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

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0
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0answers
27 views

The difference between an affine k-simplex and a rectilinear k-simplex

The notion of rectilinear k-simplex appears in Theorem 10.27 of Rudin's book "Principles of Mathematical analysis", then what is the definition of a rectilinear k-simplex? I read the proof of Theorem ...
1
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1answer
74 views

Find $\lim_{x\to\infty} \frac{(x+1)^2(2x-4)^3}{(2x-1)^4}$

Find $$\lim_{x\to\infty} \frac{(x+1)^2(2x-4)^3}{(2x-1)^4}$$ I want to divide by $x^4$, but wouldn't I need to multiply out the whole thing and make a mess? Is there a more simple way of doing this? ...
0
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0answers
32 views

Partial derivatives in linearisation.

I'm working through a linearisation of the following system of equations \begin{align} \begin{split} u_t^{+}+\gamma u_x^{+}&=\mu(u^{+},u^{-})(u^{+}-u^{-}), \\ u_t^{-}+\gamma ...
1
vote
1answer
95 views

Calculate base and coefficient for power curve through 3 non-linear points

I have a formula that takes a 0-based bounded single dimensional input and transforms it to a specific power curve. EDIT This is single dimensional. There is no $y$. In the image, I'm showing how ...
7
votes
2answers
729 views

how to compute the de Rham cohomology of the punctured plane just by Calculus?

I have a classmate learning algebra.He ask me how to compute the de Rham cohomology of the punctured plane $\mathbb{R}^2\setminus\{0\}$ by an elementary way,without homotopy type,without ...
1
vote
2answers
56 views

Integral of the Product of two Sech functions

Assume $$ u(x,t_n,x_n) = \frac{1}{\sqrt{2t_n}} \operatorname{sech}\left( \frac{x - x_n}{t_n} \right) $$ I know that $$ \int dx\;u ( x,t_n,x_m )\;u ( x,t_n,x_n ) = \frac{1}{2} ...
11
votes
3answers
145 views

Applications of functions of the form $f(x)^{g(x)}$

Early on in my calculus education, I learned how to take the derivative of $x^x$ by re-writing it in the form $e^{x\ln x}$. More generally, this technique is helpful in finding the derivative of ...
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1answer
51 views

Direction of t (Vector Space)

Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. $$x = e^{-t}\cos t, y = e^{-t} \sin t, z = e^{-t}; (1, 0, 1). $$ The ...
1
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1answer
64 views

Laplace Transformation

Question: $$ y''+ y'- 1=0$$ and the values for $y(0) = 2, y'(0) = 3$. Solution: Applying Laplace transformation; $$ L{y''} + L{y'} - L{1} = 0 $$ $$ [s^2L{y} - sy(0) - y'(0)] + [ ...
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4answers
84 views

Find maxi,minimum $f(x,y)=x^3+y^3 (\text{where} ~~~ x,y\in \mathbb{R}, x^2+y^2=1)$

I would appreciate if somebody could help me with the following problem Q: Find maximum and minimum of $f(x,y)$ that $$f(x,y)=x^3+y^3 (\text{where} ~~~ x,y\in \mathbb{R}, x^2+y^2=1)$$
0
votes
1answer
63 views

Cluster point of a set

Given a sequence $a_{n}=\sin nx$,$n=1,2,...$,where $x\in(0,\pi)$,what is the limit point of the sequence? It's non-empty by Weierstrass theorem,but is there more information we know about it? Is it ...
1
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1answer
65 views

proving $\lim_{n\rightarrow \infty}\frac{n!}{\sqrt{2\pi n}\cdot \left(\frac{n}{e}\right)^n} = 1$

Can anyone explain me to how can I prove that: $$ \lim_{n\rightarrow \infty}\frac{n!}{\sqrt{2\pi n}\cdot \left(\frac{n}{e}\right)^n} = 1 $$ $\bf{My\; Try}::$ Given: $$ \lim_{n\rightarrow ...
1
vote
2answers
67 views

Does $\sum_{k=1}^\infty(1+\frac{1}{2}+ \dots +\frac{1}{k})\frac{\sin kx}{k}$ converge absolutely?

For $x\neq k\pi,k\in\mathbb{Z}$,does the following series converge absolutely? $$\sum_{k=1}^\infty(1+\frac{1}{2}+ \dots +\frac{1}{k})\frac{\sin kx}{k}$$
2
votes
0answers
49 views

interchanging the order of $d/dt$ and $\partial/\partial x$

The situation is as follows. The variable $y$ is related to another variable $x$, by some function $f$; $f(x)=y$. Here, $x$ depends on $t$, and we define $x'(t)=v(t)$. Then the book claims that: ...
1
vote
1answer
58 views

Testing Boundary Points Of $\sum_{n = 1}^{\infty} \frac{n!z^n}{n^n}$

I'm having some trouble testing the series indicated in the title at its boundary points. I'll sketch the preliminary work, then arrive at the problem. It is clear that the series converges absolutely ...
2
votes
1answer
44 views

How to integrate this exponential function?

I'm doing probability hw, and got stuck on the following integration: $$\int_{0}^{\infty }e^{\frac{-(\ln(y))^2}{2}}dy$$ Any hint would be very appreciate!! Thanks a lot!
1
vote
1answer
55 views

Related rates problem involving angles

A picture 4ft high is placed on a wall with its base 3ft. above the eye of an observer. If the observer is approaching the wall at the rate of 4ft/sec, how fast is the measure of the angle subtended ...
2
votes
0answers
38 views

Singular Solutions of this Equation?

How would I find the singular solutions of this equation: y = $ce^{x^2}$ + $ce^{\sin x}$ (where $c$ is a constant). It should be $x^2$ if anyone gets confused by the first part of the equation. ...
3
votes
4answers
163 views

Finding the area of a region?

I sketched out the graph to get this figure but I can't seem to find the area of the shaded region... would one Y = 4 and the other Y = 8? I, in all honesty, am so flabbergasted with this question, ...
0
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1answer
32 views

Showing lack of continuity of a function

Show that the function $$ f(x) = x \cdot \mathrm{sgn}\sin(1/x), x \neq 0; f(0) = 0 $$ is not continuous on any interval containing $x = 0.$
2
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1answer
113 views

Euler Lagrange equation derivation and application of the fundamental lemma of the calculus of variations

Say we have: (1) $J(x) = \int_{\textit{to}}^{\textit{tf}} g(x(t),\dot{x}(t),t) dt$. We go through the general derivation and arrive at: (2) $\delta J(x,\delta x) = ...
1
vote
2answers
96 views

Function that satisfy the properties of the exponential function

Let $E:\mathbb{R} \to \mathbb{R}$ be an infinitely continuously differentiable function and $E$ is not zero function such that $$E(u+v)=E(u)E(v).$$ Show that $E(x)=e^{ax}$ for some $a\in \mathbb{R}$. ...
0
votes
2answers
204 views

Find the area of the shaded region?

So the question is listed below as a picture, I realize I have to subtract the min and max after inputting them into the equation. So far I think it would be integral x going from 0 and going to 11/2? ...
0
votes
2answers
41 views

Simplifying an equation

I need to differentiate $y = t^2\ln(4t)$ which i understand needs to be done using the product rule. I know that $u = t^2$ and $v = \ln(4t)$ and $du = 2t$ and $dv = \frac{1}{t}$ So $u\,dv + v\,du = ...
0
votes
7answers
66 views

Help with integration using partial fractions

I'm not sure how to get the values for $A$ and $B$ for the expression $$ \frac{3}{x^2 - 16}. $$ I've split the expression into $$ \frac{A}{x - 4} + \frac{B}{x + 4}. $$ I don't know what to do ...
3
votes
8answers
114 views

Evaluate $\lim_{x\rightarrow 0} \frac{\sin x}{x + \tan x} $ without L'Hopital

I need help finding the the following limit: $$\lim_{x\rightarrow 0} \frac{\sin x}{x + \tan x} $$ I tried to simplify to: $$ \lim_{x\rightarrow 0} \frac{\sin x \cos x}{x\cos x+\sin x} $$ but I ...
1
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3answers
124 views

Finding the higher-order derivative $\frac{\mathrm{d}^{950}}{\mathrm{d}x^{950}}(\sin x)$

Find $$\frac{\mathrm{d}^{950}}{\mathrm{d}x^{950}}(\sin x)$$ How are really high-order derivatives found? My try: $$\frac{\mathrm{d}^{950}}{\mathrm{d}x^{950}}(\sin x)= 950(\cos x)^{949}$$ ...
0
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0answers
29 views

Questions on Difference operators

Please I really need help on the following short problems on difference operators that I need even some clues on how to go by them: 1) $\sum_{t=1}^{4}{\dfrac{1}{(t+1)(t+2)(t+3)}}= ...
1
vote
2answers
48 views

calculating radius of convergence of $\sum_{k=0}^\infty\frac{1+x^k}{1+y^k}z^k$

I want to calculate the radius of convergence of $$ \sum_{k=0}^\infty \frac1{k!}z^{k^2} \qquad\sum_{k=0}^\infty\frac{1+x^k}{1+y^k}z^k$$with $z\in\mathbb C$ and $x,y\in\mathbb R^+$ For the first one: ...
2
votes
0answers
23 views

probability subspaces that make entropy function equal to a constant value

Given the entropy fucntion: $$ H = -\sum_i^n p(i) \ln(p(i))\,.$$ where $p(i)$ are probabilities and $n=4$, I would like to know all the points in the probability space that make $H = k$, being $k$ a ...
2
votes
2answers
33 views

Query on Mean Value Theorem Criteria

In the theorem why do we consider that the function is continuous on the closed interval but differentiable on the open interval? What difference it would make if in both the cases, closed interval is ...
2
votes
3answers
88 views

How to compute $\lim_{n \to \infty} \left( 1-\frac{2t}{n^2} \right)^{-n/2} $?

$$\lim_{n \to \infty} \left( 1-\frac{2t}{n^2} \right)^{-n/2} $$ How can I find the limit above? I am bit confused because of the square in the denominator. If it was $n$ instead, then the limit ...
3
votes
1answer
391 views

Volume of a surface of revolution with curve in parametric form

On Wikipedia, I recently stumbled upon a method of obtaining the volume of a solid of revolution generated by a curve in parametric form, which was useful in my case because I had a curve I had ...
5
votes
2answers
188 views

What is the average length of 2 points on a circle, with generalizations

I have earlier seen the question about finding the average length of two points and $n$ points inside the unit disk. But what about the more simple question, what happens if the points lie exactly on ...
3
votes
3answers
253 views

Examples where derivatives are used (outside of math classes)

I want to know what is the use of derivatives in our daily life. I have searched it on google but i haven't find any accurate answer. I think it is mostly used in Maths but I want to know its use in ...
3
votes
2answers
46 views

Showing that $\int_x^{\infty}\frac{1}{u^2}e^{(-u^2/2)}du=\frac{1}{x}e^{(-x^2/2)}-\int_x^{\infty}e^{{(-u^2/2)}}du$

My texbook claims that integration by parts of the integral $\int_x^{\infty}\frac{1}{u^2}e^{\frac{-u^2}{2}}du$, with the hint that $d(-\frac{1}{u})=\frac{du}{u^2}$, gives ...
2
votes
1answer
67 views

Compactly supported smooth function with Laplace transform bounded on a cone

My question is if it is possible to find a compactly supported smooth function $\varphi:\mathbf{R}\to \mathbf{R}$ s.t. the following integration $\int_{\mathbf{R}}\varphi(t)e^{itx}e^{tx}dt$ stays ...
5
votes
1answer
151 views

$\int_0^{\frac{1}{2}}\cot(\pi x)\sin(2\pi nx)\cos(2\pi mx)\,dx$ [closed]

How can I prove that $$\int_0^{\frac{1}{2}}\cot(\pi x)\sin(2\pi nx)\cos(2\pi mx)\,dx=\begin{cases}0&m>n\\\frac{1}{4}&m=n\\\frac{1}{2}&m<n\end{cases}$$ ...
1
vote
2answers
103 views

Volumes of solids of revolution

This might be a somewhat unorthodox question. I am wondering if there are any simple guidelines or tips/tricks to better understand volumes of solids of revolution. E.g. the simple assignment and my ...
1
vote
1answer
71 views

AP Calculus Derivative Slope

$xy^2-x^3y = 2$ $y' = \frac{3x^2y-y^2}{2xy-x^3}$ find the x-coordinate of each point on the curve where the tangent line is horizontal. I know that $3x^2y-y^2 = 0 \Rightarrow$ then $y=3x^2.$ If I ...
0
votes
1answer
50 views

Maximum of the function of multivariable?

I need to find the maximum of the function given by $z=x^3+xy$ in $A=[0,1]\times[0,1]$. I think I need to use partial derivatives, but I'm not sure exactly how.
2
votes
1answer
65 views

Question about Cauchy sequence convergence, and the idea behind it

Let there be $\epsilon>0$. $\{a_n\}$ is a Cauchy sequence if there is an index $k$, so that for every $n\ge k$ and for every $p \in \mathbb N$, $$|a_{n+p}-a_n|\lt \epsilon$$ And here is an ...
1
vote
1answer
45 views

how to find the partial derivation by z of f(x,y,z)= x^(y^z)?

I think the solution is f*lnx *(y^z)*lny ? But some don't agree.. I need a second opinion. Thak you.
1
vote
2answers
42 views

line of sources for differential equation

It is well known that for representation of point source with some intensity $q(t)$ in some PDE we use delta-function $\delta({\bf r} - {\bf r}')$. If problem requires usage of line of sources which ...
-1
votes
2answers
57 views

Prove the average quantum mechanical energy using l'Hôpital's rule

I am trying to prove that taking the limit as $h\to0$ for the average quantum mechanical energy $$\dfrac{hν}{e^{hν/kBT}−1}$$ yields the average classical energy, $kBT$. How would you use l'Hôpital's ...
1
vote
1answer
47 views

Exact expression for series

Can the exact expression for the following series be found, given $|x|<1$? Just curious. $f(x) = \frac{x^2}{17}+\frac{x^3}{3}+\frac{x^4}{3}+\frac{x^5}{3}+ \ldots$
2
votes
2answers
94 views

For every $n \in \mathbb{N}$, find a function $f$ which is differentiable $n$ times at $0$, but not $n+1$ times.

For every $n \in \mathbb{N}$, (in this problem $\mathbb{N}$ starts at $n = 1$) find a function $f$ which is differentiable $n$ times at $0$, but not $n+1$ times. The function I chose is the ...
0
votes
2answers
50 views

Derivative of a function wrt a scalar

Let $f$ be a function from $m \times n$ matrices to $\mathbb{R}$. Let $\alpha$ be scalar. How should I compute the derivative of $f$ wrt to $\alpha$? $$\frac{d f(\alpha X)}{d \alpha}=?$$ where $X$ is ...
0
votes
1answer
26 views

Get the closes cordinates to (0,0)

I'm making a system where I'm drawing a cube and I need to get the correct cordinates. Im trying to calculate the width and ...
1
vote
1answer
54 views

triple integral over the region bounded by 2 spheres

Evaluate the following integral $$\iiint\limits_E x \,dV$$ where $E$ is a region bounded by the spheres with radii $2$ and $3$ respectively and center in the origin. I converted the integral ...