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

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28 views

Problem 17 in chapter 3 of Spivak book

If $f(x)=0$ for all $x$, then f satisfies $f(x+y)=f(x)+f(y)$ for all $x$ and $y$, and also $f(xy)=f(x)f(y)$ for all $x$ and $y$. Now suppose that $f$ satisfies these two properties, but that $f(x)$ is ...
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0answers
13 views

Closed form of integrals containing double exponentials

Are there closed forms for the following integrals? $$\begin{align} I_1(w) & = \int_{-\infty}^{\infty} \frac{\exp(-we^y)}{y^2+\pi^2} dy, \\ I_2(w) & = \int_{-\infty}^{\infty} ...
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1answer
30 views

Question about epsilon-delta definition of limits.

In Chapter 1: Functions and limits, 1.7 The Precise Definition of a Limit, Let $f$ be a function ... the limit of $f(x)$ as $x$ approaches $a$ is $L$, and we write $$\lim_{x\to a }f(x)=L$$ if ...
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4answers
27 views

What algebraic manipulation is used to express the solution to this integral?

According to WolframAlpha, $$ \int \frac{2}{\sqrt{x^2 + 4}} dx = 2 \sinh^{-1}\left(\frac{x}{2}\right) + c.$$ I'm wondering how this was obtained, as I got the following: Let $x = 2\tan\theta$. Then ...
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2answers
31 views

Find the largest $\delta \gt 0$ such that $|{\sqrt{x - x^2} - 0}| \lt\epsilon$ when $1 - \delta \lt x \lt 1$.

Let $f(x) = \sqrt{x - x^2}$ for $0 \leq x \leq 1$. I found $L= \displaystyle\lim_{x\rightarrow1,x\leq1}=0$. Now the problem asks the following: Given a small number $\epsilon \gt 0$, find the largest ...
1
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3answers
107 views

prove this inequality with log and positive value “x”

How do I prove that for every positive $x$ , $1-x \le -\log{x}$ Can I use convexity somehow?
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3answers
32 views

Biconditionality-ish of Epsilon Delta Proof of a Limit?

I have known the precise ($\epsilon$, $\delta$) definition of a limit, $$\lim_{x\rightarrow a} f(x) = L \iff \forall \epsilon>0,\exists\delta>0 : (0<|x-a|<\delta \implies ...
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3answers
106 views

Integrating $x^2e^{-x}$ using Feynman's trick?

In the second episode of season $8$ of "The Big Bang Theory," which aired yesterday night, it is stated that one can integrate $x^2e^{-x}$ by using Feynman's trick of differentiating under the ...
3
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1answer
29 views

Construct a function $f:\Bbb{R}\to [0,\infty)$ such that every point $x\in\Bbb{Q}$ is a local strict minimum point of $f$

I got this problem: Construct a function $f:\Bbb{R}\to [0,\infty)$ such that any point $x\in\Bbb{Q}$ is a local strict minimum point of $f$ My partial solution: Define $f$ by $f(x)=1$ if ...
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2answers
32 views

Find the locus of points whose distances from the line $y=\sqrt3x$ and x-axis are equal.

Find the locus of points whose distances from the line$\hspace{0.2cm}$ $y=\sqrt3x$$\hspace{0.2cm}$ and x-axis are equal. My solution:I start with the following ...
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0answers
24 views

Evaluating convoluted integrals of complex exponentional and rational

I want to evaluate the following integral: \begin{equation} f_{abcd}(t) = \int_{-\infty}^{\infty}d\lambda\int_0^{t-\lambda} d\tau \frac{e^{i a \tau}}{ (b+i \tau)^{5/2} } \int_0^{t-\lambda} d\tau ...
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3answers
50 views

Maximum value for $x(t)$ in $x(t) = -\frac{1}{2}gt^2 + vt$

In a book I am reading it says that the maximum value of $x(t)$ in $$x(t) = -\frac{1}{2}gt^2 + vt$$ is $\frac{v^2}{2g}$ and that this happens when $t=\frac{2v}{g}$ I cannot derive this though. When I ...
2
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1answer
56 views

Maxima of this function

I am looking for extrema of the function $$g(a,b):=\frac{\pi}{2|b|}\left(\left(\text{C}\left[\frac{a-2 |b|}{\sqrt{2|b| \pi }}\right]-\text{C}\left[\frac{a+2 |b|}{\sqrt{2|b| \pi ...
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3answers
63 views

Why is the polynomial $f(x)=x^3+x^2+x+1$ monotonic?

I have to argue why the polynomial $f(x)=x^3+x^2+x+1$ has a reverse function $f^{-1}$ which is defined in on the whole of $\mathbb R$. I'm certain the argument would simply be that because $f(x)$ is ...
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0answers
15 views

Taylor expansion of a scalar function

I have an expression on the form $$ \sum_{i=1}^N{\rho_i}f(\mathbf x+\mathbf c_i)\mathbf c_i $$ where $\rho_i$ is a scalar, $f(\mathbf x+\mathbf c_i)$ a scalar function of the vector quantities ...
0
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1answer
21 views

Find $k$ s.t. $kx^2+4$, $k^2-x^2$ orthogonal

It is a problem I encounter but I don't know what it means. There are two functions $kx^2+4$,$k^2-x^2$, it asks me to find $k$ st. two graphs are orthogonal. I actually don't know what the graph ...
0
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3answers
45 views

Proving a function has real roots

I am not interested in finding roots but interested in proving that the function has real roots. Suppose a function $f(x) = x^2 - 1$ This function obviously has real roots. $x = {-1, 1}$ How could ...
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2answers
34 views

Showing If A Function Is Convex/Concave

I have to show wither the two following functions are convex: $f_1(x)=1+6x-7x^2$ and $f_2(x)=1+6x+7x^2$ Now I want to apply the condition that this is convex iff we have $f(y)\geq f(x)+\nabla^T ...
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1answer
68 views

Compute polylog of order 3 at $\frac{1}{2}$

How to compute the following series: $$\sum_{n=1}^{\infty}\frac{1}{2^nn^3}$$ I am aware this equals polylog of order 3 at $\frac{1}{2}$, but how to prove it using integral or Euler sum only (without ...
0
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1answer
25 views

Angle between tangent of Hyperbola and Xaxis

The problem is to find the angle between x axis and tangent to the hyperbola xy=9 at (3,3). The angle made by slope of the tangent of hyperbola at (3,3) is the angle between x-axis and tangent ?
2
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1answer
40 views

Integral of $\frac1{\cos^n x}$

Hi guys I have already proven for an assignment that: $$\int\cos(x)^n dx=\frac{1}{n}\cos(x)^{n−1}\sin(x) + \frac{n-1}{n}\int\cos(x)^{n−2}dx$$ Now we have been asked to calculate ...
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0answers
22 views

Extremal condition for series expansion coefficients

I want to maximize a coefficient in a series expansion, so the situation is the following. $f \in C^{\infty}$ and $f: \mathbb{R} \times \mathbb{R} \times [0,2 \pi] \rightarrow \mathbb{C}$. Now, we ...
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1answer
39 views

Multivariable calculus directional derivative

The shape of an ice hill (is given by the function $f(x,y)=x^3-3xy^2$ representing the height of the hill at a point (x,y). Suppose a person is put still at a point $(-1.6,4.2)$. In which direction ...
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5answers
78 views

is $\sin(|x|)$ differentiable

is $\sin(|x|)$ differentable at when $x$ not equals to $0$. What i know is that $|x|$ is not differentable. But does $\sin(|x|)$ also follow the same rule? I believe that it is also not ...
0
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4answers
35 views

How exactly is the squeeze theorem used in this example?

$$ \lim_{x \to \infty} \dfrac{\cos x}{x} $$ Apparently this is $0$ by the squeeze theorem, because $-\dfrac{1}{x} \leq \dfrac{\cos x}{x} \leq \dfrac{1}{x}$ for all $x>0$ I understand the squeeze ...
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2answers
34 views

Mean value/Rolle's theorem exercise: show $f'(x)=\cos x$ for infinitely many $x$

Let $f:\mathbb{R}\to\mathbb{R}$ is differentiable on $(-\infty, \infty)$ and $f(n \pi)=0$ for any integer $n$. prove there exist infintely many realy number $x$ such that $f'(x)=\cos x$ could you ...
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2answers
34 views

How find this value $\frac{a^3}{(a-p)(a-q)}+\frac{b^3}{(b-p)(b-q)}+\frac{c^3}{(c-p)(c-q)}$

let $p,q$ is a $$(x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)=0$$ roots,find this value $$\dfrac{a^3}{(a-p)(a-q)}+\dfrac{b^3}{(b-p)(b-q)}+\dfrac{c^3}{(c-p)(c-q)}$$ where $a,b,c$ is give numbers. I know this ...
0
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3answers
54 views

If $\lim_{x\to 0}\big[ \frac{f(x)-2}{x}-\frac{\sin x}{x^2}\big]=1$ then $\lim_{x\to 0}f(x)=?$

If $$\lim_{x\to 0}\big[ \frac{f(x)-2}{x}-\frac{\sin x}{x^2}\big]=1$$ then$$\lim_{x\to 0}f(x)=\,?$$ Thanks for your answer.
1
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2answers
33 views

Prove that $\lim_{x\to 0}\dfrac{a^x-1}{x}= \ln(a)$ using the definition of limit and by using the fact $\lim_{x\to 0}\dfrac{x}{\log_{a}(1+x)}=\ln(a)$

I got this problem. Prove that $\lim_{x\to 0}\dfrac{a^x-1}{x}= \ln(a)$ by using the definition of limit and by the fact $\lim_{x\to 0} \dfrac{x}{\log_{a}(1+x)}=\ln(a)$ and without using any other ...
1
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1answer
19 views

At what point does normal line intersect curve second time?

At what point does the normal line to $y=-5+4x+3x^2$ at $(1,2)$ intersect the parabola a second time? $y'=6x+4$ $m_{tangent}=6(1)+4=10$ $m_{normal}=-\dfrac{1}{10}$ $y=f'(1)(x-1)+f(1)$ ...
1
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2answers
37 views

General formula for $\sin\left(k\arcsin (x)\right)$

I'm wondering if there's a simple way to rewrite this in terms of $k$ and $x$, especially as a polynomial. It seems to me to crop up every so often, especially for $k=2$, when I integrate with trig ...
1
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1answer
58 views

understanding $\frac{\partial x}{\partial y}\frac{\partial y}{\partial z}\frac{\partial z}{\partial x}=-1$

In particular, when $x=y=z$ isn't the above expression equal to 1 and not -1?
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3answers
55 views

If $f(x)>0\;\forall x\in\mathbb{R},$ and $g(x)=f(x)+f'(x)+f''(x),$, Then prove that $g(x)>0\; \forall \; x\in \mathbb{R}$

If $f(x)$ is a quadratic expression such that $f(x)>0\;\forall x\in\mathbb{R},$ and if $g(x)=f(x)+f'(x)+f''(x),$ Then prove that $g(x)>0\; \forall \; x\in \mathbb{R}$. $\bf{My\; Trial \; ...
3
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2answers
41 views

Limits to infinity, even and odd functions

I have a couple of questions regarding a practice test I just made, so the subject might vary a little bit but most of it has to do with limits. $$ \lim_{x \to \infty} \dfrac{7x+3x^2}{1-x^3} $$ ...
0
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1answer
18 views

Using the shell method, find the volume of the solid by rotating the region bounded by the given curves

$$x=y^2+1$$ $$x=2$$ about y=-2 How would I set this up? This is what I have so far: $$V = \int_0^2 2 \pi (y+2)(y^2+1) dy$$ I am almost certain this is wrong. Especially with the limits of ...
2
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4answers
66 views

Integrating $x^3\sqrt{ x^2+4 }$

Trying to integrate $\int x^3 \sqrt{x^2+4 }dx$, I did the following $u = \sqrt{x^2+4 }$ , $du = \dfrac{x}{\sqrt{x^2+4}} dx$ $dv=x^3$ , $v=\frac{1}{4} x^4$ $\int udv=uv- \int vdu$ $= ...
0
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1answer
22 views

limsup of the product of two sequences, of which one converges

Let $\{a_n\}_{n=1 }^{\infty}$ and $\{b_n\}_{n=1}^{\infty} $ be two sequences in $\mathbb{R}$, with the first sequence convergent . Prove that $$ \limsup\limits_{n\to \infty} a_n b_n ...
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0answers
14 views

Upper and lower estimates of graphs

Copied from Webassign, "By reading values from the given graph of f, use five rectangles to find a lower estimate and an upper estimate for the area A under the given graph of f from x = 0 to x = ...
0
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1answer
21 views

Power series and zeros

When is a power series equal to zero? Example: Take $\sum_{n=0}^\infty a_n(z-z_0)^n$. Is this power series equal to zero only at $z=z_0$ if we assume that we have infinitely many nonzero $a_n$? ...
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2answers
33 views

About the integral of the area in a circle

so i start with a circle with radius $r$, and make another one with radius $r+dr$, and the area between them is $(2\times \pi \times r)\times dr$ because that area is a rectangle, and that's where my ...
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1answer
13 views

An identity derived from the Laplace transform

It seems that $$\int_0^t \int_0^l f(\tau) ~d \tau ~d l = \int_0^t z f(t-z) dz $$ since the Laplace transform of both sides is $F(s)/s^2$, where $F(s)$ is the Laplace transform of $f(t)$: the left-hand ...
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3answers
41 views

Calculus 2 Integral of$ \frac{1}{\sqrt{x+1} +\sqrt x}$

How would you find the integral of $1/(\sqrt{x+1} + \sqrt x) dx$. I used u substitution and got this far: $u = \sqrt{x+1}$ which means $(u^2)-1 = x$ $du = 1/(2\sqrt{x-1}) dx = 1/2u dx$ which means ...
1
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1answer
21 views

Imaginary first introduction $\arg (1+i)$

I have been asked to find $$\arg (1+i)$$ This is how I reason; since I know that $$\left| 1+i\right| =\sqrt{2}$$ I know that the hypotenuse is equal to $\sqrt{2}$ and I know that adjacent is equal ...
0
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2answers
22 views

Find the equation of the tangent line at the point

I need to find the equation of the tangent line at the point $(3,1)$ for $$x\ln y + 2xy = 6.$$ Can you point out a way to start.
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2answers
45 views

Integrate $\int\frac{dx}{x\sqrt{x^2+x+1}}$ [on hold]

Hello I need some help with the following integral: $$\int\frac{dx}{x\sqrt{x^2+x+1}}$$ Have been trying u-sub, and parts which do not get me to a solution!
0
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1answer
47 views

how $2x=x$ , related to differential calculus [duplicate]

can anybody please tell me what's happening here ? $$1^2=1$$ $$2^2=2+2$$ $$3^2=3+3+3$$ $$x^2 = x+x+\cdots+x \mbox{ ($x$ times)}$$ differentiating both the sides $$2x = 1 + 1 + \cdots+1 \mbox{ ...
2
votes
3answers
38 views

Integrating $\int_{-\pi}^{\pi} \frac{ d\theta}{w - sin \theta}$

I know that the integral $$\int_{-\pi}^{\pi} \frac{ d\theta}{w - sin \theta} = \frac{2\pi}{\sqrt{w^2-1}}$$ where w, is an arbitrary constant and at some point you must do the substitution $$u = tan( ...
0
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0answers
27 views

Applicantions of Newtons Method for $f(x) = \dfrac{e^x}{x^2+1}$

I´m study some applications of calculus and see that question: If $f$ and $g$ are real functions differentiable such that $g'(x)\neq 0$ for all $x$. $a)$ Show that $f(x)$ and $g(f(x))$ has the same ...
9
votes
1answer
99 views

Integral of $\sqrt{x^3 + 8}$?

I have issues solving the following integral: $$\int\sqrt{x^3+8}~dx$$ I tried substitution and integration by parts, but with no use. I'm guessing I have to use some trigonometric substitution. ...
3
votes
1answer
47 views

manipulations with Taylor expansions for log and sinh

How could we derive the equality $$ \frac14 \sum_{m=1}^\infty \frac1m \frac{1}{\sinh^2 \frac{m\alpha}2} = - \sum_{n=1}^\infty n\log (1-q^n)$$ where $q=e^{-\alpha}$ ?