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

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2
votes
1answer
81 views

Question about the derivative of distance vs displacment.

Displacement and Distance are not exactly the same things. I have seen everywhere on the Internet that the derivative of a distance function is it's velocity function, however to my understanding this ...
1
vote
2answers
25 views

what is the value of $\theta$ used in calculate volume bounded by $z=x^2+y^2$ and $x^2+y^2=2x$

This is an example from my textbook, it explain everything well except the reason why $\frac{\pi}{2} < \theta < \frac{-\pi}{2}$ but not $2\pi < \theta < 0$. It's not explained and I can't ...
1
vote
1answer
50 views

Fourier series of constant on $2\pi$ intervals

I want to find a fourier expansion of only sines representing $g(x) = 1$ on the interval $[0, \pi]$. So I extend the function on $[-\pi, \pi]$ such that it is odd, and calculate $$b_k = \frac 1\pi \...
1
vote
2answers
71 views

Maxima and Minima of Functions of Two Variables $ f(x,y) = e^{x+y^2}\cdot y $ and $ f(x,y) = e^{x^2-y^2}\cdot y $

I'm having trouble finding the local minimum and maximum of the next functions: $$1. f(x,y) = e^{x+y^2} \cdot y $$ $ f_x'= (e^{x+y^2}\cdot y) ; $ $ f_y'= (e^{x+y^2}(1+2y^2)) $ $$ 2. f(x,y) = e^{...
0
votes
3answers
74 views

How do I solve this deceleration problem?

Question: A car is traveling at 100km/hr, when the driver sees an accident 80 meters ahead. What constant deceleration is required to stop the car in time to avoid a pileup? So far I have approached ...
0
votes
2answers
218 views

Is $\int\left(\sin^2x + \cos^2x\right)\;dx = \int 1 \; dx$ ?

I have just begun my 2nd calculus course and so far have just been applying the substitution method for solving anti derivatives and other basic rules. I have a question that is probably very easy to ...
12
votes
4answers
518 views

Asymptotic behavior of the partial sums $\sum\limits_{k=1}^{n}k^{1/4} $

What is the asymptotic behavior of the sequence: \begin{equation} s_n=\sum_{k=1}^{n}k^{1/4} \end{equation} when $n\to \infty$?
9
votes
5answers
789 views

Prove that $f$ continuous and $\int_a^\infty |f(x)|\;dx$ finite imply $\lim\limits_{ x \to \infty } f(x)=0$

I'd love your help proving the following claim: If $f$ is continuous and $\int_a^\infty |f(x)|\;dx$ is finite then $\lim\limits_{ x \to \infty } f(x)=0$. Here the counter example of all these ...
2
votes
2answers
27 views

Finding an integral $\int g(x)^j dx $ from $\int g(x)^2 dx $

let $I = \int_0^1 g(x)^2 dx $, where $g$ is a real valued function. With this information is it possible to give an upper bound for $\int_0^1 g(x)^j dx $? Here $j$ is a natural number. When $j=1$ I ...
5
votes
3answers
12k views

Integrating $e^{f(x)}$

can someone tell me a way of integrating functions like $e^{f(x)}$ I have a specific case: $\int e^{-3x}\,\mathrm{d}x$ PS: I'm not looking for the answer of this, but the way of doing it. Thanks ...
0
votes
1answer
65 views

Asymptotics of function of $n^a$, $2^n$ and $\sqrt{n}$, when $n\to\infty$

I am having trouble with estimation of the following$$\frac{n^a}{2^{n-\frac{\sqrt n+1}{2}}(1-\frac{1}{2 \sqrt n})^{n-\frac{\sqrt n-1}{2}}} $$ Where $n \in N$ and $a$ is a real number greater or equal ...
-2
votes
3answers
131 views

How to bound $ax^2+2bxy+cy^2$ by multiples of $x^2+y^2$ [closed]

Let $(x,y)\in[0,1)\times[0,1)$ cum $x^2+y^2<1$. Are there any $\mu\geq\lambda>0$ such that $$\lambda\xi_1^2+\lambda\xi_2^2\leq(1-x^2)\xi_1^2+2xy\xi_1\xi_2+(1-y^2)\xi_2^2\leq\mu\xi_1^2+\mu\xi_2^...
1
vote
2answers
355 views

Limit of $x^2\cos(1/x^2)$ when $x\to0$ by squeeze theorem

How can I argue that $$\lim_{x \to 0} x^2 \cos\left(\frac{1}{x^2}\right) = 0$$ I understand I have to use a squeeze theorem and that one piece goes to zero but I'm not sure how to tackle this ...
2
votes
1answer
64 views

$0<\int_0^\infty\frac{\sin t}{\ln(1+x+t)} dt<\frac{2}{\ln(1+x)}$

This is my first time posting so please excuse me if I don't follow the proper etiquette. This one is a rather hard problem that was assigned to me for my calculus 2 class. Thank you for your help! ...
0
votes
1answer
42 views

How to prove this two separations of connectedness is equivalent?

Definition 1$\quad$ A metric space $E$ is connected if it cannot be written as the union of two nonempty separated sets (in $E$). Definition 2$\quad$ A metric space $E$ is connected if it cannot be ...
0
votes
1answer
399 views

On the pointwise limit of $\sqrt[n]{p_n(x)}$ when $n\to\infty$, for some polynomials $(p_n)$

For every $n$, I have a polynomial $p_n(x)=a^{(n)}_{n-1}x^{n-1}+a^{(n)}_{n-2}x^{n-2}+\dots+a^{(n)}_0$ (the $n$ in the exponent of the coefficients is merely an index). I can show that $\lim_{n\to\...
1
vote
2answers
117 views

Summing Lerch Transcendents

The Lerch transcendent is given by $$ \Phi(z, s, \alpha) = \sum_{n=0}^\infty \frac { z^n} {(n+\alpha)^s}. $$ While computing $\sum_{m=1}^{\infty} \sum_{n=1}^{\infty} \sum_{p=1}^{\infty}\frac{(-1)...
0
votes
1answer
50 views

How to show $G$ is a perfect set that contains no rational points?

For $E:=[0,1]$, since $\Bbb Q\cap E$ is enumerable, let it be $\{q_1,q_2,\cdots\}$. If I remove the elements of $V_1:=(q_1-\frac1{10},q_1+\frac1{10})$ from $E$, I obtain a closed (and compact) set $...
5
votes
6answers
191 views

Finding $\frac {a}{b} + \frac {b}{c} + \frac {c}{a}$ where $a, b, c$ are the roots of a cubic equation, without solving the cubic equation itself

Suppose that we have a equation of third degree as follows: $$ x^3-3x+1=0 $$ Let $a, b, c$ be the roots of the above equation, such that $a < b < c$ holds. How can we find the answer of the ...
1
vote
1answer
15 views

Question about r(t), movement along line

So I'm studying for an exam in calculus when i came across the concept of objects moving along a curve. I have a general idea of how to calculate speed, velocity and such when r(t)(position vector I ...
0
votes
2answers
54 views

Piecewise $\mathscr C^1$ and piecewise continuous

I'm a little bit confused in piecewise continuity of a function. Say, if we have an odd function like $f(x) = x$ defined on the open interval $(0, \pi)$. We then extend it to a period $2\pi$ function ...
1
vote
3answers
362 views

First principle derivative of a square root and conjugates

I'm trying to find the derivative of the equation: $$g(x)=\sqrt {x+2}-3x^2$$. I can find the solution just fine using the power rule but am finding trouble with First Principles. Essentially, I ...
1
vote
5answers
386 views

Tangent line parallel to another line

At what point of the parabola $y=x^2-3x-5$ is the tangent line parallel to $3x-y=2$? Find its equation. I don't know what the slope of the tangent line will be. Is it the negative reciprocal?
0
votes
1answer
131 views

How to use the chain rule?

Following my previews question : From the comments to the answer I feel that I don't understand how to use the chain rule. From what I understand the chain rule sais : $F(t)=f(x(t),y(t))\implies F'(t)...
1
vote
1answer
36 views

General Chain Rule

Product Rule: For a two $\mathcal{C}^{\infty}(\mathbb{R})$ functions, $u(x)$, $v(x)$ we have $$\frac{d^k}{dx^{k}}[u(x)v(x)]=\sum_{j=0}^k \binom{k}{j}\frac{d^j}{dx^j}[u(x)]\frac{d^{k-j}}{dx^{k-j}}[v(...
3
votes
1answer
179 views

How to integrate this : $ \int \cot(5x) \tan(2x) \mathrm{d}x$

Methods to integrate this integral: $$\int \cot(5x) \tan(2x) \mathrm{d}x$$ I have tried several methods, step by step, and they have led to invalid results. Helpful hints or processes are welcome. ...
1
vote
1answer
26 views

Question regarding path independence

I've been wracking my brain to try and figure out why the following works: The question is asking whether $$\int F \, dr $$ is independent of path. We have a hint, that is (compute $$\int_a F \, dr $$...
0
votes
1answer
18 views

Coefficients of general Fourier Series

I know how to compute coefficients of Fourier Series on an interval of $2\pi$. But in this case I need to find the sine series of $f(x)=b$ on the interval $x \in [-L,L]$. Can someone please let me ...
0
votes
1answer
70 views

basic calculus/analysis question. why does the multivariable chain rule work?

Say $f$ is a function of $x(t)$ and $y(t)$ $$\frac{ df}{dt} = \frac{ \partial f}{\partial x} \frac{ dx}{dt} + \frac{ \partial f}{\partial y} \frac{ dy}{dt}$$ why is it so additively symmetric? (The ...
10
votes
3answers
96 views

Global Minimum of $f(a) = \int _{-\infty}^{\infty} \exp\left(-|x|^a\right)dx, a\in(0,\infty)$

Playing around with the Standard Normal distribution, $\exp\left(-x^2\right)$, I was wondering about generalizing the distribution by parameterizing the $2$ to a variable $a$. After graphing the ...
4
votes
2answers
65 views

Example of a non-polynomial function $f: \mathbb{R} \to \mathbb{R}$ such that $f'(x)$ is negative for $x<0$ and positive for $x \ge 0$.

I have a bunch of polynomial functions example easily (e.g. $x^2$), but have trouble coming up with a non-polynomial function. I was thinking of defining $f(x) = e^{-x}$ for $x<0$ and $f(x) = e^{x}...
7
votes
4answers
229 views

Taking the half-derivative of $e^x$

While attempting to teach myself the fractional calculus, I encountered a tragically early roadblock. For non-power rule fractional derivatives, I am having a lot of trouble evaluating for a closed ...
2
votes
1answer
54 views

What is the number of distinct elements in $S$?

Allow for these values: $$A = \begin{pmatrix} \cos \frac{2 \pi}{5} & -\sin \frac{2 \pi}{5} \\ \sin \frac{2 \pi}{5} & \cos \frac{2 \pi}{5} \end{pmatrix} \text{ and } B = \begin{pmatrix} 1 &...
0
votes
1answer
35 views

Need help with this question concerning compact spaces

Let the set be given like in the following manner: $$\{x_n: n\in\mathbb N\}\subset \mathbb{R^n}$$ $$l^2=\left\{\{x_{n}\}_{n=1}^{\infty}\,\Big|\, \sum_{n=1}^{\infty}|x_n|^2<\infty\right\}.$$ Prove ...
2
votes
1answer
109 views

Definite integration by induction

$U_n= \int\frac{x^n}{((x(1-x))^{0.5}}$ where $0<x<1$ Prove that $2nU_n=(2n-1)U_{n-1}$ My work I did $U_0=\pi, u_1=\pi/2$ so its true for $n=1$
0
votes
2answers
57 views

Moment of inertia about the origin of an ellipsoid?

Find the moment of inertia about the origin of an ellipsoid. Heres what I did but I believe it is incorrect: $$I_o= \iiint_{V_e}{(x^2 +y^2 +z^2)\rho dx dy dz} $$ Making Substitution of $aX=x \ bY=y \ ...
0
votes
0answers
16 views

End-point as point of tangency

Can an end-point be a point of tangency? For example in the function $f(x)=8x^{3/2}$ can the point $(0,0)$ be a tangent point?
4
votes
7answers
316 views

Differentiate the Function: $y=2x \log_{10}\sqrt{x}$

$y=2x\log_{10}\sqrt{x}$ Solve using: Product Rule $\left(f(x)\cdot g(x)\right)'= f(x)\cdot\frac{d}{dx}g(x)+g(x)\cdot \frac{d}{dx}f(x)$ and $\frac{d}{dx}(\log_ax)= \frac{1}{x\ \ln\ a}$ $(2x)\cdot [\...
4
votes
2answers
195 views

Evaluating $\sum_{n=0}^{\infty } 2^{-n} \tanh (2^{-n})$

Reading in some tables pages I found $$\sum _{n=0}^{\infty } 2^{-n} \tanh \left(2^{-n}\right)=\tanh (1) \left(1+\coth ^2(1)-\coth (1)\right)$$ I try to split in two sum using the roots of the $\tanh$...
1
vote
2answers
51 views

Upper and lower Riemann sums problem

Let $c > 0$ and $f(x) = x, x\in [0,c].$ Let $P = \{x_0, x_1, x_2,...,x_n\}$ be a partition of $[0,c]$ with $x_i = \frac{i}{n}c, i = 0,1,2,...,n.$ Find $U(P,f).$ Find $\lim_{n \to \infty} U(P,f).$ ...
3
votes
2answers
119 views

If $|u+v| = |u| + |v|$ then $u = \lambda v$. How do I prove $\lambda \ge 0$?

I'm trying to prove that if $|u+v| = |u| + |v|$ implies $u = \lambda v$ for $\lambda \ge 0$. To this end I have $|u + v|^2 = (|u| + |v|)^2 \Rightarrow |u|^2 + |v|^2 + 2|u||v| = |u|^2 + |v|^2 + 2u\...
1
vote
2answers
56 views

How do I solve a double integral with an absolute value?

Given the following integral $$\int_{y=0}^1 \int_{x=0}^1|x-y|(6x^2y) \, dx \, dy$$ how do I change the limits of integration? According to my textbook, it is $$\int_{y=0}^1 \int_{x=y}^{1}(x-y)(6x^...
2
votes
1answer
720 views

What does 'monotonically related' mean?

I am reading this paper. On page 2, it says that "the likelyhood is monotonically related to the average per-symbol log likelyhood." I know what a monotonic function is. But what does 'monotonically ...
1
vote
2answers
64 views

Maximizing area under curve

I came across this problem in TMH mathematics for jee.I tried finding the derivative to the curve but I got stuck while evaluating the area of triangle in terms of tangent to the curve.How should I ...
2
votes
1answer
169 views

Check my proof of a property of the greatest integer function?

Prove that $\forall n \in \mathbb{Z}, \lfloor x + n \rfloor = \lfloor x \rfloor + n $. Proof: Let $K = \{\ k\ |\ k\in\mathbb{Z},\ k \leq x+n\}$. Then, by definition, $$ \lfloor x + n \rfloor = j = \...
4
votes
2answers
131 views

The converges of $ \sqrt { 2 } +\sqrt { 2-\sqrt { 2 } } +\sqrt { 2-\sqrt { 2+\sqrt { 2 } } } + …=$

I would like to know wheather this series converge or diverge? $\sqrt { 2 } +\sqrt { 2-\sqrt { 2 } } +\sqrt { 2-\sqrt { 2+\sqrt { 2 } } } +\sqrt { 2-\sqrt { 2+\sqrt { 2+\sqrt { 2 } } } } +......
39
votes
3answers
3k views

Addition is to Integration as Multiplication is to ______

Addition is to Integration as Multiplication is to ______ ? Everyone knows that definite integration is "a way to sum continuum-many terms" in a rough sense. Can we "multiply continuum-many factors" ...
0
votes
6answers
137 views

Factoring $ x^2 + x +1 > 0$ from Spivak Calculus exercise

Hi!! I found me in trouble when I saw the solution of a simple inequality, that can be found at the end of the first chapter, that is the exercise 4 - (viii): $x^2+x+1 > 0$. Very easy to solve I ...
1
vote
4answers
151 views

Is $|z-i| = |z+i|$?

I computed a Mobius transformation $-\frac{z-i}{z+i}$ that maps the upper half plane to a disk, with i mapping to the center of the disk, $w = 0$. How do I know that the disk is a unit disk and not ...
1
vote
1answer
80 views

All $f(x)$ on $[0,1]$ such that center of mass of the function (uniform density) is on its graph

So, as the title describes, I'm trying to find a way to express all $y=f(x)$ differentiable on $[0,1]$ such that the center of mass of the function, assuming it has uniform density, will be a point on ...