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

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0
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0answers
12 views

Does this process of finding $\delta$ is wrong?

Find $\delta$ such that $$|x^n - a^n| < \epsilon $$ whenever $$|x-a|<\delta$$. My thought is factoring $|x^n-a^n|$ so we have $|(x-a)|\cdot|x^{n-1}+x^{n-2}a+x^{n-3}a^2+...+xa^{n-2}+a^{n-1}|$ ...
1
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1answer
28 views

A tough integral and its generalization:

I happened to encounter an integral, a definite while I was walking the other day: $$ \int\limits_0^{\pi} \frac{ \sin ( 100 t ) }{\sin t } dt $$ I have tried the usual methods and nothing. I have ...
1
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2answers
23 views

How to get the radii for this volume integral?

I need to find the volume of the solid obtained by rotating the region bounded by the curves $y=x, y=0, x=2$, and $x=4$ around the line $x=1$ I know I need to integrate $\pi*((\text{outer radius})^2 ...
3
votes
1answer
26 views

Find $ \lim_{x\rightarrow 0}\frac{\tan^2 x+2x}{x+x^2} $ without L'Hôpital's Rule

I need to find $$\lim_{x\to 0}\frac{\tan^2 x+2x}{x+x^2}$$ This is what I did: Let ...
0
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0answers
5 views

Show that the function of 2 variables in greater or equal than 0 on specific set.

$D=\{(x,y) \in \mathbb{R^2} : \frac{x^2}{2} + y^2 \le 1 \} $ $ f : D \rightarrow \mathbb{R}$ is continuous and differentiable inside $Int(D)$ $ (x,y) \in \partial D \Rightarrow f(x,y) \ge 0$ $ ...
0
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1answer
17 views

Limit of a summation, using integrals method

I have seen an interesting question on stackexchange, which I would like to requote so that I can understand the answer =) $\lim\limits_{n\to\infty} \dfrac{1^{99} + 2^{99} + \cdots + ...
0
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3answers
31 views

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. [on hold]

$x^{2/3} + y^{2/3} =4$ $(3\sqrt{3}, 1)$ (asteroid) Use implicit differentiation to find an equation of the tangent line to the curve at the given point.
-1
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1answer
29 views

Special case of higher order derivatives

I am trying to translate the expression on the left side of this equation u varies with x and z. What should the left side equal ? $\frac {d^2}{dz^2}( u \frac{dT}{dx}) $
1
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1answer
17 views

If $u$ and $\partial_n u$ along a curve are known, then the full gradient $\nabla u$ is also known

I just stumbled above this statement. If a curve $\Gamma$ in $\mathbb{R}^2$ is given. Then if $u$ and $\partial_n u$ on $\Gamma$ are known. Then the full gradient $\nabla u$ is also known. So far, I ...
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0answers
5 views

proof of a special case of discrete-time tower property

I'm reading a book on stochastic process and the first chapter is about properties of conditional expectation. One of the example the book gives is the proof of a special case of tower property in ...
2
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0answers
23 views

Taylor polynomial converging pointwise but not uniformly?

Many standard examples of Taylor series $(\exp(x), \sin(x), \cos(x))$ converge uniformly, others don't converge to its original function at all, e.g. $\exp(-x^{-2})$. I couldn't think of any smooth ...
-3
votes
2answers
51 views

Is $z^2 = x^2 + y^2$ a function of two variables? [on hold]

$z$ is $f(x,y)$, where f is a relation that assigns to every ordered pair of input values $(x,y)$ a unique output value denoted by $f(x,y)$ which can be called a function of two variables. ...
0
votes
2answers
37 views

Prove that the sequence $(f(x_n))_{n\geqslant1}$ is Cauchy.

Let $f:[0,2]\to\mathbb{R}$ be a regulated function. Let $(x_n)_{n\geqslant1}$ be a sequence in $[0,1)$ with $\lim_{n\to \infty}x_n=1$. Prove that the sequence $(f(x_n))_{n\geqslant1}$ is Cauchy. I ...
1
vote
1answer
19 views

Continuity of a function defined as the limit of another function

I'm trying to prove that if the limit of f always exists then the function g defined as the limit of f, is continuous. I think that because the limit of f always exists then f has to be continuous ...
0
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3answers
62 views

How to evaluate this $1/n$ infinite sum?

How to evaluate$$\sum ^{\infty}_{n=1} {e}^{-n}$$ without using the easy-formula. We easily notice a pattern. $$\begin{align} S_1 &= e^{-1} \\ S_2 &= e^{-2} + e^{-1} = \frac{1 + e}{e^2} \\ ...
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0answers
18 views

Explanation of the Leibniz formula

I am reading the book Solving Ordinary Differential Equations I - Nonstiff Problems (1987) by Hairer et al. My question is from Section II, chapter 2 (Order conditions for RK methods), equation 2.4. ...
5
votes
2answers
45 views

How to evaluate $\int_0^{\infty} \bigg(\frac{e^{-x}}{\sinh(x)} - \frac{e^{-3x}}{x}\bigg) \; dx$

Evaluate the integral below $$\int_0^{\infty} \bigg(\frac{e^{-x}}{\sinh(x)} - \frac{e^{-3x}}{x}\bigg) \; dx$$ Using Wolfram I get the integral is $\gamma + \log\bigg(\frac{3}{2}\bigg)$, where ...
2
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0answers
18 views

$\int \sec ^m(x) \tan ^n(x) \, dx$

$$\sec ^2(x)=\tan ^2(x)+1$$ $$\csc ^2(x)=\cot ^2(x)+1$$ We can evaluate integrals of the form: $$\int \sec ^m(x) \tan ^n(x) \, dx$$ $$\int \csc ^m(x) \cot ^n(x) \, dx$$ with substitution unless m ...
11
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5answers
113 views

Finding $\sum \frac{1}{n^2+7n+9}$

How do we prove that $$\sum_{n=0}^{\infty} \dfrac{1}{n^2+7n+9}=1+\dfrac{\pi}{\sqrt {13}}\tan\left(\dfrac{\sqrt{13}\pi}{2}\right)$$ I tried partial fraction decomposition, but it didn't work out after ...
2
votes
0answers
11 views

Fluid Flow: lubrication, integration, ODE

Basically, I'm modelling the flow of a "coating" process -- a fluid flow between a flat moving plane and a stationary cylinder, 2D, cartesian coordinates. Subscript 0 is the at the minimum height b/w ...
1
vote
1answer
23 views

Fourier transform (properties)

I have a function $f$ such that $|f(x)|\leq e^{-x^2/2}$ hence in $\mathcal{L}^2(\mathbb{R})\cap\mathcal{L}^1(\mathbb{R})$ and thus we can compute the Fourier transform $$\hat{f} (\xi) = ...
0
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0answers
23 views

A simple question in calculus (equivalence of limits).

So I want to prove the next equivalence: where D-lim, is $\lim_{n\rightarrow \infty , \ n \notin M \subset \mathbb{N}}$. The easy part, mainly $\Rightarrow$ I did I think good. I am having ...
0
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1answer
22 views

Change of limits in definite integral - non constant limit

I have the following definite integral: $\displaystyle \int^{g(a)}_{a} f(x) \, dx$ and I am asked to perform a shift of the variable x, so that it transforms in $x + T$ (T is just some constant). ...
1
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1answer
38 views

Prove that f(x) is regulated.

Define $f:[0,1]\to \mathbb{R}$, $f(x):=0$ if $x\notin \mathbb{Q}$, $f(p/q):=1/q$, $q>0$, $p, q$ coprime integers. Prove that $f$ is regulated. A function $f:[a,b]\to\Bbb R$ is a regulated ...
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1answer
33 views

Is $f(x)=f(x+2\pi)$? [on hold]

Is $f(x)=f(x+2\pi)$ if ‎ ‎$f(x)=‎ ‎\left \{ \begin{array}{lr}‎ ‎x-\pi& -\pi<x<0\\‎ ‎\pi-x& 0<x<\pi ‎\end{array}‎ ‎\right.$‎ Every hint is appreciated.
-1
votes
1answer
39 views

Given $f(x)=x+\int_{0}^1 t(t+x)f(t) dt $ , what is $f(0) $? [on hold]

Let $f:\mathbb R \to \mathbb R$ be such that $f(x)=x+\int_{0}^1 t(t+x)f(t) dt $ , then how do we find $f(0) $ ?
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0answers
17 views

I'm confused in the definition of derivative could someone help me?

A function's rate of change may change or not change or may have no rate of change at all.The way I see it is their are three possible way a function may change its rate. Case 1: It change ...
1
vote
1answer
31 views

straightforward calculus problem

Find the arc length of the graph of $\displaystyle \large x^{\frac{2}{3}}+y^{\frac{2}{3}}=1$. Hint: Use symmetry with respect to the line $y=x$. Let $y=x$ intersect at $a$. So, $\displaystyle ...
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0answers
14 views

Evaluate the line integral with Euler.

Need some help evaluating this line Intergral. $\int$$_c$ xy${e^y}$$^z$ dy Where C: x = 4t ; y = 3t$^2$ ; z = 3t$^3$ ; 0$\le$t$\le$1 Any help would be great. Thanks.
0
votes
2answers
32 views

Finding limit of sequence

I have to find the limit of sequence ${a_{n}}$ such that $${a_{n}} = \frac {n^\frac{2}{3} \sin(e^n)} {n+1}$$ I have no idea where to start. Any hints on where to start?
1
vote
3answers
35 views

Finding the Limit using L'Hospital's rule

Need help with this: $$\lim_{x\to 0} {1-x\over\sqrt{1-\cos x}}-{1+x\over\arctan x}$$ First I combine the fractions: $$\lim_{x\to 0} {\arctan x(1-x)-\sqrt{1-\cos x}(1+x)\over\arctan x\sqrt{1-\cos ...
2
votes
0answers
21 views

How is the interchange of the limit and the maximum valid at this point in Erwin Kreyszig?

In 1.5-5 in Erwin Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS, the author shows completeness of the space $C[a,b]$ of all (real- or complex-valued) functions defined and continuous ...
1
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2answers
22 views

Continuous differentiability of atan2

Consider the function atan2 defined on the plane, minus the origin and the negative $x$-axis, as the unique $\theta$ such that $-\pi<\theta<\pi$ and $$ x = r \cos \theta, \qquad y=r \sin ...
5
votes
2answers
44 views

$ \lim_{n\rightarrow \infty}n^{-\frac{1}{2}\left(1+\frac{1}{n}\right)}\cdot \left(1^1\cdot 2^2…n^n\right)^{\frac{1}{n^2}}$

Evaluation of $\displaystyle \lim_{n\rightarrow \infty}n^{-\frac{1}{2}\left(1+\frac{1}{n}\right)}\cdot \left(1^1\cdot 2^2\cdot 3^3\cdot................n^n\right)^{\frac{1}{n^2}}$ $\bf{My\; Try}::$ ...
0
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0answers
16 views

Maximum value of $ f(y) = \int_{0}^{y}\sqrt{x^4+(y-y^2)^2}dx\;,$ where $y\in \left[\;0,1\;\right]$

The Maximum value of $\displaystyle f(y) = \int_{0}^{y}\sqrt{x^4+(y-y^2)^2}dx\;,$ where $y\in \left[\;0,1\;\right]$ $\bf{My\; Try::}$ Given $\displaystyle f(y) = ...
1
vote
1answer
25 views

Calculate limit $\lim_{n\rightarrow\infty}\dfrac{(4n-100)^{4n-100}n^n}{(3n)^{3n}(2n)^{2n}}?$

The limit $$\lim_{n\rightarrow\infty}\dfrac{(4n)^{4n}n^n}{(3n)^{3n}(2n)^{2n}}$$ can be calculated as ...
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2answers
20 views

Sketch a graph of the following parametric curve $\:C:\left\{x=3+2cos\theta \right\}\left\{y=5+2sin\theta \right\}$

(a) Sketch a graph of the following parametric curve $\:C:\left\{x=3+2cos\theta \right\}\left\{y=5+2sin\theta \right\}$ HINT: use the fact that $\left(2cos\theta \right)^2+\left(2sin\theta ...
0
votes
2answers
26 views

find the equation of the tangent line to the curve y=x^4-6x and perpendicular to the line x-2y+6=0

this is all I got right now: $y=x^4-6x$ $y'=4x^3-6$ $x-2y+6=0$ $y=(1/2)x+3$
0
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0answers
21 views

What's the general procedure to find roots of a funcion using numerical methods?

When I'm facing functions for which no formula exist to calculate the roots directly, what can I do with calculus to analyse it so that I can obtain information about the function's behavior? Suppose ...
1
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0answers
35 views

Harmonic number identity

I search for an elementary proof of the following identity: $$ \sum_{i=1}^{n-k} \frac{(-1)^{i+1}}{i}\binom{n}{i+k}=\binom{n}{k}\left(H_n-H_k\right) $$ I have found the following proof: $$ ...
1
vote
1answer
22 views

Prove f(x)=glb{|x-a| : a in A} is continuous

Let $A \subset R$, let $f(x)=glb{ |x-a| : a \in A}$ -Prove $f$ is well defined -Prove $f$ is continuous (Ok, here's the deal, because of the absolute value the greatest lower bound is always going ...
0
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0answers
15 views

Closed-forms of real parts of special value dilogarithm identities from inverse tangent integral function

The inverse tangent integral is defined as $$\operatorname{Ti}_2(x)=\Im\operatorname{Li}_2\left(ix\right)$$ Because this we have some special value identitiy. Let $c_1 = \operatorname{Li}_2(i)$, ...
5
votes
2answers
58 views

Why is $e^x=\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n$

While looking through an example in Carothers' Real Analysis, I came across the following: $$e^x=\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n$$ and of course I noticed that it looks similar to ...
7
votes
1answer
37 views

Newton's way of getting a Taylor expansion

I don't understand how Newton find the Taylor expansion of $\frac{a^2}{b+x}$ by the following method : **This screenshot is from : The method of fluxions and infinite series Any idea ?
1
vote
1answer
36 views

Proving that $F(x)$ is a constant

This was on a test and i know i was supposed to use 2nd ftoc to prove that $F(x)$ was a constant when $x>0$ $$ F(x) = \int_{0}^{x} \frac{1}{t^2 +1} dt + \int_{0}^{\frac{1}{x}} \frac{1}{t^2 +1} ...
0
votes
0answers
24 views

Differential inequality involving derivatives

I'm having trouble with a differential inequality. Consider a smooth function $f(x)$ defined for $x>0$ with $f'>0$. Given $0< a < b$, show that there exists smooth functions $g(x)$ and ...
0
votes
0answers
23 views

Writing solution to an arbitrary ODE with arbitrary initial values as the sum of a power series?

Let $f(t), g(t)$ be polynomials, and let $y$ be a function of $t$. Given the ODE $y'' + f(t) y' + g(t) y = 0$ with initial conditions $y(0) = \alpha$ and $y'(0) = \beta$, write $y$ as the sum of a ...
0
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0answers
26 views

Change of Variables in Cauchy-Euler equations

So I'm working on the change of variables in the Cauchy-Euler equation. And I understand everything except one step. It's the same one step skipped in the answer to the very related question here. I ...
0
votes
1answer
25 views

Derivation of the volume of a prism.

How can we prove that the volume of a prism with any plane region as its base is the area of its base multiplied by its height? By volume I mean the measure of the quantity of three-dimensional space ...
7
votes
2answers
84 views

Prove that $\int_0^\infty \frac{\ln x}{x^n-1}\,dx = \left(\frac{\pi}{n\sin\left(\frac{\pi}{n}\right)}\right)^2$

This question inspired me to ask the following. Prove that $$I_n = \int_0^\infty \frac{\ln x}{x^n-1}\,dx = \left(\frac{\pi}{n\sin\left(\frac{\pi}{n}\right)}\right)^2,$$ for $\Re(n)>1$. For some ...