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

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0answers
4 views

Simplify exponential equation

I really need your help to solve exponential equation. look so simply but I couldnt find any solution until now, Thank you very much for your kidness :)equation
-1
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0answers
13 views

Express $\prod \frac{a_i}{x+b_i}$ in terms of known functions

I am interested if there is a way to express the finite product $$ \prod_{i=1}^{S} \frac{a_i}{x+b_i} $$ in terms of known functions like Gamma, Beta, etc.
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0answers
10 views

Volumes by Cylindrical Shells - What am I doing wrong?

I am trying to solve this exercise from a textbook: $y = x^4, y = 0, x = 1;$ rotated about $x=2$ This is my attempt at solving the problem: Shell radius: $2 - x$ Shell height: $x^4$ $a = 1$ $b = 2$ ...
0
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4answers
56 views

find $\lim_{n\to\infty}(1+\frac{1}{3})(1+\frac{1}{3^2})(1+\frac{1}{3^4})\cdots(1+\frac{1}{3^{2^n}})$

$$\lim_{n\to\infty}\left(1+\frac{1}{3}\right)\left(1+\frac{1}{3^2}\right)\left(1+\frac{1}{3^4}\right)\cdots\left(1+\frac{1}{3^{2^n}}\right)$$ You have to find the given limit when $n$ tends to ...
1
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3answers
87 views

Showing that this sequence is eventually decreasing

I'm trying to show that this sequence $$a_n = \frac{3^n-7}{4^n+5}$$ is decreasing for all $n$ greater than some $N\in \Bbb N$. All I can see to do is something like $$a_{n+1} = ...
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1answer
13 views

Finding volume of a solid of revolution

I need to find the volume of the solid that is formed when the (x>0, y< -1) region of the curve y= -1/x is rotated about the y-axis. If I'm correct, this volume can be calculated by: Evaluating ...
1
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1answer
23 views

Inequality with three unknowns

Consider: \begin{equation} \Big(e^x-1\Big)\mathbb{1}_{(x\geq0)} \leq \lambda_1+\lambda_2e^x + \lambda_3x^2 \end{equation} where $\lambda_1$, $\lambda_2$, $\lambda_3$ are three unknown constants. By ...
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0answers
40 views

Explain a physical problem by mathematics

I am not very good at physics, so I don't understand how to solve the following problem with vector calculus. A perfect incompressible fluid moves steadily under gravity around the outside of a fixed ...
10
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2answers
217 views

Function that is the sum of all of its derivatives

I have just started learning about differential equations, as a result I started to think about this question but couldn't get anywhere. So I googled and wasn't able to find any particularly helpful ...
1
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2answers
25 views

Asymptotes of $\arctan (2x)$

My book tells me the horizontal asymptotes of $\arctan2x$ is either at positive or negative $\frac{\pi}{2}$, yet the vertical asymptotes of $\tan2x$ occurs at positive or negative $x=\frac{\pi}{4}$, ...
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1answer
15 views

How to solve a boundary value problem of a Laplace equation?

Suppose $x,y$ are in the range $0 \leqslant x \leqslant 2,0 \leqslant y \leqslant 1$, I can use separation of variables to get $\frac{{{\partial ^2}u}}{{\partial {x^2}}} + \frac{{{\partial ...
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1answer
35 views

Finding the First Derivative ( 1 question)

Using the Definition of a limit: [ Of form $\lim_{x\to a} \frac{f(x)-f(a)}{x-a}$] Find $f'(x)$ when $x=9$ for $f(x)=\frac{2}{\sqrt{x}}$ I tried simplifying it but got jumbled when trying to multiply ...
1
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0answers
22 views

Path to self learning Calculus

I am currently self-learning Calculus from an old and cheap edition of Calculus (by James Stewart) which contains both single and multivariable calculus.I am aware that Stewart's book is regarded as ...
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5answers
41 views

find the area of a kite with integration

A stunt kite has the shape in the diagram below: How can I find the area using calculus integration. Can anyone help me start this question, I am not looking for the full answer. I assume I only ...
4
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3answers
94 views

Limit of $a_{n+2}=a^2_{n+1}+\frac{1}{6}\cdot a_n+\frac{1}{9}$

Find a limit of sequence: $$a_{n+2}=a^2_{n+1}+\frac{1}{6}\cdot a_n+\frac{1}{9}$$ $$a_1=0,a_2=0$$ I tried to prove that $a_n$ is bounded and monotonic, but I couldn't prove that $a_n$ is monotonic (by ...
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2answers
38 views

Minimum and maximum of a two variable function

I have to study the type of critical points of the function $$ f(x,y)=(2x^2+y^2-1)(x^2+y^2-1)+1 $$ and find minimum and maximum on the generic circle centered in $ (0,0) $ and radius $ r>1 $. I ...
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0answers
38 views

Integration of $\frac{x^2}{2\left(e^x+1\right)}$

Let: $$f(x) = \int \frac{x^2}{2\left(e^x+1\right)}dx $$ Is there a way to find $f(x)$? I've tried through integration by parts, but that didn't work out. If substitution is the answer, I can't see ...
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0answers
48 views

Prove that a person can't touch a wall with Integral Calculus [on hold]

Is it possible to prove that a if a person walked straight at a wall that person would never actually get to the wall, with Integral Calculus? If so, how? For more details refer to this website: ...
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0answers
27 views

Closed form of an integral $\int_0^{\pi/2} \ln^n (\sin x) \, dx$

Let $n \in \mathbb{N}$. May we have a closed form for the integral: $$\mathcal{J}=\int_0^{\pi/2} \ln^n (\sin x) \, {\rm d}x$$ One obvious approach would be to go through beta functions and ...
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2answers
28 views

Area between two curves, which curve is on top?

Given a question like this: Find the area between ${y = x^2 + 2x - 3}$ and ${y = 2x^2 -5x -3}$. I know how to find the area ${\int y_1 - y_2}$ but how can I tell which one is the top curve? Are ...
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0answers
25 views

Determine equation of a continuous function by value at axis of symmetry and area of the graph?

(My apologies in advance for any confusing terms, please excuse my stumbling through what I am trying to ask:) I am trying to figure out if it is possible to determine (derive?) the equation of a ...
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5answers
66 views

integration of $1/x$ a counterexample to the rule

We know that the integration of $\displaystyle\int\frac{1}{x}\,dx=\log\left(|x|\right)$+$c$ with $x\neq 0$ , but if we go by normal rule then it becomes $\infty$. Is this a counterexample to the rule ...
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2answers
20 views

Need help in verifying if I am taking the derivative of $f(x) = \frac{x}{\cos(x)}$ correctly

I need to take the derivative of $f(x) = \frac{x}{\cos(x)}$. What I am doing: $$f'(x) = \frac{d\ (x\cos(x)^{-1})}{d \ x} + (\frac{d\ (x\cos(x)^{-1})}{d\ (\cos(x))} * \frac{d\ \cos(x)}{d\ x})$$ ...
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1answer
30 views

unsure how the 1/2 gets in this problem [on hold]

can someone explain how the 1/2 gets in there I don't see how enter image description here
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3answers
67 views

Prove $\lim_{n \to \infty} \frac{\ln(n)}{n}=0$ without L'Hospital's Rule

Prove the following without using L'Hospital's Rule, integration or Taylor Series: $$\lim_{n \to \infty} \frac{\ln(n)}{n}=0 $$ I began by rewriting the expression as: $$\lim_{n \to ...
0
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0answers
11 views

Taylor's expansion and remainder of $f(x)=0, -1\le x\le0$ and $f(x)=x^4, 0<x\le 1$

Let $f(x)=0, -1\le x\le0$ and $f(x)=x^4, 0<x\le 1$ If $$f(x)=\sum_{k=0}^n\frac{f^{(k)}(0)x^k}{k!}+\frac{f^{(n+1)}(\xi)x^k}{(n+1)!}$$ is the Taylor's formula for $f$ about $x=0$ with maximum ...
3
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2answers
32 views

A simple problem on first order differential equations

An ODE (Ordinary Differential Equation) of order $n$ becomes a relation: $$F(x,y,y^{(1)},...,y^{(n)})=0$$ Then $F(x,y,y^{(1)})=0$ defines an ODE of order one. In "basic standard texts", for purposes ...
2
votes
2answers
71 views

3 body problem using only math

This question was suggested to be placed in the math forum. 3 particles are at the corners of an equilateral triangle with side $a$. Assume that particle 1 is at $(0,0)$, particle 2 is at $(a,0)$ and ...
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1answer
99 views

Why do we teach Calculus in High School instead of, say, programming? [on hold]

I was wondering "Why do we teach Calculus in High School instead of programming?" 'Calculus' only goes up to about partial derivatives, then its called different things like real analysis and other ...
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2answers
27 views

Integration using substitution and reduction formula?

Use substitution and the reduction formula to find: $$\int x^4e^{2x}\,\mathrm{d}x$$
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1answer
26 views

First Order Differential Equation for a Harmonic Oscillator

A box with mass $m$ is attached to a spring with spring coefficient $k$. This system is then placed into a glass case filled with a liquid with drag coefficient $\alpha$. Now I have the following ...
1
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2answers
76 views

Proof for $log\left(\sum_{n=1}^{\infty} \frac{1}{n}\right)$ diverging.

Proof for $log\left(\sum_{n=1}^{\infty} \frac{1}{n}\right)$ diverging. I know that the harmonic series diverges. What is the quickest way to prove the logarithm of it diverges? I have not used any ...
0
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2answers
48 views

Find the minimum of the function

I was trying to solve a problem that is as follows: Find the minimum value of $$ (a+b+c+d+e)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}\right) ,\qquad a,b,c,d,e>0.$$ I have ...
0
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1answer
20 views

Finding correct variation for $\rho$ in spherical coordinate integration

I am having some trouble and looking for help on calculating the moment of inertia about the z axis of the region bound by the cone $z=\sqrt{3(x^2+y^2)}$ and the sphere $x^2+y^2+z^2=a^2$ if the ...
0
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2answers
62 views

Expansion $f(x)=1/(x-1)$

How to expand $f(x)=1/(x-1)$ into the form $1/x+1/x^2+1/x^3+...+1/x^n$ for x>1 I know f(x) can be rewritten as $f(x)=\frac{(1-1/x)^{-1}}{x}$. Next step is to expand $(1-1/x)^{-1}$ to ...
-1
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1answer
38 views

Piece wise function continuity [on hold]

Find all values of $a$ and $b$ so that the following function is continuous for all value of $x$. ($x\in\Bbb R$). $$ f(x)=\begin{cases}-3a+4x^5b&\text{when }x\le -1\\ ax-2b&\text{when ...
0
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1answer
59 views

Solve $x^2 = 2^x$. [duplicate]

One can see that the solutions are $x=2, 4$ and $x=-0.77$(approximately) seen from the graph. I am posting this to find if there is a way to solve this and find solutions like polynomial equations. ...
0
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2answers
40 views

Flaw in the technique I am using to find the area between line and curve

I am asked to find the area between ${y = 7}$ and ${x^2 -5x + 13}$ Combining these equations together I get ${-x^2 - 5x + 6 = 0}$. Factorising into ${(x - 3)(x - 2)}$ I am taking ${y = 7}$ to be ...
0
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1answer
17 views

What is the maximum of the following function?

Let $f(x,y) = \frac{xy^\alpha}{x+y},\alpha\in(0,\infty)$. How to compute $$\sup_{(x,y)\in[a,b]\times [0,c]}\frac{xy^\alpha}{x+y},$$ with $b>a>0$?
2
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0answers
49 views

Conjecture: $\int_0^{\infty}dx\frac{e^{i\alpha\sqrt{x^2+1}}}{\sqrt{x^2+1}}J_1(Qx)=\left(e^{i\alpha}-e^{i\sqrt{{\alpha}^2-Q^2}}\right)/Q$

Here $\alpha>0$, $Q>0$, and $J_1$ is a Bessel function. I'm fairly certain the closed form in the title is accurate for a couple of reasons. First, I've evaluated the integral numerically in ...
2
votes
1answer
42 views

Proving that a function grows faster than another

I'm told to prove or disprove that $4^{\sqrt{n}}$ grows faster than $\sqrt{4^n}$ As n tends to infinity. From my Previous years Calculus I know that if I take the derivative of two functions, and one ...
0
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2answers
18 views

Rotational Volume

I have to find the volume of the region bounded by $ y= \sqrt{x-1} $, y=3, the y-axis and the x-axis rotated around y=5 I set up $\int_1^{10} $ $\pi((5-(\sqrt{x-1}))^2 - (5-3)^2)$dx + $\int_{0}^1$ ...
2
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1answer
65 views

The closed form of $\int^\infty_{B}e^{-(x+\frac{A}{x})}\,dx$, where $A>0$, $B>0$.

What tools, ways would you propose for getting the closed form of this integral? $$\int^\infty_{B}e^{-\left(x+\frac{A}{x}\right)}\,dx,$$ where $A>0$, $B>0$. When $B=0$, from Table of ...
2
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5answers
102 views

Quick integral question

Sorry about the formatting, but how would I go about this question: $$\frac{d}{dx} \int_{\cos x}^1 \sqrt{(1 + e)^t} dt$$ What I've learned in class is that the derivative of an integral is just the ...
0
votes
1answer
21 views

Maximum slope of a function related to a signal

A signal x(t) inceases linearly to the value 2 at $t=2$, starting from $t=1$. It stays constant for $t \in [2,3]$ then decreases linearly to 0 at $t=5$. Let $y(t)=x(2t-1)$. What is the maximum ...
0
votes
0answers
13 views

Prove Bernoulli Function is Constant on Streamline

I have an incompressible, inviscid fluid, under the influence of gravity, with a velocity potential: $$ \mathbf{u} = (-\cos(x)\sin(y), \sin(x)\cos(y), 0) $$ Using Euler's equations, $$ \mathbf{u} ...
1
vote
4answers
71 views

Proving that the exponential inequality $e^x \ge x^e$ holds for all $x \ge 0$ [duplicate]

How does one prove that $$e^x \ge x^e$$ for all $x \ge 0$? I tried to do this by setting $f(x)=e^x-x^e$ Plotting this function shows this easily, as seen here. However, when I tried to prove ...
2
votes
2answers
100 views

Integral of $\int_{-\infty}^{\infty} \left(\frac{1}{\alpha + ix} + \frac{1}{\alpha - ix}\right)^2 \, dx$

I'm having trouble integrating $$\int_{-\infty}^{\infty} \left(\frac{1}{\alpha + ix} + \frac{1}{\alpha - ix}\right)^2 \, dx$$ where $\alpha$ is a real number and $i = \sqrt{-1}$. I'm guessing that I ...
0
votes
1answer
18 views

Country ranking by combination of factors [on hold]

I'm trying to find the most correct way of ranking countries based on multiple factors with measurements in different units. Take the following example: I am comparing $4$ countries nl.: United ...
1
vote
2answers
45 views

What does third derivative tell about inflection point?

I was trying to find the nature (maxima,minima,inflection points) of the function $$\frac{x^5}{20}-\frac{x^4}{12}+5=0$$ But I faced a conceptual problem.It is given in the solution to the problem ...