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

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0
votes
1answer
23 views

Trigonometric equation

I have the equation $B\sin(ka)=0$. I must find what value has ($ka$). I was thinking about the trigonometric formula $\sin x=a$ which includes $x=(-1)^k \arcsin(x)+k\pi$. My final result must be: ...
1
vote
1answer
22 views

Area surrounded by a curve

I would need help to calculate the area surrounded by a curve. The curve is given with the following polar coordinates: I know we need need to integrate with respect to r and theta but am stuck ...
1
vote
0answers
28 views

Counting the sum $\sum^{\infty}_{k=0} q^{k^{2}}$

Is it possible to obtain explicit form of the sum $\sum^{\infty}_{k=0} q^{k^{2}}$ (without using elliptic functions)? It is well known that $\sum^{\infty}_{k=0} q^{k} = \frac{1}{1-q}$ for all $q \in ...
0
votes
1answer
20 views

Why is $F'(x) = 2x·\tan(x^2)-\tan x$ if $F(x) = \int_{x}^{x^2}\tan u\, \mathrm du$?

Evaluate $F'(x)$ if $$F(x) = \int_{x}^{x^2}\tan u\, \mathrm du$$ I tried to do this by the change of variables formula and hence, $$F(x) = \int_{x}^{x^2}\tan u\, \mathrm du=\int_{\sqrt x}^{x}\tan ...
-1
votes
0answers
12 views

$f$ is smooth and periodic function,$\exists \lambda$ such that $f^{(4)}=\lambda f$ prove:$\exists \lambda$ such that $\lambda= (\frac{2\pi n}{T})^4$

Given $f:\mathbb{R} \rightarrow \mathbb{C}$ is smooth and periodic with period $T>0$ and exists $\lambda \in \mathbb{C}$ such that $f^{(4)}(x)=\lambda f(x)$ for any $x \in \mathbb{R}$. prove: ...
3
votes
1answer
21 views

How to prove define integrate from f(sin x)

i need help for prove this problem , i dont have idea for this prove, i very appreciate your sugerences. $$ \int ^{\pi }_{0}xf(\sin x)\,dx = \int ^{\pi }_{0}\frac{\pi }{2} f(\sin x)\,dx $$
4
votes
1answer
39 views

Proving an equivalent definition of the $\lim_{x\to a}f(x)$ exists [duplicate]

Prove that the following statements are equivalent. (a) $\lim_{x\to a}f(x)$ exists (b) Given $\epsilon \gt 0$, there is a $\delta \gt 0$ such that if $0\lt |x-a| \lt \delta, 0\lt |y-a| \lt \delta$, ...
1
vote
4answers
67 views

how to prove that $\ln(1+x)< x$

I want to prove that: $\ln(x+1)< x$. My idea is to define: $f(x) = \ln(x+1) - x$, so: $f'(x) = \dfrac1{1+x} - 1 = \dfrac{-x}{1+x} < 0, \text{ for }x >0$. Which leads to $f(x)<f(0)$, ...
1
vote
2answers
27 views

Finding the derivative of a multivariable integral?

Okay so I'm asked to find $\frac{df}{dt}$ of $f(t)=\int_{1}^{t^2}{\frac{1}{s}e^{s^2t}ds}$. Letting $g(s)=\frac{1}{s}e^{s^2t}$ I get: $$f'(t)=G'(t^2) \cdot 2t - G'(1) \cdot 0 =g(t^2) \cdot 2t = ...
1
vote
4answers
55 views

Calculus Optimization problem [on hold]

An oil company wishes to construct a pipeline from its offshore facility $A$ to its refinery $B$. The offshore facility is $2$ miles from shore, and the refinery is $1$ mile inland. Furthermore, $A$ ...
3
votes
1answer
43 views

Derivation of Gradshteyn and Ryzhik integral 3.876.1 (in question)

In the Gradshteyn and Ryzhik Table of Integrals, the following integral appears (3.876.1, page 486 in the 8th edition): \begin{equation} \int_0^{\infty} \frac{\sin (p \sqrt{x^2 + a^2})}{\sqrt{x^2 + ...
0
votes
0answers
24 views

To construct a power series such that the radius of convergence of the power series $\sum_{n=0}^{\infty} a_n b_n x^n$ is $2R$.

Let $\sum_{n=0}^{\infty} a_n x^n$ is a power series with radius of convergence $R(>0)$. To construct a power series $\sum_{n=0}^{\infty} b_n x^n$, other than $\sum_{n=0}^{\infty} (\frac x2)^n$, ...
0
votes
0answers
33 views

Epsilon delta limit to show that [on hold]

show that $$\left|\frac{28}{3x+1}-4\right| = \left|\frac{12}{3x+1}\right| \cdot |x-2| $$ using $\epsilon$-$\delta$ definition of a limit. I have no idea where to start since the question is not ...
5
votes
4answers
106 views

Find $\int_0^1(\ln x)^n\hspace{1mm}dx$

I am not a big fan of induction, it's just a personal preference. Is there a method other than induction. Answer is $n!$ by the way
2
votes
0answers
16 views

Change of variables missing definition

A book says to consider the transformations $x^* = \phi(x,y,y')$ and $y^* = \psi(x,y,y')$ and consider the functions $y(x)$ and $y^*(x^{*})$. I'm confused because I don't have a definition for ...
1
vote
0answers
35 views

Calculating in closed form $\int_0^{\infty} \frac{\text{PolyLog}^{(1,0)}(1,-x)}{1+x^2} \, dx$

Can you confirm the following result? Mathematica and other computational stuff I used seem unable to do anything about this result. Maybe to confirm it numerically? $$\int_0^{\infty} ...
3
votes
5answers
71 views

Limit calculation: $\lim_{x\to 0}\frac{1}{x}\ln\left(\frac{e^x − 1}{x}\right)=$?

For some reason I'm having trouble calculating the limit of the following function : $$\lim_{x\to 0}\frac{1}{x}\ln\left(\frac{e^x − 1}{x}\right)$$ The function might, or might not converge. I've ...
0
votes
1answer
33 views

Help with Euler Equations

This is from my textbook. Can someone give me a better explanation of what to do here? What does part (a) mean, i.e., how am I supposed to write $x = ln(t)$ in terms of $\frac{dy}{dx}$ and ...
1
vote
1answer
45 views

Evaluation of Real Integral

Given the following definition:$$I=\int\limits_{0}^{2\pi}e^{-i\theta n}\left(\frac{1}{n}\right)^{\rho e^{i\theta}}d\theta$$ Is there an analytic method for evaluating this integral? Best Regards
0
votes
0answers
77 views

Conjecturing the closed form $\frac{\pi ^2}{8}-\frac{\pi ^2}{8 \sqrt{2}}+\frac{\pi \log (2)}{4 \sqrt{2}}$

I conjecture that $$\small \int_0^{\pi/2} \frac{\cos ^2(x) \left(-2 \log \left(4^{-\sin ^2(x)} \sin ^{-4 \sin ^2(x)}(x)\right)-4 \log (\cos (x))+\cos (2 x) (4 \log (\cos (x))+\pi +\log ...
1
vote
3answers
193 views

Calculus: Finding Arc Length--Squaring the Derivative Where did the -1/2 come from?

Math Example about finding the arc length. I have gotten the derivative of the equation. Here is the equation. $$f(x)=\frac{x^5}{5} + \frac{1}{12x^3}$$ Derivative of the equation is: $$f'(x) = x^4 - ...
2
votes
1answer
80 views

can you find four distinct positive integers in which their sum in pair is square? [on hold]

square number is a number which is obtained by multiplying a number by itself.
2
votes
2answers
22 views

Differentials where the variable undergoes a percentage increase. Where am I wrong?

Let $R = \frac{k}{r^4}$, where $k$ is some constant. Find the change in $R$ as $r$ is increased by 10%. $R$ is the resistance of blood flow, $r$ is the radius of a vein. This problem seems easy ...
3
votes
6answers
695 views

L'Hospital rule, exponental ratio

$$\lim_{x\to ∞} \frac {x^{1000000}} {e^x}$$ could anyone please provide some hits with what result I will end up? After all applyings of L'Hospital rule, I will get $\frac {n} {e^x}$, where $n$ is ...
5
votes
2answers
87 views

Substitution for limits [duplicate]

How does substitution for limits exactly work? I see often answers that use the substitution $t=\frac1x$, then changing $x\rightarrow\infty$ to $t\rightarrow0^+$. I have seen this question, this ...
1
vote
2answers
15 views

Exponential Growth and Decay Question: A Bacteria Culture Contains 100 Cells and Grows at a Rate Proportional to its Size

A bacteria culture contains 100 cells and grows at a rate proportional to its size. After an hour the population has increased to 420. a) Find and expression for the number of bacteria after $t$ ...
1
vote
2answers
36 views

Minimum value of an an expression

Find the minimum value of $(\alpha-\beta)^2+(\sqrt{2-\alpha^2}-\frac{9}{\beta})^2$ where $ 0<\alpha<\sqrt{2}$ and $\beta>0$ My attempt: In my view,this minimum value is the shortest ...
2
votes
3answers
17 views

Simple equation for $E_c$

Can someone hint me a clue for: I have the form of $E_c$=$\frac{mv^2}{2}$ and for p=mv; I want to have the form for $E_c$ in case when $E_c$ will depend by p. I was thinking to multiply the initial ...
0
votes
4answers
63 views

Find the volume of the parallelepiped.

Find the volume of the parallelepiped having vectors $a=(1,4,-7)$, $b=(2,-1,4)$, and $c=(0,-9,18)$ as adjacent edges. What conclusions can you make about vectors $a$,$b$, and $c$? So to solve this I ...
4
votes
4answers
38 views

limit of $x \cot x$ as $x\to 0$.

I was asked to calculate $$\lim_{x \to 0}x\cot x $$ I did it as following (using L'Hôpital's rule): $$\lim_{x\to 0} x\cot x = \lim_{x\to 0} \frac{x \cos x}{\sin x} $$ We can now use L'Hospital's ...
1
vote
4answers
43 views

Find the value of $dy/dx$ at $x=8$

Given that variables $xy=40$, find $dy/dx$ at $x=8$. I used $40/8$ to get $y=5$. So why is the answer $-5/8$ and not $5/8$?
0
votes
1answer
17 views

How to prove this “local invertibility” theorem for bounded linear operators?

The theorem states that, suppose $X,Y$ are complete normed vector spaces, if $\mathscr A_0\in \mathscr L(X;Y)$ is invertible (i.e., $\exists \mathscr A_0^{-1}\in\mathscr L(Y;X)$ s.t. $(\mathscr ...
0
votes
1answer
33 views

How to solve integral with natural logarithm and product

I am trying to solve the following integral: $$\int{\frac{x}{4} \ln\left(\frac{4}{x}\right)}$$ Using this integral table, the more close case is (43). However, this is not the right one to use. Do ...
0
votes
1answer
19 views

From 1D gaussian to 2D gaussian

I read this: The Gaussian kernel for dimensions higher than one, say N, can be described as a regular product of N one-dimensional kernels. Example: g2D(x,y,$\sigma_1^2 + \sigma_2^2$) = ...
0
votes
0answers
29 views

How to find best plane passes through the center of $n$ points

Consider $n$ points $x_1,\ldots,x_n$ in $\mathbb{R}^n$. How to find the best plane passes through the center of these points with following approach: If we assume that the such plane has form ...
0
votes
0answers
14 views

Finding limit and maximizer

Let $f(x):=x^\alpha - k \cdot (x+c)^\alpha$, defined for $x>0$, where $k,c>0$ and $0<\alpha<1$. Question: solve $\max_{x>0} f(x)$. Below are my thoughts: Calculate $f'(x) = \alpha ...
0
votes
0answers
25 views

Find distance between a plane and some points [on hold]

Consider points $x_1,\ldots,x_n$ and plane $w\cdot x-\gamma=0$ in $\mathbb{R}^n$ and let $A=[x_1,x_2,\ldots,x_n]^T$. Is correct following formula to find the distance between these points and the ...
1
vote
0answers
68 views

Optimal allocation in network

Given a network (N,g). We want to analyse specializaton matters. Nodes are individuals, and they can product goods and services just like in our usual economy. Individuals can be consumers too. This ...
-1
votes
1answer
33 views

prove $\max \mathbf{a}^T \mathbf{b}_i \leq \mathbf{a}^T \mathbf{c}_i$

Can I prove the following (with or without assumptions, e.g. all the elements in $\mathbf{a}$ or $\mathbf{b}$ are positive? $\max \mathbf{a}^T \mathbf{b}_i \leq \mathbf{a}^T \mathbf{c}_i$ where ...
0
votes
1answer
45 views

If $f \le g$ and f, g are integrable, decreasing functions, then$\int_{x}^{\infty} f \le \int_{x}^{\infty} g$?

If $f \le g$ and $f, g$ are integrable, decreasing functions, then $\int_{x}^{\infty} f \le \int_{x}^{\infty} g$? Intuitively, I suppose it holds, but I have not found any such theorem in the ...
1
vote
1answer
42 views

Reasons for different answers when finding area using Simpsons rule and numerical integration?

I have a function $\sqrt{x^4(x+4)}$ to be integrated from 0 up to -4. Using Simpson's will give me 19.02 but using normal numerical methods giving me -19.5 ! What's the reason behind this difference ...
0
votes
0answers
15 views

Is it possible to prove $\arg\min_a f(\max(\mathbf{a}^T\mathbf{b}_i)) = \arg\min_a f(\mathbf{a}^T\max(\mathbf{b}_i))$

I have the following optimization problem, $ \arg\min_\mathbf{a} f(\max(\mathbf{a}^T\mathbf{b}_i))\;\; i=1, \dots ,N$ where $\mathbf{a}$ and $\mathbf{b}_i$ are vectors of dimension $d$. Let $B = ...
2
votes
3answers
84 views

$\lim_{x\to 0}\frac{e^x-1}{\sin x}$ equal to $\lim_{x\to 0}\frac{e^x-1}{x}$ because $x$ and $\sin x$ tend both to $0$ for ${x\to 0}$

I'm stuck in this limit: $$\lim_{x\to 0}\frac{x(e^x-1)}{\cos x-1}$$ I tried to solve it using special limits, so: $$\lim_{x\to 0}\frac{x(e^x-1)}{\cos x-1}=$$ $$=\lim_{x\to 0}(e^x-1)\frac{x(\cos ...
0
votes
1answer
35 views

Show that $y=\frac{4\sin\theta}{2+\cos\theta}-\theta$ is increasing function when $\theta \in [0,\frac\pi2]$

Show that $$y=\dfrac{4\sin\theta}{2+\cos\theta}-\theta$$ is increasing function when $\theta \in [0,\frac\pi2]$ What I have done If $\theta_1,\theta_2\in[0,\frac\pi2]$ then $$\sin\theta_1 < ...
-1
votes
0answers
34 views

$\int_{1}^{\infty} \frac{\omega^2-x^2}{(\omega^2+x^2)^2}(x^2-1)^{-(2/3)}dx$

Could someone kindly evaluate $\int_{1}^{\infty} \frac{\omega^2-x^2}{(\omega^2+x^2)^2}(x^2-1)^{-(2/3)}dx$ for me? Cheers, Allen
2
votes
4answers
74 views

Solve $\int\frac{8x+9}{(2x+1)^3}\,dx$.

Do I split $\displaystyle\int\frac{8x+9}{(2x+1)^3}\,dx$ into partial fractions? Or do I use $(2x+1)^{-3}$ by itself? Not sure what to do. Please advice. The answer given is ...
3
votes
1answer
44 views

If the Fourier Transform of $f(x)$ is known, can one deduce the Fourier Transform of $|x|f(x)$? [on hold]

If the Fourier Transform of $f(x)$ is known, can one deduce the Fourier Transform of $|x|f(x)$ ? I've been trying to find the Fourier Transform of $|x|^{7/6}K_{-1/6}(x)$. I know the transform of ...
1
vote
2answers
32 views

computing maclaurin series for $(\sin x)^3$ , order $3$

I have a clarification to ask: I want to compute $f(x)=(\sin x)^3$ by maclaurin series, order $n=3$. I know that: $\sin x=x-\dfrac{x^3}{3!}+R_3(x)$. So can i say that: $\sin^3x=(\sin ...
0
votes
2answers
44 views

How is $ \frac{\sqrt{a}}{a+1} (0^{a+1}+1^{a+1}) $ equal to $ \frac{\sqrt{a}}{a+1} (-1)^a $

I am trying to integrate this equation $$ y = \int_{-1}^0 \sqrt{a} x^{a} $$ $$ y = \sqrt{a} \int_{-1}^0 x^{a} $$ $$ y = \frac{\sqrt{a}}{a+1} \int_{-1}^0 x^{a+1} $$ $$ y = \frac{\sqrt{a}}{a+1} ...
2
votes
1answer
38 views

Comparing $\text{tr}(A^{-1})$ and $\text{tr}(A(B+A)^{-2})$ for pd $A$ and psd $B$

Suppose that $A$ is positive definite and $B$ positive semidefinite, both with dimension $n\times n$. Is there some inequality between $$ \text{tr}(A^{-1})\quad\text{and}\quad\text{tr}(A(B+A)^{-2})? ...