ ; then is $(f(x))^2+(f'(x))^2 \le 1$?

Let $f \in C^2(\mathbb R)$ be such that $$(f(x))^2 \le 1 ; (f'(x))^2+(f''(x))^2 \le 1 , \forall x \in \mathbb R$$ Then is it true that $(f(x))^2+(f'(x))^2 \le 1 , \forall x \in \mathbb R$ ? I ...
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### Continous function

There are two functions g(x)= ($(2x+1)^{1/2}$-$1$)/x , where x is not equal to zero = 1 , x=0 h (x) = $x^9 - 6x^8 -2x^7 + 12x^6 +x^4 -7x^3 + 6x^2 + x-7$ ...
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### Find the maximum and minimum values of $x-\sin2x+\frac{1}{3}\sin 3x$ in $[-\pi,\pi]$

Find the maximum and minimum values of $x-\sin2x+\frac{1}{3}\sin 3x$ in $[-\pi,\pi]$. Let $f(x)=x-\sin2x+\frac{1}{3}\sin 3x$ $f'(x)=1-2\cos2x+\cos3x$ Put $f'(x)=0$ $1-2\cos2x+\cos3x=0$ gives ...
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### Convolution of Gaussian and error function

I am trying to evaluate the following integral: $$\int_{-\infty}^{\infty}e^{-\frac{x^2}{2}}\Phi(x-t)dx$$ where $$\Phi(y) = \frac{1}{2} + \frac{1}{2}erf\left(\frac{y}{\sqrt{2}}\right)$$ I have ...
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### The limit as x approaches 0 of xsin(3/x). [closed]

Could someone show me how to get to the answer for the question: The limit as x approaches 0 of xsin(3/x).
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### Unable to choose functions for evaluating a limit using the Squeeze Theorem

Evaluate $$\lim_{n\to \infty}\dfrac{1}{\sqrt{n^2}}+\dfrac{1}{\sqrt{n^2+1}}+...+\dfrac{1}{\sqrt{n^2+2n}}$$  I'm supposed to solve this problem using the Squeeze Theorem. I had selected the ...
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### Applying the chain rule to compute $\frac{d}{dx}(\cos^6 x)$

$$\frac{d}{dx}(\cos^6x)$$ Using the chain rule $M'(N(x)).N'(x)$, I'm deconstructing the $\cos$ function \begin{align*} &M= \cos^6 \\ &N= x\end{align*} End result should be ...
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### Evaluating the arc length integral $\int\sqrt{1+\frac{x^4-8x^2+16}{16x^2}} dx$

Find length of the arc from $2$ to $8$ of $$y = \frac18(x^2-8 \ln x)$$ First I find the derivative, which is equal to $$\frac{x^2-4}{4x} .$$ Plug it into the arc length formula ...
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### Dual of a maximization problem

We have a positive, smooth, increasing concave function $f:\mathbf{R}^n\to \mathbf{R}^+$ and $k$ smooth, increasing constraint functions $f_i:\mathbf{R}^n\to\mathbf{R}$. I've recently encountered two ...
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### Integral of $\frac{1}{(x^2+2)^3}$

Ive been struggling to find the integral $\frac{1}{(x^2+2)^3}$ by using the integral $I_n=\frac{1}{(x^2+1)^n}$. (assume I know how to solve $I_n$ by a recursive way. Ive tried to make it to the form ...
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### Approximation of $\sin\left(\dfrac{x}{n}\right)$

I wrote in my analysis notes the following: $\sin\left(\dfrac{x}{n}\right) = -\dfrac{x}{n} + \omicron\left(\left| \dfrac{x}{n} \right|\right)$. I'm guessing it comes from Taylor's formula but I ...
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### All real values of $k$ in rational function

All real values of $k$ for which the range of function $\displaystyle f(x) = \frac{x-1}{k-x^2+1}$ does not contain the interval $\displaystyle \left[-1,-\frac{1}{3}\right].$ $\bf{}My\; Try::$ Let ...
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### Differentiation of an integral in regards to different variables

It is known by the second fundamental that $$\frac{d}{dx}\int_0^x{\sin{(a \cdot t)}\ dt}=\sin{(a \cdot x)}$$ But what can we say about $$\frac{d}{da}\int_0^x{\sin{(a \cdot t)}\ dt}=\ ?$$
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### Complex logarithms when computing real-valued integral

My question arise when I try to calculate real-valued integral, specifically, I want to evaluate the integral $$\int_0^1 \frac{\ln \left(\frac{x^2}{2}-x+1\right)}{x} dx$$ ...
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### I have to find $I=\int_{0}^{\pi}\ln(1-2a \cos x+a^2)\, dx$ [duplicate]

I have to find $$I=\int_{0}^{\pi}\ln(1-2a \cos x+a^2)\,dx$$ Can someone help me to solve it?
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### $\int_0^\infty \ \frac{(x\cdot\cos x - \sin x)^3}{x^6} \ dx$

What is the value of $$\int_0^\infty \ \frac{(x\cdot\cos x - \sin x)^3}{x^6} \ dx$$ I have no idea how to start with this integral, any hint?
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### How to read $\frac{dy}{dx}$ when the term is only given?

When the term $\frac{dy}{dx}$ (not $\frac{d}{dx}y$) is only given, how to read the term between "the derivative $y$ with respect to $x$" and "the quotient of the differential $dy$ by the differential ...
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### For which values of real $\alpha, \beta$ does $\sum_{n,m \ge 1} \frac{1}{n^{\alpha}+ m^{\beta}}$ converge?

I was wondering how does the series $$\sum_{n,m \ge 1} \frac{1}{n^{\alpha}+ m^{\beta}}$$ behave for real $\alpha, \beta > 0$. My approach: firstly I considered the case $\alpha = \beta > 2$. ...
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### Find the derivative of $\frac{3}{x} - \frac{x}{2}$

I must find the derivative for: $\frac{3}{x} - \frac{x}{2}$ I know the answer is$\frac{-3}{x^2} - \frac{1}{2}$ But I can't figure out why the 3 is negative and where the 1/2 came from Any help ...
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### Mean curvature submanifold

Consider $S^{N-1}$ the unit sphere and let us focus our attention on the cap $$G=S^{N-1}\cap\{x_N>0\}$$ with boundary $\partial G= S^{N-2}\times\{0\}$: it is quite obvious to see that $G$ is a ...
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### How to solve $f'(x)=f'(\frac{x}{2})$

How do we solve this given $f'(0)=-1$. It does not look separable. I can integrate both sides but end up with a functional equation with is not helpful.
### Integrate $\int_0^\infty \frac{e^{-x/\sqrt3}-e^{-x/\sqrt2}}{x}\,\mathrm{d}x$
I can't solve the integral $$\int_0^\infty \frac{e^{-x/\sqrt3}-e^{-x/\sqrt2}}{x}\,\mathrm{d}x$$ I tried it by using Beta and Gamma function and integration by parts. Please help me to solve it.