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

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0
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0answers
11 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 ...
-2
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0answers
19 views

Prove that a person can't touch a wall with Integral Calculus

Is it possible to prove that a person can't touch a wall, even though he touches it, with Integral Calculus? If so, how?
-1
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0answers
16 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 ...
1
vote
2answers
27 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 ...
0
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0answers
16 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 ...
0
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5answers
60 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 ...
-2
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2answers
19 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: $$f'(x) = \frac{d\ (x\cos(x)^{-1})}{d \ x} + (\frac{d\ (x\cos(x)^{-1})}{d\ (\cos(x))} * \frac{d\ \cos(x)}{d\ x})$$ ...
-6
votes
1answer
27 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
1
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3answers
65 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
9 views

Similarities between Pade approximations and ARMA($p,q$) in time series

I am wondering if someone would mind explaining what Pade approximations in calculus and ARMA($p,q$) in time series have in common.
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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 ...
2
votes
2answers
28 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 ...
1
vote
1answer
52 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 ...
-6
votes
1answer
94 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 ...
0
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2answers
24 views

Integration using substitution and reduction formula?

Use substitution and the reduction formula to find: $$\int x^4e^{2x}\,\mathrm{d}x$$
0
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1answer
22 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
vote
2answers
74 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
votes
2answers
47 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
votes
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
votes
2answers
53 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
votes
1answer
37 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
votes
1answer
57 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
votes
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
votes
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 $b>a>0$?
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0answers
45 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
41 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
votes
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
votes
0answers
48 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
votes
5answers
101 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
68 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
94 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
17 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
39 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 ...
4
votes
5answers
70 views

Finding $\lim_{x\to -2}{\frac{x+2}{\sqrt{-x-1}-1}}\;$ without L'Hospital

I have been trying to find $$\lim_{x\to -2}{\frac{x+2}{\sqrt{-x-1}-1}}$$ without L'Hospital's Rule, but I am stuck. I tried Rationalizationg the denominator Factoring out $\,x$ But it did not ...
0
votes
0answers
22 views

Prove equivalence between two Bessel functions relations

Given the following equation $$\frac{J_{n - 1} (u)}{uJ_n (u)} - \frac{K_{n-1}(w)}{wK_n(w)} = 0$$ (where $J$ is the Bessel function of the first kind, $K$ is the modified Bessel function of the ...
7
votes
1answer
57 views

Find all $f:\mathbb {R} \rightarrow \mathbb {R}$ where $f(f(x))=f'(x)f(x)+c$

Recently, while studying calculus, I have come across multiples problems which asked the following: If $f(x)$ is a polynomial, find all $f(x)$ that $f(f(x))=f'(x)f(x)+c$, where $c$ is a constant. ...
0
votes
1answer
19 views

A question in limit matrix polynomial

Suppose ${A_j},\,{\Delta _j} \in {\mathbb C^{n \times n}},\quad\big(\,j = 0,\,1,\,2,\,\ldots,\,m\,\big)$ ${P_\Delta }\left(\lambda\right) = \left({A_m} + {\Delta _m}\right){\lambda ^m} + \, \cdots ...
1
vote
4answers
54 views

Give that $f$ is a decreasing continuous function and that $f(x+y) = f(x) + f(y) -f(x)f(y)$ and $f'(0)=-1;$ Then find $\int_{0}^{1}f(x)dx$

Give that $f$ is a decreasing continuous function and that $$f(x+y) = f(x) + f(y) -f(x)f(y)$$ and $f'(0)=-1;$ Then it is to be found what is $\int_{0}^{1}f(x)dx$ I am at a loss on how to approach ...
-1
votes
0answers
22 views

Solid of revolution problem [on hold]

how do I find the Volume of the solid of revolution of $y = x^2$ rotated around the $x$-axis on the interval from $0$ to $1$ using double integrals and triple integrals
0
votes
1answer
26 views

If $Y = (\mathcal{N}(\mu_1,\sigma_1^2) + \mathcal{N}(\mu_2,\sigma_2^2))^2$, what is $\Pr(Y>\mathrm{E}[Y])$?

Given $X_1 \sim \mathcal{N}(\mu_1,\sigma_1^2)$ and $X_2 \sim \mathcal{N}(\mu_2,\sigma_2^2)$, with $X_1$ independent of $X_2$, as well as $Y = (X_1 + X_2)^2$, what is $\Pr(Y>\mathrm{E}[Y])$? ...
-1
votes
0answers
17 views

Graph transformations (g in terms of f)

I am wondering how to describe the graph g in terms of the graph of f for these cases: $g(x)=f(1/x)$ $g(x)=|f(x)|$ $g(x)= f(|x|)$ $g(x)=\max(f,0)$ $g(x)=\min(f,0)$ $g(x)=\max(f,1)$
0
votes
1answer
30 views

Why does this follow from the triangle inequality?

Proving that differentiability implies continuity.
1
vote
2answers
33 views

meaning of definite integral

So to my knowledge a definite integral's significance is how it shows the "intensity" or area under the curve for a function. However, I am confused then why the definite integral for x from 0 to 1 ...
0
votes
0answers
29 views

Show that $|g'(x)|\le\frac{1}{2}$ whenever $x^2>2|c|$

Consider the fixed point iteration $$ x_{n+1}=-b-\frac{c}{x_n}=g(x_n)$$ How would I show that $|g'(x)|\le\frac{1}{2}$ whenever $x^2>2|c|$?
0
votes
1answer
22 views

How to find centroid of this region bounded by surfaces

I am having difficulty find the centroid of the region that is bound by the surfaces $x^2+y^2+z^2-2az=0$ and $3x^2+3y^2-z^2=0$ (lying above $xy$ plane, consider the inner region). I know the first ...
0
votes
0answers
30 views

When is the limit of an infinite product equal to the infinite product of the limit?

For a finite case we have $\lim\limits_{n\rightarrow\infty}f(n)\cdot g(n) =\lim\limits_{n\rightarrow\infty}f(n)\cdot\lim\limits_{n\rightarrow\infty}g(n)$ however when is it possible to interchange the ...
0
votes
2answers
30 views

Whats bigger? lim n->infinity n^x or lim n->infinity x^n

What is bigger? lim n->infinity n^x or lim n->infinity x^n I have a relationship where I am trying to find the lim n->infinity (2^n + n^20) / 3^n and am having a hard time deciphering it.
4
votes
1answer
34 views

if $r,s$ are rational numbers, then $r+s\sqrt2$ is irrational unless $s=0$?

if $r,s$ are rational numbers, Prove $r+s\sqrt2$ is irrational unless $s=0$? I need to prove this simple question, but not sure if my method is acceptable I'm trying to prove it by ...