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

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votes
0answers
57 views

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

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: ...
2
votes
1answer
53 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
33 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
votes
4answers
76 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 ...
-1
votes
2answers
25 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: $$f'(x) = \frac{d\ (x\cos(x)^{-1})}{d \ x} + (\frac{d\ (x\cos(x)^{-1})}{d\ (\cos(x))} * \frac{d\ \cos(x)}{d\ x})$$ ...
-4
votes
1answer
36 views

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

can someone explain how the 1/2 gets in there I don't see how enter image description here
1
vote
3answers
84 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
votes
0answers
19 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
votes
3answers
45 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
3answers
123 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
117 views

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

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
votes
2answers
29 views

Integration using substitution and reduction formula?

Use substitution and the reduction formula to find: $$\int x^4e^{2x}\,\mathrm{d}x$$
0
votes
1answer
31 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
83 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
52 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
28 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
81 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
40 views

Piece wise function continuity [closed]

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 ...
1
vote
1answer
67 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
43 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
19 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
65 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
50 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
1answer
73 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
109 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
24 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
20 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
74 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
107 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
20 views

Country ranking by combination of factors [closed]

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 ...
4
votes
5answers
72 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 ...
1
vote
3answers
47 views

Derivative of given $f(x)$ at $x=0$

If given this function: $$f(x) = \begin{cases} e^x, & x \le 0 \\[2ex] -e^{-x}+2, & \text{x > 0} \end{cases} $$ How do I calculate the derivative at $x=0$? Shall I calculate by the normal ...
0
votes
0answers
27 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 ...
8
votes
1answer
136 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
63 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 ...
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votes
0answers
23 views

Solid of revolution problem [closed]

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
33 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 $\Pr(Y>\mathrm{E}[Y])$? ...
0
votes
1answer
31 views

Why does this follow from the triangle inequality?

Proving that differentiability implies continuity.
1
vote
2answers
34 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
32 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
24 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
33 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
31 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
37 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 ...
2
votes
3answers
27 views

Substitution and Partial Fractions (Integration)

$$\int\frac{dx}{x-\sqrt[4]{x}}$$ given the substitution $x=u^{4}, dx=4u^{3}du$ $$=\int\frac{4u^{3}du}{u^{4}-u}=\int\frac{4u^{3}du}{u(u^{3}-1)}=\int\frac{4u^{2}du}{(u^{3}-1)}$$ At this point I ...
1
vote
1answer
53 views

How do you differentiate the integral from $ \int_{e^{-x}}^{e^x} \sqrt{1+t^2}\,dt$ [duplicate]

How do you differentiate the integral from $e^{-x}$ to $e^x$ of $\sqrt(1+t^2)$ with respect to t? $$ \int_{e^{-x}}^{e^x} \sqrt{1+t^2}\,dt $$ I know the answer is $$ e^x\sqrt{1+e^{2x}} + ...
-1
votes
0answers
44 views

Closed form for $\left(\sum_{k=0}^n\frac{x^k}{k!}\right)^p$

The expression for the p-th power of the sum of the first $n+1$ powers of x is given analytically by ...