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

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8 views

Let ${a_n}{x^n} + … + {a_1}{x^1} - f(t) = 0$ and $f(t) $ is decreasing function of $t$. Can we say that $y(t)$ is decreasing function of $t$?

Let $t\in (0,1)$ and ${a_n}{x^n} + .... + {a_1}{x^1} - f(t) = 0$ $f(t) $ is continuous decreasing function of $t$. $a_i\ge0$ for all $i$. $y(t)$ is zero of equition. Can we say that $y(t)$ is ...
0
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1answer
19 views

Find the volume of the solid generated by the region

Find the volume of the solid that is generated when the region enclosed by $ y = \cosh 2x, y = \sinh 2x, x = 0, $ and $ x = 5 $ is revolved around the x-axis.
0
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0answers
21 views

Prove that the following function is convex?

I am trying to prove that the function $$g(\alpha)=\ln\Big(\sum_{i=1}^{n}t_i^\alpha+A(\alpha)\Big) ~~t_i, \alpha>0,$$ where $A(\alpha)=\int_{a}^{b}x^{\alpha+\beta}e^{-\lambda x^\beta}\,dx$,is ...
1
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1answer
26 views

Finding the absolute maximum and minimum within an interval?

For some arbitrary function: $f(x)$ within the interval $a<x<b$, should I just calculate the roots for $f'(x)$, and the points $f(a)$ & $f(b)$, and make deductions based around what is going ...
2
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1answer
54 views

Find $\int\limits^{\infty}_{0}\int\limits^{\infty}_{0}{\frac{1}{(x+y)^{3/2}}\exp\left\{-\frac{a^2}{2(x+y)}\right\}}\,dy\,dx$.

In my posterior probability computation, I got the following integration and I could not figure it out. ...
6
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0answers
41 views

Confusion with Courant: Which of his two calculus books is THE one?

Since I've worked my way through Spivak's Calculus book, I thought I'd give Courant's allegedly fantastic exposition of the subject a go as well. However, I've run into a problem. People in ...
0
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1answer
14 views

Are difference and differential operators commutative?

I was searching information, theorems, etc... about commutativity between difference and differential operators but I dont found explicit statements so Im unsure about my assumptions (so I ask here in ...
0
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1answer
23 views

Creating an integral for finding the volume of this revolution

I need to find the volume of a solid that is created by rotating the area within the following boundaries: $y=x^3$ $y=8$ $x=0$ which is rotated over $x = 3$. I thought I had the correct integral ...
0
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1answer
16 views

Bifurcation Diagram question for Population harvesting model $P' = rP (1-\frac{P}{K}) - hP$

A deer population grows logistically and is harvested at a rate proportional to its population size. The dynamics of population growth is modeled by $P' = rP (1-\frac{P}{K}) - hP$ where $r$ (the ...
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0answers
7 views

Partial derivative of polynomial dependant on previous time values

I have not touched calculus for a few years, and I am not sure what is going on here. Any help would be greatly appreciated :) Essentially, let $p_{t} = \log p(y_{t}|h_{t},h_{t+1})$, where ...
3
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0answers
43 views

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that $f'(x)$ is continuous and $|f'(x)|\le|f(x)|$ for all $x\in\mathbb{R}$

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that $f'(x)$ is continuous and $|f'(x)|\le|f(x)|$ for all $x\in\mathbb{R}$. If $f(0)=0$, find the maximum value of $f(5)$. $f'(x)=f(x)$ ...
3
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2answers
282 views

If a function is discontinuous at one point, then filled in, is it now continuous?

I am looking at the continuity of the following function $f(x) = \sin(1/|x|), f(0) = 0$ So this is $f(x) = \sin(1/|x|)$ filled in at $x = 0$ Clearly, $\lim\limits_{x \to 0} f(x) = 0 $ by squeeze ...
-1
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3answers
61 views

Without computing, is the integral of $\int_0^1 t(t-1)(t-2)\,dt$ positive or negative? [on hold]

I have to graph the function, but I don't think I'm doing it right. Here is a picture of it Sorry, this is my first time using this site and I don't know how to use MathJax yet.
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0answers
33 views

Differentiating a matrix product

In one of the books I found that given that for a linear system $x'=Ax$, there exists a matrix $Q:=\int\limits_0^\infty B(t)dt$, where $B(t)=e^{tA^T}e^{tA}$, and $V(x) = x^T Q x$, ...
0
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2answers
24 views

Confusing result obtained taking second derivative of ye^y

I was doing my calculus homework, and one of the questions asked for the first and second derivative of $ye^y=x$, I did the computations and arrived at $-(x+1)^{-2}$, which was a lot neater and ...
1
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4answers
51 views

Solving $\lim_{n \to \infty} \sqrt{n} \sin\left({\sqrt{n+3}-\sqrt{n-2}}\right)$

I have trouble finding the value of the following limit: $$\lim_{n \to \infty} \sqrt{n} \sin\left({\sqrt{n+3}-\sqrt{n-2}}\right)$$ For now I have rewritten the term into: $$ \lim_{n \to \infty} ...
0
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0answers
13 views

Function with both easy to find Fourier and Hermitian coefficient

I'm writing some notes on Spectral theory and I would like to make a simple example finding the generalized fourier coefficient of a function in respect of two different bases. I was thinking about ...
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3answers
31 views

Integrate the following equation. (exponential function)

Integrate $$\frac{e^x -2}{e^{x/2}}$$ This is my calculation: but it is wrong....
2
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2answers
16 views

Vector-Valued Functions and Continuity

Why is it that when a vector-valued function $r(t)$ is continuous at some time $t$ then $\|r(t)\|$ is also continuous at that time $t$, but the converse is not true? That if $\|r(t)\|$ is continuous ...
0
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0answers
30 views

Shortest distance between two functions/curves

I'm completely stumped with this, I've tried looking at other questions asking the same kind of thing without success. I am given two functions, $f(x)=x^2+4x+6.2 $ and $h(x)=-3x^2-5$ and I am asked ...
0
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2answers
35 views

Why is $\cos\left(\frac{3\pi}{2}-t+2k\pi\right) = -\sin(t)$ [on hold]

Why is this true? $$\cos\left(\frac{3\pi}{2}-t+2k\pi\right) = -\sin(t)$$
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2answers
46 views

Evaluating $\lim_{x\to 0}{\frac{\sin^2x}{2x^2}}$ without L'Hospital

I have been trying to evaluate $$\lim_{x\to 0}{\frac{\sin^2x}{2x^2}}$$ Finally, I used the L'Hospital's Theorem and I got the answer $1/2$, but I wonder if there is a way to solve this without this. ...
0
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3answers
36 views

Area of a rectangle within a curve

The cargo space of a bulk carrier is 60m long. The shaded part of the diagram represents the uniform cross-section of this space. It is shaped like a parabola with equation ${{1\over 4}x^2, - 6 \le ...
0
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1answer
17 views

An “extra” solution to an initial value problem

So I came up with this example when I was teaching: consider the IVP $$ y'(x) = xy-x-5y+5, y(0)=1. $$ The standard approach is to separate variables: $y'(x) = (x-5)(y-1)$, which allows me to ...
0
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2answers
49 views

Find the inverse $\dfrac{x}{\|x\|}$ in $\mathbb{R^2}$

I wish to find the inverse of $\dfrac{x}{\|x\|}$, where $x \in \mathbb{R}^2$ Let's do this. Let $$y_1 = \dfrac{x_1}{\sqrt{x_1^2+x_2^2}}$$ $$y_2 = \dfrac{x_2}{\sqrt{x_1^2+x_2^2}}$$ Then $$y_1 = ...
1
vote
2answers
32 views

Can I prove a function is continuous by looking at the domain?

I came across the following question in a calculus book: For the function $$f(x)=1-\sqrt{1-x^2}$$ show that it is continuous on the interval $$-1≤x≤1$$ The solution in the book showed that the one ...
1
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0answers
13 views

Combining two results from partial integration

I have a set of two PDEs: $$\partial_{\tau}\theta+\partial_{\eta}\psi=0$$ $$\partial_{\tau}\psi=-\partial_{\eta}\theta+\alpha\partial_{\eta}^{2}\psi$$ These can be combined into a wave equation of ...
1
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1answer
17 views

Computing a line integral where the curve is in polar coordinates

Compute $\int \limits_{C} F.dr$ for $F(x,y)=(y,x)$ and $C$ is the curve given by $r=1+\theta$ for $\theta \in [0,2\pi]$ My Attempt Am I correct in saying that $F$ is a conservative vector field ...
0
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2answers
24 views

How to simplify inverse trigonometric function

How to simplify the following equation: $$\sin(2\arccos(x))$$ I am thinking about: $$\arccos(x) = t$$ Then we have: $$\sin(2t) = 2\sin(t)\cos(t)$$ But then how to proceed?
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0answers
21 views

Fourier coefficients of the Gaussian.

I would need to find the fourier coefficient of this gaussian for a problem. I'm now stuck with this integral, \begin{equation} c_{n}=\int_{-1}^{1}e^{\frac{x^{2}}{2}}\left(\cos\left(\pi ...
0
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3answers
20 views

How do we find more appropriate constants for expansions of functions?

We all knonw that the expansion of $e^x$ is $$1+x+x^2/2+...$$. But what if I want to find more approximate expansion of $e^x$. I try that $$e^x-1-c_0(x)+(c_0+c_1)(x^2/2)-(c_0+c_1+c_2)(x^3/3)=0$$ and ...
3
votes
2answers
62 views

How to derive: $\left(1 + \frac{1}{n}\right)^n < 1 + 1 + \frac{1}{2}+…+\frac{1}{2^{k-1}}$

In my textbook they write the following inequality: $\left(1 + \frac{1}{n}\right)^n < 1 + 1 + \frac{1}{2}+...+\frac{1}{2^{k-1}}+...+\frac{1}{2^{n-1}}$ They say that they derive this inequality by ...
4
votes
2answers
40 views

Prove that $\lim\limits_{x\to\infty} f'(x)=0$

Let $f$ be a function in $(0,\infty)$ such that $f'(x)$ exists. In addition, $\lim\limits_{x\to \infty} f'(x)=L$ (finite) and $f(n)=0$ for every $n \in \Bbb N$. Prove that ...
0
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1answer
16 views

How to represent y as a function of w?

Assume : $F(y)=G(w)$ where $F,G$ are two real-valued functions from $R \to R$. We want to find the function $C(w)$ such that : $F'(y)=C(w)$ and C should be built based on F and G. Thanks so much.
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0answers
16 views

Recursively enumerable VS recursively sets [on hold]

How can I show that are many recursively enumerable sets than recursively sets?
0
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0answers
17 views

normal plane to a level curve [on hold]

$\ f(x,y,z)=(x^2 + y^2 - z^2, x + y + 2z)$ $\ C: f(x,y,z)=(1,0). $ Find the cartesian equation of the normal plane to C at $\ (1,1,-1)$ Where do I start here?
0
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1answer
35 views

Find $\lim_{x\to 1} \frac{|x-1|}{\sqrt{2x^2+2}-(x+1)}$

$$\lim_{x\to 1} \frac{|x-1|}{\sqrt{2x^2+2}-(x+1)}$$ I have multiply by $\frac{\sqrt{2x^2+2}+(x+1)}{\sqrt{2x^2+2}+(x+1)}$ and got: $$\lim_{x\to 1} ...
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0answers
22 views

Function Limit & Continuity [on hold]

What is Function Limit & Continuity? I'm a little bit silly.Is there anyone to explain those terms precisely? Thanks in Advance...
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2answers
56 views

Help with continuity [on hold]

Could you please clarify these questions to me. Find all the numbers for which the given function is discontinuous. $F(x)=[x-1]$ I think the solution is $\Bbb Z$ numbers right ? $F(x)= ...
1
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3answers
58 views

Prove $\ln x \ge \frac{x-1}{x}$

Prove that for every $x>0$: $$\ln x \ge \frac{x-1}{x}$$ What I did: $$f(x) = \ln x, \text{ } g(x) = \frac{x-1}{x} $$ $$f(1) = g(1) = 0 $$ So it's enough to prove that $$ f'(x) \ge g'(x)$$ ...
-3
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3answers
54 views

Find the following integrals [on hold]

I am really having a hard time trying to solve these integrals and I would be very thankful if you would help me solve them:
5
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1answer
34 views

$\lim_{n \to \infty}(\frac{a_n}{\sqrt{a_n^2+1}})=\frac{1}{2}$ - show that $a_n$ is convergent sequence

Problem: Show that $a_n$ is convergent sequence and find a limit of $a_n$. $$\lim_{n \to \infty}(\frac{a_n}{\sqrt{a_n^2+1}})=\frac{1}{2}$$ I tried to look at this as normal limit problem so I wrote ...
1
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4answers
62 views

Minimum distance between the curves $f(x) =e^x$ and $g(x) =\ln x$ [on hold]

What is the minimum distance between the curves $f(x) =e^x$ and $g(x) = \ln x$? I didn't understand how to solve the problem. Please help me.
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0answers
8 views

When is the Stieltjes integral of bounded variations?

I was trying to figure out when a Riemann or Lebsgue Stieltjes integral is of bounded variation. For simplicity let $f$ be a increasing RCLL function; when is that $$\int_0^t g(x) df(x)$$ is of ...
2
votes
3answers
16 views

Finding the formula for acceleration from $v=2s^3+5s$, where $s$ is the displacement at time $t$

This is the question: I first found $\frac{dv}{ds}=6s^2+5$, then I tried to find $\frac{ds}{dt}$ by messing about a little with implicit differentiation, but I had no luck and I therefore couldn't ...
1
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1answer
38 views

solve $\frac{\partial u^2}{\partial x\partial y}=0$

I need to solve $$\frac{\partial u^2}{\partial x\partial y}=0$$ with the boundary conditions: $u(x,y=x^3)=\sin(x^6)$ and $\frac{\partial u}{\partial x}(x,y=x^3)=0$. I got a particular solution, I ...
1
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1answer
45 views

Weird indefinite integral homework questions

I'm solving a couple of integration problems using the method of changing variables, and would like assistance with two particular problems that I can't seem to solve. I completed rest of the problems ...
0
votes
0answers
25 views

Range of Derivative

Let $g(x) = f(x)/(x+1)$, where $f(x)$ is differentiable on $x\in[0,5]$, such that $f(0)=4$ and $f(5)=-1$. What is the range of values $g'(c)$ for a $c$ belonging to $[0,5]$? Considering values of ...
1
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0answers
41 views

Find $\sum_{i=1}^{2000}\gcd(i,2000)\cos\left(\frac{2\pi\ i}{2000}\right)$

What is the value of the following sum? $$\sum_{i=1}^{2000}\gcd(i,2000)\cos\left(\frac{2\pi\ i}{2000}\right)$$ where $\gcd$ is the greatest common divisor.
1
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1answer
19 views

Finding a limit on multiple square roots in a row?

Here are basically my two problems, which I have the answer from WolframAlpha: $$ \lim_{n\to\infty}(1-\sqrt 2-\sqrt{n+1}+\sqrt{n+2})=1-\sqrt 2 $$ $$ \lim_{n\to\infty}(\sqrt n-2\sqrt{n+1}+\sqrt{n+2})=0 ...