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

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0
votes
1answer
44 views

Calculate the the integral $\int_1^2(\sqrt{1+4v}-2\sqrt{v})dv$

I need to calculate the following integral : $$\int_1^2(\sqrt{1+4v}-2\sqrt{v})dv$$ This is what I did : $$\int_1^2(\sqrt{1+4v}-2\sqrt{v})dv=|t=1+4v,dt=4dv,\frac{dt}{4}=dv|=\frac{1}{4}\int_5^9\sqrt{...
4
votes
1answer
137 views

Finding the closest point to a set of lines in 2D

I would need to write an algorithm to find the closest point to a set of lines. These lines are infinite and are not parallel between each other. Closest point means that point where the sum of the ...
1
vote
1answer
66 views

Negative volume of solid of revolution around x axis.

There was a question on my exams yesterday about the volume of revolotion of the solid generated when the circle $x^2+(y-3)^2=1$ rotates around x axis. That moment I thought it would be best if I used ...
1
vote
0answers
25 views

asymptotic smooth kernel log(|x-y|)

I am currently trying to show that the function $\log(|x-y|)$ is an asymptotic smooth kernel function, in the sense that: for $x,y \in \mathbb{R}^2$ there exist constants $C_{1},C_{2} > 0$ and an ...
0
votes
4answers
59 views

How to find $\frac{dy}{dx}$ from $x^3y^6 = (x+y)^9$ using implicit differentiation?

How to find $$dy/dx$$ from $$x^3y^6 = (x+y)^9$$ using implicit differentiation? I tried solving but I ended up with solution that does not agree with my textbook answer. How can I get $$dy/dx = y/x$$?
2
votes
1answer
52 views

Reduction formula doubt.

If $$I_n = \int{(\frac{1}{a^2+x^2})^{n}}dx$$ Prove that:$$I_n = \frac{x}{2a^2(n-1)(a^2+x^2)^{(n-1)}}+\frac{2n-3}{2(n-1)a^2}I_{n-1}$$ I used Ibp but couldn't get such a relation. Please help me. Also,...
2
votes
2answers
67 views

If $\lim_{x \rightarrow 1} f'(x)=2015$. Prove that $f$ is differentiable at $x=1$. [duplicate]

Let $f:[0,2] \rightarrow \mathbb R$ be a continuous function such that, $f$ is differentiable everywhere on $[0,2]$ except possibly at $x=1.$ If $\lim_{x \rightarrow 1} f~'(x)=2015$. Prove that $f$ is ...
1
vote
2answers
29 views

A solution of a differential equation of first order in the large-variable limit

The differential equation reads: $ \dfrac{\partial R (t)}{\partial t} = \dfrac{c_2}{R^2} + \dfrac{c_3}{R^3} + O(R^{-4})$, Where $c2 > 0$ and $c3 > 0$, how to get the solution of the ...
2
votes
1answer
42 views

Prove an inequality for a specific Ordinary Differential Equation

Let $a \in \mathbb R$ Consider the differential equation $$\frac{d^9y}{dt^9}-\dfrac{dy}{dt}+ay=0 \tag 1$$ suppose $\varphi : \mathbb R \rightarrow \mathbb R$ is a solution of $(1)$ on $\mathbb R$. ...
0
votes
0answers
64 views

Distance traveled by a projectile

Like the title says, I want to find the distance traveled by a particle in projectile motion (2-D). I tried rectifying the curve above the $x$-axis and finding the length of the curve as $$\int \sqrt{...
1
vote
1answer
62 views

Finding the limit that involves Fourier coefficients,

Given the function $f(x) = 1 - \dfrac{|x|}{\pi}$, I had computed its Fourier coefficients, using integration by parts and got: $$ a_n = \begin{cases} 0, & \text{for $n$ even}, \\[6pt] \dfrac{2}{n^...
0
votes
1answer
116 views

Suppose f(x) is continuous and differentiable on (a,b) and lim|f(x)|= infinity as x approaches a+. Prove that f'(x) is unbounded on (a,b)

so far for this problem i have said let c,d be in (a,b) and $d\gt c$ suppose f'(x) is bounded by M( i want to eventually get a contradiction) $\mid f'(x)\mid= \frac{f(d)-f(c)}{d-c} \le M$ but not ...
2
votes
2answers
121 views

How can I use a precise definition to find values of delta that correspond with given epsilon values

I have been given this problem: For the limit $$\lim_{x\to 2}({x^3-3x+4})=6$$ illustrate "Definition 2" (I have included this below) by finding values of $\delta$ that correspond to $\varepsilon=0.2$ ...
1
vote
2answers
63 views

How to compute the following double integral?

I would like to solve the following: $$ \int_0^1 \int_0^2 \sqrt{1+4x^2+4y^2} \,dy \, dx$$ I tried to use trig substitutions by letting: $$y= \sqrt {\frac{1+4x^2}{4}} \cdot \tan(\theta)$$ $$ \partial ...
3
votes
4answers
165 views

How to compute the derivative of $\sqrt{x}^{\sqrt{x}}$?

I know have the final answer and know I need to use the natural log but I'm confused about why that is. Could someone walk through it step by step?
0
votes
4answers
51 views

Compute the derivative of the function $f(x)=\left(1-\frac1x\right)^x$, $x > 1$, and conclude that $f(x)$ is monotone increasing

So for this question the derivative for this function is $$ f'(x)= \left(1-\frac1x\right)^x\left[\log\left(1-\frac1x\right)+\frac{x^2}{(x-1)}\right] $$ but I am not sure how to use the derivative to ...
7
votes
3answers
266 views

How to solve the Few Scientists Problem (big word problem) in its general form?

I'm trying to figure out how to solve this word problem. I'm pretty sure it involves calculus or something even harder, but I don't know how to solve the general form. Let me start with the concrete ...
0
votes
1answer
23 views

Is the path with the highest average value the same as the path with the lowest total difference from the function's maximum?

If you have a function $f(x, y)$ and you draw two paths (curved lines) from points A to B where: The first path is the path with the highest average value (if the value at distance $d$ from the ...
2
votes
1answer
89 views

How can the limit exist for a function on an interval (a,b) but not be continuous on that interval?

This is from a practice test true/false question. The statement I was given is that If $\lim_{(x,y)→(a,b)} f(x, y)$ exists, then $f(x, y)$ is continuous at $(a, b)$. I put True but the answer ...
1
vote
0answers
30 views

Zeta 3 quick convergent series

I found in this paper.. $$\zeta (3)-\left(\frac{16}{3} \sum _{k=0}^1 \frac{\text{csch}\left(\frac{\pi (2 k+1)}{\sqrt{2}}\right)}{(2 k+1)^3}-\frac{8}{3} \sum _{k=1}^1 \frac{(-1)^k}{\left(e^{\sqrt{2} \...
5
votes
4answers
533 views

Limit of a geometric series

I am trying to practice for a Precalculus exam, but there's this black sheep that I just can't figure out. $$\lim_{n\to \infty} \dfrac{\sqrt[n]{e}+\sqrt[n]{e^2}+\sqrt[n]{e^3}+...+\sqrt[n]{e^n}}{n}$$ ...
5
votes
2answers
167 views

Prove that, $f'(0) \ge -\sqrt{2}$ for a function $f$ satisfying some conditions on $(-1,1)$.

Let $f:(-1,1)\to \mathbb{R}$ be a twice differentiable function such that, $f(0)=1$, $f'(x)≤0$, $f(x)≥0$ and $f''(x)≤f(x)$ for all $x≥0$. Prove that, $f'(0)≥-\sqrt2$ Progress: I was able to prove $f'...
0
votes
2answers
25 views

Is there a systematic way of dealing with functions without a well defined limit approaching infinity (such as $\sin(x)$) as part of a limit problem?

Consider $\lim\limits_{x \to \infty} e^{-x}\sin{x} $, by the product rule of limits, we can split this up as: $(\lim\limits_{x \to \infty} e^{-x})(\lim\limits_{x \to \infty}\sin{x})$, but while $\lim\...
1
vote
2answers
42 views

How do I prove that this sequence converges? $\sum_{0}^{\infty} \frac{n-3}{n+2}^{n^2-n}$

I've been having trouble checking whether this sequence converges or not: $$\sum_{n=0}^{\infty} \frac{n-3}{n+2}^\left({n^2-n}\right)$$ At a first glance I thought I should try the root test but that ...
0
votes
6answers
267 views

What does the dx mean in an integral? [duplicate]

I know dy/dx for example means "derivative of y with respect to x," but there's another context that confuses me. You will generally just see a dx term sitting at the end of an integral equation and I ...
0
votes
2answers
48 views

Prove that the function $f(n)=n^3+(\frac{n}{2^n})^5$ satisfies some property (where $n\in\Bbb{N}$)

I am stucked at this problem: Prove that the function $f(n)=n^3+(\frac{n}{2^n})^5$ satisfies the property (where $n\in\Bbb{N}$): there exists $c\in(0,1)$ and $n_0\in\Bbb{N}$ such that for all $n_0\...
1
vote
1answer
54 views

If $f:\mathbb{R}\rightarrow\mathbb{R}$, $f\in C^{\infty}(\mathbb{R})$ and $f(0)=0$ then $\frac{f(x)}{x}\in C^{\infty}(\mathbb{R})$

If $f:\mathbb{R}\rightarrow\mathbb{R}$, $f\in C^{\infty}(\mathbb{R})$ and $f(0)=0$ then $\frac{f(x)}{x}\in C^{\infty}(\mathbb{R})$. Following ther is what i did: the Maclaurin series for $f(x)$ is $$...
-3
votes
1answer
32 views

What is the radius of the next convergence

What is the radius of convergence of: $$\sum_{n=2}^\infty \frac{(1-x)^{5n}}{n5^n\ln(n)}$$ I know that I should use power series but how?
0
votes
1answer
40 views

Points in a given volume/Area

I have a rectangular prism(3D bounding box) for which i have the point(i.e center of gravity) and the height,width,depth dimensions . Given these parameters, is it possible to find all the points that ...
1
vote
1answer
49 views

Issues with solving PDE

It's been a while since I've had to solve the heat equation, and so I am having a slight issue. The question is as follows: A long, hollow, rigid tube, of length $L$ and constant cross section is ...
0
votes
1answer
130 views

Finding coefficients of a function, given a list of points on the function

Given $f(x) = ax^n + bx^{n-1} + ... + cx + d$, a list of points, and a specification of a tangent line (point $p_t$ and equation) find $a, b, ..., c, d$ s.t. $f(x)$ passes through each point the ...
3
votes
2answers
84 views

How to prove that this sequence converges? $\sum_{n=1}^{\infty} \frac{1}{n\ln^2(n)}$

I'm trying to prove this sequence converges: $\sum_{n=1}^{\infty} \frac{1}{n\ln^2(n)}$ I noticed that this is continuous function which its derivative is always less than $0$ for $ x \gt 1 $, so I ...
-1
votes
1answer
17 views

Uniformly converge

fn(x) is a sequence of continuous function at [a,b] that uniformly convergent to f(x) at [a,b]. Prove that for all p>0: Lim(integral |fn(x)-f(x)|^p from a to b)dx when n to infinity =0 Sorry for my ...
0
votes
1answer
26 views

showing $\int_a^x |f(t)-g(t))|dt \leq (x-a) \max_{a\leq t\leq b} |f(t)-g(t)|$

How to show this? $x\in[a,b]$, $f,g$ are continuous. $$1)\quad\int_a^x |f(t)-g(t))|dt \leq (x-a) \max_{a\leq t\leq b} |f(t)-g(t)|$$ Someone suggested: $$2)\quad\int_a^x |f(t) - g(t)| dt \leq \int_a^...
0
votes
2answers
38 views

The volume of a cone is $18π m^3$ Find the minimum length of the slant edge

Using pythagoras theorem, I received.. $$l (Slant)=\sqrt{r^2+h^2}$$ Using the volume of a cone formula in terms of $h$..$$h=\dfrac{54}{r^2}$$ I then subbed this into the 1st equation and diffrenciated ...
0
votes
2answers
83 views

Comparing the greatest values of two functions (Derivatives)

I've tried doing this task, and for this kind of task I should be using derivatives. When I done all the calculus, everything I got were some weird result which I do not know how to compare. Task ...
1
vote
1answer
38 views

Find antiderivative of $8\sin^3(2x)\cos(2x)$

I was tasked with finding the antiderivative of $8\sin^3(2x)\cos(2x)$ This is what I have $$4\sin^4(2x)-\int24\sin^3(2x)\cos(2x)\,dx$$ I don't know the step after that.
0
votes
1answer
29 views

Implication Of limit f(x)=L>0

I know that there is a theorem saying that if $lim_{x\to a}f(x)=L>0$ there is $\epsilon$ neighborhood that is $>0$. Than I came across the following: if $lim_{x\to a}f(x)=L>0$ there is $\...
0
votes
1answer
32 views

what is the limit of the following sequence when $n \longrightarrow \infty$?

what is the limit of the sequence $a_n=\frac{n!(0.5)^{n-1}}{\sqrt{n}\{[(n-1)/2]!\}^2}$ when $n \longrightarrow \infty$. It seems that the right answer is $\frac{1}{\sqrt{2\pi}}$, but I cannot figure ...
2
votes
2answers
160 views

Slow decreasing function that exhibits asymptotic behaviour.

I am currently doing some work on modelling the effects of treated nets usage on mosquito populations. Nets do not retain their maximum efficacy forever. They lose their chemical efficacy after about ...
0
votes
2answers
67 views

calculate the volume

There is a triangular prism with infinite height. It has three edges parallel to z-axis, each passing through points $(0, 0, 0)$, $(3, 0, 0)$ and $(2, 1, 0)$ respectively. Calculate the volume within ...
24
votes
4answers
376 views

How to compute $\int_0^\infty \frac{1}{(1+x^{\varphi})^{\varphi}}\,dx$?

How to compute the integral, $$\int_0^\infty \frac{1}{(1+x^{\varphi})^{\varphi}}\,dx$$ where, $\varphi = \dfrac{\sqrt{5}+1}{2}$ is the Golden Ratio?
0
votes
1answer
707 views

Find $dy/dx$ by implicit differentiation: $x = \sec (1/y)$

I tried to solve it by: Taking the derivative of both sides using Chain Rule: $1 = \sec(\frac{1}{y})\tan(\frac{1}{y}) \frac{1}{y'}$ Multiplying both sides by the derivative of $y'$ to isolate $y'$: ...
2
votes
2answers
69 views

evaluate $\int_0^{2\pi} \frac{1}{\cos x + \sin x +2}\, dx $

This is supposed to be a very easy integral, however I cannot get around. Evaluate: $$\int_0^{2\pi} \frac{1}{\cos x + \sin x +2}\, dx$$ What I did is: $$\int_{0}^{2\pi}\frac{dx}{\cos x + \sin x +...
0
votes
1answer
104 views

How treat integral of an absolute value function, over a symmetric interval,

I am computing Fourier coefficients for some function f and have a question about how to treat the integral of |x|*cos(nx), over the interval [-$\pi$, $\pi$]. Is there symmetry to apply here, ...
0
votes
0answers
45 views

About Bâle's problem Euler's proof [duplicate]

Can someone explain me with the more details as possible the factorisation in the Euler's proof of $$\sum_{k=1}^{\infty} \frac{1}{k^2}=\frac{\pi^2}{6}$$ In wikipedia they only say that $$\frac{\sin(x)...
1
vote
1answer
111 views

Choosing a surface that makes the flux of F maximal,

For a closed surface S in $R^3$, consider the flux of F, given by the usual flux integral. For what choice of S will the flux be maximal? So, I want to apply the divergence theorem and instead look ...
3
votes
2answers
39 views

proof with complex integration by u-substitution

If $f$ is continuous in $[0,\pi]$, use the substitution $u = \pi - x$ to show that $\int_0^{\pi} xf(\sin x)dx = \frac{\pi}{2}\int_0^{\pi} f(\sin x)dx$ Not having much idea where to begin, I naively ...
1
vote
0answers
13 views

parts per million offset

If the maximum allowable deviation from a frequencyset to be (60 GHz) is 20 parts per million (ppm). Then what is the deviation in Hz? My solution: $\frac{20}{1000000}* 60 *10^6 = \pm 1200 Hz$ But ...
2
votes
5answers
78 views

Evaluate $\lim_{x\to 0}\frac{\sqrt[m]{\cos\alpha x}-\sqrt[m]{\cos\beta x}}{\sin^2x},m\in \mathbb{N}$

Evaluate $$\lim_{x\to 0}\frac{\sqrt[m]{\cos\alpha x}-\sqrt[m]{\cos\beta x}}{\sin^2x},m\in \mathbb{N}$$ I used L'Hospital's rule, but that didn't work. Could Taylor series be used? I don't know how ...