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

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-1
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0answers
26 views

To show that $\int_{0}^{\infty} \frac{e^{-ax}\sin(bx)}{x}=\arctan \left( \frac{b}{a}\right)$

To show that $\int_{0}^{\infty} \frac{e^{-ax}\sin(bx)}{x}$ = $\arctan \left( \frac{b}{a}\right)$ I have to use Frunalli integral to do this. But i am not able to start.
0
votes
0answers
15 views

Determination of a formula of an area of shape limited by curves using integrals?

I did two examples of determining an area limited by curves and I'm not sure if I did them right. I would really appreciate if someone checked my solution and fixed any possible mistakes. Ex.1 Area ...
0
votes
0answers
29 views

Integral of $\vec \nabla f(x)$

I was trying to prove a theorem and I came across this integral as a part of the theorem: $$ \int d^3x \left(\, \psi \vec r \,\nabla^2 \psi^* - \psi^* \vec r \,\nabla^2 \psi \right)$$ I was thinking ...
2
votes
1answer
31 views

Elementary differentiation problem involving logarithms: What am i missing here?

Consider the finite sum $S=$ $\sum_{k=2}^n \log k - \log(k-1)$. Differentiating $S$ w.r.t $k$, we have $S'= \sum_{k=1}^n \dfrac{1}{k} - \dfrac{1}{k-1}=-\sum_{k=1}^n \dfrac{1}{k(k-1)}<0$. But ...
3
votes
2answers
76 views

Prove that $\lim_{x\rightarrow \infty} \frac{x^2 - 1}{x^2 + 1} = 1$ using definition of limit.

Ok, so if I have to use definition, then I should prove something like this: $(\forall \epsilon >0)(\exists k>0)(\forall x \in X)$ then if $ x>k$ then $|f(x) - L| <\epsilon$ $L$ is ...
-1
votes
1answer
38 views

CALCULUS: Sketching a function by given conditions [on hold]

Pls help. I'm currently on a struggle with this calculus problem. Thanks in advance.
1
vote
0answers
8 views

maximum likelihood estimators of a shifted gamma distribution?

i had this question in my exam but didn't know how to solve this apart from constructing the likelihood function and differentiating .but got stuck in the middle of nowhere.please help .
2
votes
5answers
51 views

Finding $\lim_{n \to \infty} \frac{1}{2^{n-1}}\cot(\frac{x}{2^n})$

Find: $$\lim_{n \to \infty} \frac{1}{2^{n-1}}\cot\left(\frac{x}{2^n}\right)$$ Can L' Hopital's rule be used to solve this? And differentiate it with respect to $x$ or $n$? What I've found is ...
0
votes
1answer
17 views

Convergence of $f_n = \frac{x}{3-5n|x|}$

Study the convergence of the sequence $$f_n = \frac{x}{3-5n|x|}$$ The domain of $f_n$ is $\operatorname{dom} f_n = \mathbb R \backslash \{\pm\frac3{5n}\}$ and $$\lim_{n \to +\infty} f_n = 0$$ ...
6
votes
1answer
58 views

Is it to the students' advantage to learn the language of infinitesimals?

A colleague of mine asked an interesting question reproduced below with his permission. It is reasonable to ask whether it is to the students' advantage to learn the language of infinitesimals - ...
0
votes
2answers
55 views

To evaluate $\lim_{x \to 0^+} \frac{\log(x)}{\sqrt x}$ using inequality

To evaluate $$\lim_{x \to 0^+} \frac{\log(x)}{\sqrt x}$$ I know that $\log(x) < x$ for $x > 0$. So dividing by square root of $x$ and taking limits gives me nothing. Which inequality should I ...
2
votes
1answer
47 views

integral involving greatest integer function

Let $S_n = \sum_{k=1}^n \frac{1}{k}$ and $I_n=\int_1^n \frac{x-[x]}{x^2}dx$. Then, what is $S_{10} + T_{10}$? The only clue that i can get is break the limits of integration according as the ...
0
votes
0answers
61 views

If $y(t) = t\left(1-\int_0^ty(x)\,dx\right)+4\int_0^tx\,y(x)\,dx,$ then $\int_0^{\pi/2} y(t)\,dt$ is equal to?

Leibniz rule or Laplace transform? Let $y(t)$ be a continuous function on $[0,\infty)$. If $$y(t) = t\left(1-4\int_0^ty(x)\,dx\right)+4\int_0^tx\,y(x)\,dx,$$ then $\int_0^{\pi/2} y(t)\,dt$ ...
2
votes
4answers
30 views

Limit of a function of multiple variables: $\lim_{(x,y) \to 0}\dfrac{x^2y}{17x^2+y^2}$

$$\lim_{(x,y) \to 0} \dfrac{x^2y}{17x^2+y^2}$$ I want to obtain this limit but don't know how to. The most general advice I've found is to convert this function into polar coordinates, so when I ...
0
votes
0answers
26 views

$f(x) = 2x \mod 1$ not equal to zero for all $x$?

If any number $\mod 1$ is zero, then how can $f(x) = 2x \mod 1$ be a Baker's map? For any $x\in \mathbb{R}$, shouldn't $f(x)=0$?
0
votes
1answer
14 views

Examine the end behavior of $f(x)$ and find any horizontal asymptotes.

$f(x)=\frac{10x^3-3x^2+8}{\sqrt{25x^6+x^4+2}}$ and this is what I have done so far I divided the numerator and denominator by $x^6$, ...
-6
votes
0answers
31 views

Instantaneous rate of change help please [on hold]

USING ALTERNATIVE DEFINITION FORM OF LIM F(x) = x/x-1, given x=2 Im stuck mid way through of simplfying! Someone help! I plugged in -(x/x-1) - 2/2-1 which is all over x-2 -common dinominator ...
3
votes
1answer
54 views

How to solve integrals where you can't factor a polynomial?

Hi there guys I don't know if the title of the question should be the one for this but the thing is that I'm trying to solve this integral $\int \frac {\frac 12-u^2}{2u^4-2u^2+1}$$du$ and I have this ...
0
votes
1answer
24 views

Optimizing the area of a rectangle with one side against a wall using the am-gm inequality

Given 300 meters of fence, how can I find the dimensions of a rectangle that is built against a wall the encloses the maximum area. I found this question in a calculus book and saw a simple solution ...
1
vote
2answers
31 views

How to find approximate value of $1.01e^{1.01({0.99) }^2} $?

I want to find the approximate value of $1.01e^{1.01({0.99) }^2}$ by using derivative. I tried choosing x=1 and $\delta x=0.01$ it didnt work. How can I start?
2
votes
3answers
36 views

how to calculate derivative of $f_n(x)=f \circ f … \circ f(x)$? Derivative on $f \circ f_{n-1}$ or $f_{n-1} \circ f$?

Denote $f_n(x)=f \circ f ... \circ f(x)$, the $n$th power of composition multiplication of $f(x)$. Assume $f(x)$ is differentiable for any order. $f(1)=1$, $f^{'}(1)=p$, $f^{''}(1)=q$ Question: Get ...
0
votes
1answer
39 views

$\lim \frac{\sqrt{x}-1}{x-1}=\frac{1}{2}$ as $x \rightarrow 1$ Epsilon-Delta proof help. [on hold]

I'm trying to sort out how to define epsilon-delta proofs and this one is a tricky one. Any ideas?
1
vote
1answer
24 views

concentration of volume of hypersphere

I am reading about features of volume of hyperballs, where I see two theorems, Most of the volume of the d-dimensional ball of radius r is contained in an annulus of width $O(r/d)$ near the ...
0
votes
3answers
43 views

Find the derivative of $y=\frac{\tan(x)}{1+\tan(x)}$

$$y=\frac{\tan(x)}{1+\tan(x)}$$ $$\frac{(1+\tan x)(\sec^2x)-(\tan x)(\sec^2x)}{(1+\tan x)^2}$$ I understand this first step but I struggle with simplifying to end up with only $$\sec^2x$$ in the ...
2
votes
1answer
57 views

Differential equation: $\ddot{y}(x) + \alpha\dot{y}^2(x) + \beta y(x) = 0$

I am interested in finding an approximate solution for this differential equation, since the exact analytic solution seems to not exist. I tried with Mathematica and it spits out nothing. ...
2
votes
2answers
61 views

$\int_{0}^{10} \int_{0}^{10} \int_{0}^{10} (x+y+z)$

I am trying to understand the $\int_{0}^{10} \int_{0}^{10} \int_{0}^{10} (x+y+z) dx dydz$ It has been a while since I took calculus $3$ but I thought the answer should be $150$ however its not. The ...
1
vote
2answers
31 views

For a series, can you always find a subseries whose sum is smaller in magnitude?

Let's say you got a series $a_n$, such that $\sum^\infty_{n=0}a_n=L>0$. For any $0 \lt K \le L$, can you always find a subsequence $b_n$ of $a_n$, such that $\sum^\infty_{n=0}b_n=K$? If not always, ...
0
votes
0answers
11 views

Integral of dot product of unit vector

I am having trouble with the following integral. $$\int \left(\bar{A} \cdot \hat{ F\left(\lambda\right)}\right)^p\mathrm ds$$ Note that the right hand side of the dot product is normalised. Where: ...
0
votes
1answer
28 views

Simplify the last step of this differentiation

$$y = (3x+4)^4(5x+3)^{-3}$$ I don't know how they simplify the last step to get to the answer of: $$\frac{3(3x+4)^3(5x-8)}{(5x+3)^4}$$
0
votes
2answers
61 views

How to calculate $\lim\limits_{x \to 0}((-2^x+1)^{\sin(x)})$

How can I solve the following limit problem? $\lim\limits_{x \to 0}((-2^x+1)^{\sin(x)})$ I can't find any approach to this one, although it really doesn't look that bad. Somehow my intuition ...
0
votes
0answers
9 views

Convolution using Integration

Using integration, how would I solve f(t) convolve g(t) given that $$f(t)=u(t)-u(t-5)$$ and $$g(t)=2[u(t)-u(t-1)]$$ I know it should be $$\int_0^6 f(\tau) \ast g(t-\tau)~ d\tau = ...
0
votes
0answers
13 views

Critical point of an ODE

I have been asked to deduce if an ODE has a critical point from drawing its isoclines and then sketching the integral curve. What exactly is a critical point of an ODE, and how would I deduce it from ...
1
vote
1answer
10 views

Volume Exponential Function

I should find the Volume received by rotating the region bounded by: $y = e^x $, $ y = 0 $,$ x = 0 $, $ x = 1 $ rotated around the x axis. I know how to find it by using the disc method but I could ...
6
votes
6answers
74 views

Closed form of $\sum\limits_{n=1}^\infty \frac{4^n(x+4)^{2n}}n$

Let $$S(x) = \sum_{n=1}^\infty \frac{4^n(x+4)^{2n}}n$$ 1. Find the radius of convergence. 2. Calculate $S(x)$. 3. Find $S^{(n)}(x)$ without computing the derivatives of $S(x)$. From the ...
1
vote
3answers
30 views

How is this limit solved?

The limit I need to solve is the following. I don't know if it's correct to treat the $+2$ and the $+1$ as insignificant, or if there's another way around it. $$ \lim \limits_{x \to \infty} ...
0
votes
1answer
18 views

Continuity of the function $f(x)=\lim\limits_{n \to \infty}\frac{x}{1+(2\sin(x))^{2n}}$

I was studying the continuity of the function: $f(x)=\lim\limits_{n \to \infty}\frac{x}{1+(2\sin(x))^{2n}}$ I understood that the function behave as $ f(x)=x \quad2 \sin(x) \leq 1 \\ f(x)=0 ...
1
vote
1answer
48 views

Integrating $\int^1_0 \dfrac{x^2e^{\arctan x}}{\sqrt{x^2+1}}$

This is a very hard integral that I am trying to solve. I’ve tried many substitutions, integration by parts, but I cannot evaluate this. Are there any other approaches I can take to solve this ...
2
votes
3answers
75 views

integrate $\int\frac{\sin x}{1+\sin^{2}x}dx$

$$\int\frac{\sin x}{1+\sin^{2}x}\mathrm {dx}$$ $$\int\frac{\sin x}{1+\sin^{2}x}\mathrm {dx}=\int\frac{\sin x}{2-\cos^{2}x}\mathrm {dx}$$ $u=\cos x$ $du=-\sin x dx$ $$-\int\frac{\mathrm ...
0
votes
0answers
20 views

Fourier limits integral

I´m trying to set up the limits of the Fourier Integral of this periodic function. My doubt is if the interval is from $-\infty$ to $\infty$ or from $0$ to $\infty$.
1
vote
3answers
21 views

Laplace transform for $-t\cos(2t)$

This Laplace transform exercise is giving me a headache. I was trying to use the definition of the Laplace transform but when I make the $u$ and $dv$ substitutions for the integration by parts I never ...
2
votes
2answers
43 views

Find $f$, when $f(1)=f(2)=1$ and $f'(x)\leq (x^2-x-1) e^x, \forall x\in [1,2]$

Let $f:[1,2]\to\mathbb{R}$ be a differentiable function such that $f(1)=f(2)=1$ and $f'(x)\leq (x^2-x-1) e^x, \forall x\in [1,2]$. Find $f(x)$. I am pretty sure that from things I have tried ...
-1
votes
3answers
59 views

Problem with Indefinite Integral $\int \frac {\cos^5x}{ 16(\cos^4x+\sin^4x)}dx$

Hello guys I'm totally lost in this indefinite integral, i'm just looking for advices/tips $$\int \frac {\cos^5x}{ 16(\cos^4x+\sin^4x)}dx$$ Should I begin with universal substitution? or there is ...
0
votes
0answers
13 views

Singularities of composite function

Given a smooth, compact manifold $M$ (of dimension much less than $n$) and two maps $f:\mathbb{R}^n \rightarrow M$, $g:M\rightarrow \mathbb{R}$, I want to understand the topology of the critical set ...
2
votes
1answer
119 views

Calculate $I=\int_0^{1}\frac{1+x}{x^2+x+1}\log\left({\frac{x}{1-x}}\right)\,\mathrm dx$ without using complex analysis

Calculate $$I=\int_0^{1}\frac{1+x}{x^2+x+1}\log\left({\frac{x}{1-x}}\right)\,\mathrm dx$$ without using complex analysis. How to calculate without using the residue theorem? The correct answer ...
1
vote
2answers
36 views

Speed of a parametric function?

I know speed = |velocity| Why is speed of parametric defined as $$speed = \sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}$$ How is this derived? What is the principle here? Is ...
0
votes
2answers
56 views

A weaker than “zero derivative” condition implies that a function is constant?

Let $f:(a,b)\to R$ be a continuous function such that $\limsup\limits_{n\to \infty}\frac{f(x_n)-f(x_0)}{|x_n-x_0|}\leq 0$ for every $x_0\in(a,b)$ and sequence $x_n$ converging to $x_0$ such that ...
0
votes
1answer
26 views

Changing the order of a double integration ? $\int_{-5}^{5}dx\int_{-7}^{\sqrt{25-x^2}}f(x,y)dy$

I've been doing an example of changing the order of a double integral and I'm not sure if I did it right. I would really appreciate if someone would check if my solution is right and correct any ...
1
vote
1answer
35 views

How to show that this is a martingale?

Let $H_s$ be a predictable and bounded process. How can I show that $$M_t = \int_0^t H_s \, dW_s$$ is a martingale? Clearly since $H_s \in L^2_\text{loc} (W)$ we have that $M_t$ is a local ...
0
votes
0answers
11 views

Quasi-linear partial differential equations. Solving them.

This is what I have as a quasi-linear partial differential equation:$$u(x_1,...,x_n), \ \ \ \ \sum_{i=1}^{n}A_i(X,u) \frac{\partial u}{\partial x_i}=A_{n+1}(X,u) \ \ \ (1)$$ Then it says let ...
-6
votes
0answers
57 views

Let the limit be limit of $(\arctan(x))^x$ as $x\to0$ then [on hold]

Let the limit be $\lim_{x\to0}(\arctan(x))^x$ then (A) evaluate the limit using technology (B) justify your result using "known special limits" (c) justify your result by using the "exponentiating ...