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

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0
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2answers
30 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
0answers
8 views

How to find radius of hemisphere in applied problem

I don't know how to solve it. If the stem of a mushroom is a right circular cylinder with diameter $1$ and length $2$ and its cap is a hemisphere of radius $a$, where the mushroom is a homogenous ...
0
votes
1answer
29 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
0answers
12 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
48 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 ...
0
votes
1answer
33 views

Piece wise function continuity

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
54 views

Solve $x^2 = 2^x$.

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
37 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
35 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
39 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
15 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
42 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
98 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
67 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
66 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
20 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
18 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
53 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
25 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
15 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
29 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
18 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
29 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
28 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
33 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
22 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
42 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
37 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 ...
0
votes
1answer
21 views

Why Can't I find the Volume of a Rotated Graph by Average Value Theorum?

I am wondering why I get an incorrect answer when trying to find the volume of a rotated function about the x-axis when using the Average Value Theorem. I want to find the volume of $y=\sqrt{x-2}$ as ...
0
votes
0answers
18 views

Riemann-Stieltjes Integral Substitution

I want to prove $\int^b_a\,f(g(x))\,dg(x) = \int^{g(b)}_{g(a)}\,f(x)\,dx$ for all f continuous. Firstly, $\int^b_a\,f(g(x))\,dg(x) = \int^b_a\,f(g(x))g'(x)\,dx$, since g is continuous and ...
1
vote
1answer
16 views

Error Bounds with Trapezoidal Formula

I know there are some posts about the same thing but I am unable to do my specific question or at least, I don't think I'm doing it the right way. The question asks me to use the Trapezoidal Error ...
3
votes
0answers
74 views

tough definite integral: $\int_0^\frac{\pi}{2}x\ln^2(\sin x)~dx$

Any ideas on $\int_0^\frac{\pi}{2}x\ln^2(\sin x)\ dx$ ? Best numerical approximation I can get is $0.2796245358$ Is there even a closed form solution?
5
votes
4answers
83 views

application of L'Hopital's rule?

I am trying to evaluate the following limit: $$ \lim_{x \to 0} \frac{e^x}{\sum_{n = 1}^\infty n^k e^{-nx}}, $$ where $k$ is a large (but fixed) positive integer. I am unsure how to proceed. Can this ...
1
vote
2answers
42 views

Calculate the area enclosed by the curve and line

Calculate the are enclosed by ${y = 2x - 1}$ and ${y= x^2 + 6x + 2}$ First of all I combine the equations into: ${x^2 + 4x + 3 = 0}$ ${(x + 3)(x + 1), x = -3, x = -1}$ They intersect at ${(-3 ...
1
vote
3answers
39 views

application of the inequality $\|fg\|_1 \leq \|f\|_p\|g\|_q$

application of the inequality $\|fg\|_1 \leq \|f\|_p\|g\|_q$ where $1/p + 1/q = 1$ I know this is a straight application of the inequality, but how am I assured that the integral of ...
0
votes
1answer
28 views

A limit of integral

How can I prove that $ \lim_{ n\to \infty} \int_{R}^{} \cos(nt)f(t)dt = 0 $ for any $f \in L_{1}(R) $? I believe that I should use a fact that cosinus is a cyclic function and divide this integral ...
0
votes
3answers
39 views

integrate the following integral $\int \sqrt{\frac}{a-2y^2}{3} dy$

How do I integrate the following.I tried by substituting y=sin t but I did not get anything proper. $\int \sqrt{\frac}{a-2y^2}{3} dy$
0
votes
0answers
23 views

Changing the order of integration? - $ \int_{-5}^{5}dx\int_{-7}^{\sqrt{25-x^2}}f(x,y)dy$

I'm trying to change the order of the integral and I don't understand it very well. I am trying to understand it by following solution to another example but still - I would appreciate any hints or ...
0
votes
1answer
19 views

Double integral over the set with an absolute value of $y$

I need to calculate an integral over the set: $$D \colon 0\leq x\leq \pi\text{ and }|y|\leq x$$ from the set (definite integral) $D \int \cos(y)dA$ I don't understand what $|y| \leq x$ means. Can ...
0
votes
1answer
28 views

Derivative of integral - two functons of different variables

Let's say we have two functions $f(x)$ and $g(z)$ where $x$ and $z$ are two different, unrelated variables. Could anyone tell me why the following equality holds (if if does)? $$\frac{d}{dx}\int ...