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

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0
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1answer
7 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 ...
0
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0answers
6 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 ...
1
vote
0answers
30 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 $f(5)$. $f'(x)=f(x)$ ...
1
vote
2answers
66 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 ...
11
votes
1answer
237 views

Closed-form of $\int_0^1 \left(\ln \Gamma(x)\right)^3\,dx$

From the amazing result by Raabe we know that $$LG_1=\int_0^1 \ln \Gamma(x)\,dx = \frac{1}{2}\ln(2\pi) = -\zeta'(0).$$ We also know that $$LG_2 = \int_0^1 \left(\ln \Gamma(x)\right)^2\,dx = ...
1
vote
4answers
46 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} ...
1
vote
0answers
28 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
votes
3answers
57 views

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

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.
3
votes
0answers
103 views

How do find the numerical average of $x^x$ from $(-4,-2)$?

I wanted to find the approximate average of all real points in $(x)^{x}$ from $[-4,-2]$. This means I am ignoring all complex points and need average to be a real number. To first solve this I found ...
1
vote
3answers
30 views

Integrate the following equation. (exponential function)

Integrate $$\frac{e^x -2}{e^{x/2}}$$ This is my calculation: but it is wrong....
0
votes
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
votes
2answers
21 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 ...
2
votes
2answers
15 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
votes
0answers
9 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 ...
0
votes
0answers
26 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
votes
2answers
34 views

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

Why is this true? $$\cos\left(\frac{3\pi}{2}-t+2k\pi\right) = -\sin(t)$$
0
votes
1answer
16 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
votes
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} ...
0
votes
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 = ...
0
votes
3answers
33 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
votes
1answer
14 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.
0
votes
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 ...
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 ...
5
votes
3answers
130 views

In the definition of a limit, why do we care about all $\epsilon > 0$?

Definition of $\lim_{x \to a} f(x) = L$: $\forall \epsilon > 0, \exists \delta > 0 s.t. |f(x) - L| < \epsilon \ if \ 0 < |x-a| < \delta$ Why can't we weaken the assumption to ...
0
votes
1answer
39 views

Limit of a sequence with binomial coefficient. Can I use Stirling?

I was trying to solve this limit: $\lim_\limits{n\to \infty} \binom {3n}{n}^{1/n} $ I solved it with Cesaro theorem: $\lim_\limits{n\to \infty} \binom {3n}{n}^{1/n} $= $\lim_\limits{n\to \infty} ...
1
vote
1answer
16 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
votes
2answers
23 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?
1
vote
0answers
12 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 ...
3
votes
2answers
60 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 ...
1
vote
0answers
20 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 ...
4
votes
2answers
38 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 ...
3
votes
1answer
893 views

Proof that the product of two differentiable functions is differentiable

Let $f:A\subset \mathbb{R^p}\rightarrow\mathbb{R^q}$ and $\phi:A\subset \mathbb{R^p}\rightarrow\mathbb{R}$ differentiable in $c\in A$. I have to prove that $g(x)=\phi (x)f(x)$ is differentiable, ...
-4
votes
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)= ...
0
votes
0answers
15 views

Recursively enumerable VS recursively sets [on hold]

How can I show that are many recursively enumerable sets than recursively sets?
-1
votes
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?
-2
votes
0answers
21 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...
23
votes
2answers
338 views
+50

Prove $|P(0)|\leq 2n+1$

Let $P(x)$ be a polynomial with degree $\leq n$ and $|P(x)|\leq\frac{1}{\sqrt{x}}$ for $x\in(0,1]$. Prove that $|P(0)|\leq 2n+1$. The idea should be that if $|P(0)|$ is too large, then the polynomial ...
1
vote
3answers
56 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 $$ f'(x) \ge g'(x)$$ ...
-2
votes
3answers
52 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
votes
1answer
33 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 ...
0
votes
1answer
36 views

2nd Order Differential Equation, particular integral query.

What would the form of the particular integral be of the following differential equation: $$\frac{d^2y}{dx^2} -4 \frac{dy}{dx} +5y=8 \sin x$$ Should the particular integral be of the form; ...
0
votes
1answer
35 views

Is the gap between successive real roots of $x(t) = \frac{1}{30000 e^t} + \frac{2 e^{t/2} \cos (\sqrt{3}t/2)}{30000} $ eventually less than $5$?

Consider the function $x : \mathbb{R} \to \mathbb{R}$ given by $$x(t) = \frac{1}{30000} \frac{1}{\mathrm{e}^t}+ \frac{2}{30000} \mathrm{e}^{\frac{t}{2}} \cos \left(\frac{\sqrt{3}}{2}t\right), \quad ...
1
vote
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.
0
votes
2answers
68 views

Expansion $f(x)=1/(x-1)$

How to expand $f(x)=1/(x-1)$ into the form $1/x+1/x^2+1/x^3+...+1/x^n$ for x>1 I know f(x) can be rewritten as $f(x)=\frac{(1-1/x)^{-1}}{x}$. Next step is to expand $(1-1/x)^{-1}$ to ...
1
vote
2answers
40 views

Finding the First Derivative ( 1 question)

Using the Definition of a limit: [ Of form $\lim_{x\to a} \frac{f(x)-f(a)}{x-a}$] Find $f'(x)$ when $x=9$ for $f(x)=\frac{2}{\sqrt{x}}$ I tried simplifying it but got jumbled when trying to multiply ...
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
vote
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 ...
3
votes
1answer
60 views

Circle is similar to a polygon with infinite number of sides

It is know from the time of Euclid, that a circle is similar to a polygon with infinite number of sides. But this ^^ is informal. Do you know any formalization where it appears that a circle is a ...
1
vote
1answer
36 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 ...
6
votes
3answers
300 views

Is there a formula for the area under $\tanh(x)$?

I understand trigonometry but I've never used hyperbolic functions before. Is there a formula for the area under $\tanh(x)$? I've looked on Wikipedia and Wolfram but they don't say if there's a ...