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

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1answer
36 views

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 $\lim\limits_{x\to0} P(n)/ x$ ...
1
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1answer
19 views

$\mathcal{T_B}$ 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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0answers
22 views

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 \begin{equation} ...
-2
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0answers
24 views

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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0answers
6 views

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 ...
2
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2answers
50 views

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 ...
0
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2answers
45 views

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 ...
1
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3answers
161 views

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 ...
0
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0answers
31 views

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 ...
1
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1answer
55 views

Show that the series converges uniformly

Show that $\sum_{n=1}^{\infty} \sin \left(\dfrac{x}{n}\right)^2$ converges uniformly on $[a,-a]$ , $a\in \mathbb{R}$. My attempt: Using Taylor's formula, we have:$$ \sin\left(\dfrac{x}{n}\right)^2 ...
0
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1answer
22 views

Help Computing integral of continuous function

Consider the Dirichlet function $F:R \rightarrow R$ given by $f(x):=\begin{cases} x &\text{if } x < 1 , \\{}\\ x+1 &\text{if } 1<= x <= 2, \\{}\\ -x+5 &\text{if } 2<x ...
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3answers
74 views

Evaluating the inverse trigonometric limit $\lim_{x \to \frac{1}{\sqrt{2}}} \frac{\arccos \left(2x\sqrt{1-x^2} \right)}{x-\frac{1}{\sqrt{2}}}$

$$ \lim_{x \to \frac{1}{\sqrt{2}}} \frac{\arccos \left(2x\sqrt{1-x^2} \right)}{x-\frac{1}{\sqrt{2}}} $$ I was doing some questions on limits, I saw one in which there is $\arccos x$. I am stuck ...
0
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0answers
27 views

problem solving these differential equations

$$(\sin^2(x) D^2 +\sin(2x) D+\cos^2 x+1)y=\sin^3 x$$ and there is another $$(xD^2-x(x+2)D+(x+2))y=x^3-2x+1$$
0
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0answers
20 views

Intuition for second frechet derivative

I am now used to thinking of the first derivative of a map between vector spaces $f:V\to W$ in the "proper" Frechet sense, as being "the assignment to each point $v$ of $V$ of the linear map ...
2
votes
3answers
40 views

Changing Sides of Limits Property

Is the following property about limits correct?$$\lim_{x+b\to a}\,f(x)=\lim_{x\to a-b}\,f(x)$$ and does it also mean $$\lim_{x\to a}\,f(x)=\lim_{nx\to na}\,f(x)=\lim_{x^2\to a^2}\,f(x)=\lim_{g(x)\to ...
1
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0answers
60 views

$f \in C^2(\mathbb R)$ , $(f(x))^2 \le 1$ ; $(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 ...
0
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3answers
28 views

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) = $x^9 - 6x^8 -2x^7 + 12x^6 +x^4 -7x^3 + 6x^2 + x-7$ ...
0
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3answers
56 views

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 ...
0
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1answer
30 views

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 ...
0
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2answers
70 views

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).
0
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1answer
25 views

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 ...
2
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3answers
97 views

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 ...
1
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1answer
39 views

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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0answers
44 views

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 ...
0
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1answer
58 views

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 ...
1
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1answer
41 views

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 ...
0
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1answer
32 views

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 ...
1
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0answers
27 views

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 $$\frac{d}{da}\int_0^x{\sin{(a \cdot t)}\ dt}=\ ?$$
0
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2answers
45 views

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 \begin{equation} \int_0^1 \frac{\ln \left(\frac{x^2}{2}-x+1\right)}{x} dx \end{equation} ...
2
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2answers
65 views

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?
8
votes
1answer
72 views

$ \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?
1
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1answer
32 views

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 ...
1
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0answers
35 views

Clarification of meaning of dx in an integral [duplicate]

I would like to have some clarification on the physical meaning of $dx$. I already know the following in the context of the area under the curve: $\lim_{\Delta x \rightarrow 0} \sum f(x) \Delta x ...
1
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1answer
53 views

Find the derivative of $\left(\frac{4x+2}{x-2}\right)^5$

Hey helpful people I have one more question I am stuck on! $$f(x) = \left(\frac{4x+2}{x-2}\right)^5$$ I know the answer is $$\frac{-50(4x+2)^4}{(x-2)^2(x-2)^4}$$ But I really can't figure out how ...
0
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0answers
17 views

A inequality concerns with the Legendre polynomial of n-th degree of $\cos\theta$

I am reading a paper, where the author concluded that $${\left| {{P_n}(\cos \theta )} \right|^2} \leqslant \frac{2} {{n\pi \sin \theta }},\,\,\forall \theta \in \left( {0,\pi } \right).$$ Here ...
0
votes
1answer
35 views

The bases for the set of all functions f:[0,1]→[0,1]

Let $X = [0, 1]^{[0,1]}$, the set of all functions $f : [0, 1] \rightarrow [0, 1]$. Given a subset $A \subseteq [0, 1]$, let $U_A = \{ f \in X : f(x) = 0 \forall x \in A \}$ . Show that $B := \{U_A : ...
0
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2answers
39 views

What will the value of an account be after $12$ years if the account earns $4.91\%$ a year and if someone invests $\$20,000$?

Second National Bank offers an account that earns $4.91\%$ per year, compounded continuously. If a person invests $\$20,000$ in this account, what will be the value of the account at the end of $12$ ...
4
votes
2answers
51 views

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 $\alpha = \beta > 2$. ...
1
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3answers
50 views

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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0answers
18 views

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 ...
2
votes
4answers
158 views

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.
6
votes
3answers
119 views

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.
0
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0answers
41 views

Convergence of $f_n(x)= \frac{x^n}{1+x^n}$

I have a sequence of functions: $$f_n(x) = \frac{x^n}{1+x^n}$$ I use ratio test, for finding convergence: $$\Large \lim \frac{\frac{x^{n+1}}{1+x^{n+1}}}{\frac{x^n}{1+x^n}}=\lim ...
-1
votes
0answers
50 views

How should I calculate $\int_a^b x \cdot \sin x \cdot \sqrt{1-x^2}\, dx$? [closed]

Could anyone please show me how to calculate $$ \int_a^b x \cdot \sin x \cdot \sqrt{1-x^2}\ dx? $$
4
votes
3answers
132 views

If $f(\frac{x+y}{2})=\frac{f(x)+f(y)}{2}$, then find $f(2)$

Let $f(\frac{x+y}{2})=\frac{f(x)+f(y)}{2}$ and it is given that $f(0)=1$ and $f'(0)=-1$, where $f'$ denotes first derivative. Find the value of $f(2)$ Could someone tell me how to use $f'(0)=-1$ ...
1
vote
1answer
24 views

Where does this come from? or how do I derive it?

$$\delta \vec{x} = \frac{\partial\vec{x}}{\partial r}\delta r + \frac{\partial\vec{x}}{\partial \theta}\delta \theta+\frac{\partial\vec{x}}{\partial \phi}\delta \phi$$
0
votes
1answer
44 views

Does the integral converge

How to prove that the following integral doesn't converge? $$\int_0^\infty \frac{1}{(\ln^4x + \ln^2x)\ln^2(1-x^{1/3})^2(x + \sqrt{x} + 1)}dx$$ I suppose it doesn't converge because of quick growth ...
11
votes
1answer
101 views

Integral with arithmetic-geometric mean ${\large\int}_0^1\frac{x^z}{\operatorname{agm}(1,\,x)}dx$

The arithmetic-geometric mean$^{[1]}$$\!^{[2]}$ of positive numbers $a$ and $b$ is denoted $\operatorname{agm}(a,b)$ and defined as follows: $$\text{Let}\quad a_0=a,\quad b_0=b,\quad ...
0
votes
4answers
37 views

Find particular solution to nonhomogeneous DE $y'+y=x^2+\sin{x}+\cos{x}$

I'm new to nonhomogeneous DE's and I have come across this DE: $$y'+y=x^2+\sin{x}+\cos{x}$$ which I'm supposed to provide a general solution to. However, I get stuck with the particular solution. The ...
0
votes
3answers
40 views

Is there any way to calculate the roots of this polynom?

I need to calculate the roots of the real function $f$: $$ f(x)=\frac{-{x}^{3}+2{x}^{2}+4}{{x}^{2}} $$ But I am not able to decompose the numerator. There should be only one real solution and two ...