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

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4 views

The space of symmetric 2x2 matrices?

The space of symmetric 2x2 matrices? please help me with this question. I think this question is asking to find basis of the matrix
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2answers
22 views

How to prove functions are odd and even

Show that any function f on [-a,a] where a is a positive constant, can be written as the sum of an even and an odd function?
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1answer
19 views

how to show this manipulation in the integral

Let we have: $$G(t)=y_1(t)\int y_2(s)ds$$ when we take the limits as: $$G(t)=y_1(t)\int^t_{t_0} y_2(s)ds$$ then is it possible to write it as: $$G(t)=y_1(t)\int^t_{t_0} y_2(s)ds=\int^t_{t_0} ...
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0answers
18 views

Simplifying Gamma functions yet having a complication while graphing when the function was able to be graphed previous to simplification?

According to the Euler's duplication formula: $$ \Gamma(z) \Gamma(z+\frac{1}{2}) = 2^{1-2z} \sqrt{\pi} \Gamma(2z) \therefore $$ $$ \Gamma(2z) = \frac{\Gamma(z) \Gamma(z+\frac{1}{2}) ...
2
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0answers
23 views

Integral involving hyperfactorial

I'm trying to prove that: $$ \int_0^1 \ln\left(K(x)\right)\space dx =-\zeta'(-1)=\ln(A)-\frac{1}{12} $$ Where $A$ is Glaisher Kinkelin's constant and $K(x)$ is a generalization of the hyperfactorial ...
2
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3answers
41 views

Find derivative of $f(x)=\frac{1}{\sqrt{x+2}}+2x$ by definition

Use the definition of a derivative to find the derivative of: $$f(x)=\frac{1}{\sqrt{x+2}}+2x$$ my work: $$\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$$ ...
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1answer
15 views

Derivative of complex conjugate

In general, two different mathematical operations need not commute. Let f(x,y) be a complex valued function, taking in two real-valued inputs x and y. Then under what circumstances is the partial ...
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1answer
10 views

Volume of a region after some transformation.

Consider $$R=\{(x,y,z):x^2+y^2\leq 1,0\leq z\leq 2\}$$ and the transform $$T:(x,y,z)\to(x,y+\tan \alpha z,z)$$ where $0<\alpha<\pi$ Then what is the volume of $T(R)$? I tried myself but I ...
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0answers
19 views

Reference request regarding calculus exam

I'm currently a first year computer science student and I'm deeply interested in calculus . That being said, what we studied so far consists of: Cantor sets, sequences and a brief introduction to ...
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3answers
11 views

Integrate the differential equation of a simple rate equation

Could somebody please show me how to integrate the following: $dA/dt = -kA$ I'm told that the answer is: $A(t) = A(0)e^-kt$ but I do not know why. Could you be explicit in your answer and explain ...
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1answer
13 views

how to find the interval at which a derivative function is increasing

Alright, so here's the deal. I need to find the interval of this derivative function: f(x)= −5x2+12x−7 So far, I've gotten that the derivative is this: ...
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1answer
13 views

Summation of arithmetic-geometric series of higher order

There is a closed formula for the summation of arithmetic-geometric series: $\sum_{x=1}^{+\infty }(ax+b)r^x=\frac{(a+b)r-br^2}{(1-r)^2}$ when $-1<r<1$ But consider: $\sum_{x=1}^{+\infty ...
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0answers
10 views

How do I find the relative coordinates of a picture of a plane in 3d space.

Important info (ABCD) is a normal coordinate system perpendicular to the plane I am examining at (EAGB). Plane EAGB does not appear in a way where I can just use a ruler to measure the change in 'y' ...
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1answer
22 views

What are the rules being used to compute $\lim\limits_{x\rightarrow \frac{\pi}{2}} (1-\cos x)^{\tan x}$?

I am given $\lim\limits_{x\rightarrow \frac{\pi}{2}} \frac{\ln(1-\cos x)}{\cos x} = -1$ So, $(1-\cos x)^{\tan x} = e^{(\tan x) \ln(1-\cos x)}$ and as $x\rightarrow \frac{\pi}{2}$, we have: $(\tan ...
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3answers
46 views

Function composition: $f^{653}(56)=?$

Let $f(x) = \frac1{(1-x)}$. Define the function $f^r$ to be $f^r(x) = f(f(f(...f(f(x)))))$. Find $f^{653}(56)$. What I've done: I started with r=1,2,3 and noticed the following pattern: $$f^r(x)= ...
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1answer
32 views

Is there any way to solve this inequality by hand?

I have had this one equality problem I've been stuck on for a while. Could anyone share on what methods I would have to use to solve this by hand? No need for full answer, just a hint is okay. ...
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2answers
14 views

What is $1 + \sum_{k=1}^{\infty} \frac{(it)^k}{k!}a^{2k+1}$?

I want to express $$1 + \sum_{k=1}^{\infty} \frac{(it)^k}{k!}a^{2k+1}$$ in terms of standard functions (exp, cos, sin, etc.), but I just don't see what this function is. Does anybody here have an ...
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1answer
18 views

Given that $x(\theta)=5\cos \theta,y(\theta)=5\sin \theta,z(\theta)=\theta$

Given that $x(\theta)=5\cos \theta,y(\theta)=5\sin \theta,z(\theta)=\theta$ $L(\theta)$ is the arclength at the point $P(x(\theta),y(\theta),z(\theta))$ and $D(\theta)$ is the distance from origin to ...
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1answer
17 views

Rate of Change with Derivatives

We just started with rate of change while using derivatives and I am stuck on a question, hope you can help. A particle moves on a vertical line so that its altitude at time $t$ is $y=t^3−12·t+3$, ...
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0answers
18 views

Maturity and Proficiency in calculus, linear algebra for successful research

Will the high level maturity and proficiency in basic calculus, linear algebra (both calculation and theorem aspects) be required or recommended as an important factor to be successful in mathematical ...
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23 views

Proving $\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}}{e^{\cos(x)}\cot(x)} dx < \frac{1}{e}$

While i was playing around with very weird functions and came across this: $$ \int_{\frac{\pi}{4}}^{\frac{3\pi}{4}}{e^{\cos(x)}\cot(x)}dx \approx 0.3676932086...\approx \frac{1}{e} - ...
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0answers
21 views

Area of a circle shall equal the area of a square

How can I, using bolzanos theorem, discuss the equal areas of a circle and a square? How can this be shown in a graph? Would be really grateful if any could help me! :)
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1answer
10 views

Show that the z-coordinate of the center of mass is 2/3 and so independent of the parameter a.

For a>0, consider the family of solids bounded below by the paraboloid z=a(x^2+y^2) and above by the plane z=1. If the solids all have constant mass density 1 gm/cm^3, show that the z-coordinate of ...
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2answers
64 views

Why “$\lim\limits_{x\rightarrow \infty} \frac{x+\sin x}{x}$ does not exist” is not an acceptable answer?

Find the limits: $\lim\limits_{x\rightarrow \infty} \frac{x+\sin x}{x}$ Since the numerator and denominator tends to infinity as $x$ tends to infinity, then applying Lhopital's rule: ...
3
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1answer
13 views

Writing the integral $ \int_{t}^{\infty}(\frac{1}{4\pi s^{3}})^{1/2}\frac{r(|x|-r)}{|x|}e^{-\frac{(|x|-r)^{2}}{4s}}ds$ in simpler form?

I was wondering if $\int_{t}^{\infty}(\frac{1}{4\pi s^{3}})^{1/2}\frac{r(|x|-r)}{|x|}e^{-\frac{(|x|-r)^{2}}{4s}}ds$ can be written more simply, where $x,r\in \mathbb{R}$ ? wolfram alpha doesn't ...
0
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2answers
45 views

integral of ∫xdx where x is constant

I am trying to convince my friend that : ∫xdx where x is constant ⇨The result of integration is zero. This is important because by believing that the result is non zero, he believes that one can ...
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1answer
18 views

Double integral and integration by parts

Let $f:[0,b]\to[0,d]$ be a continuous bijection. If $h:[0,d]\to \mathbb{R}$ is a Riemann integrable function, how to prove that $$\int_{0}^b\left(\int_{0}^{f(x)}h(y)dy\right)dx = \int_0^d ...
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0answers
7 views

Looking for an alternative solution for optimal control problem

Let's say we have the following function ; $\intop_{0}^{\infty}\int_{0}^{N}V\left(C(t,\tau\right)dtd\tau$ and we want to maximise it according to the following constraint ; ...
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0answers
23 views

Does there exist such function?

Fix an integer value $k\geq 1$. Let $[0,1]$ the unit interval and let $s\in [0,1]$. Does there exist a function $f$ (which depends on $k$ of course but not on $s$) such that $$\int_s^1 \left( ...
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0answers
23 views

how can I give an elementary proof of Maximum Modulus Theorem for polynomials?

how can I give an elementary proof of Maximum Modulus Theorem for polynomials? I got proof, but not elementary. This question in this book Conway.
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0answers
16 views

how to calculate $f(D(0,\delta) - \{ 0 \})$ with $f(z)=z\sin(\frac{1}{z})$?

how to calculate $f(D(0,\delta) - \{ 0 \})$ with $f(z)=z\sin(\frac{1}{z})$ ?. I know that zero is an essential singularity, and so $f(D(0,\delta)-\{ 0 \})$ is dense in $\mathbb{C}$. This question ...
2
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1answer
19 views

Improper integral: is it convergent?

Is this integral finite? $$\int_s^t \frac{dx}{x^{1/2} - s^{1/2}}$$ where $s,t \in (0,\infty)$. More generally, I have the following integral $$\int_s^t ...
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1answer
37 views

$\int_y^x \cos(e^t)\,dt=(\sin(e^t))]_y^x$

I have this problem: Originally, I had $F(x,y)=\int_y^x \cos(e^t)\,dt$. I want to find the partial derivate of x. So this is the solution given to me $F_x(x,y)=\frac{d}{dx} \int_{t=y}^{t=x} ...
2
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1answer
86 views

Find the limit of recursive sequence, if it exists (Analysis, calculus)

My goal is to to test this recursive sequence if it's convergent and if yes, find the limit. $$a_1=3,\:a_{n+1}=\frac{7+3a_n}{3+a_n}$$ I know how to do this with normal sequences, but this is the ...
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0answers
25 views

Prove whether a particular function is concave

Given the following equation: $$V(w) = - \frac{\alpha}{2} \left[ y_1(w) + y_2(w) + \int _{-\infty}^{+\infty} \vert y_1(w) - y_2(w) - x\vert f_{T1}(x)dx\right] \\- \beta \int _{w - y_1(w)} ^{+\infty} ...
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2answers
5 views

between what two disjoint sections we can do a Union in order to get this group of solutions?

between what two disjoint sections we can do a Union in order to get this group of solutions? $0<|x+6|\leq{0.4}$ in other words, in what values should I fill the blankets: (____),(____) $\cup$ ...
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1answer
16 views

polynomial solution of second order differential equation

Find the polynomial solution $$u_n(x) = x^n + a_1x^{n-1}+...+a_n$$ of the differential equation $$u_n'' + xu_n' - nu_n = 0$$ satisfied by u_n(x). Note that this is entry-level calculus, so in my ...
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1answer
51 views

Limit problem involving cosine

Simply, how would you calculate $$\lim_{x \to 0}x^3\cos\left(\frac{1}{x}\right)$$ I gave this problem by accident to my calculus 1 class.
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5answers
76 views

Find $\int_{ - \infty }^{ + \infty } {\frac{1} {1 + {x^4}}} \;{\mathrm{d}}x$

How can we find the integral: $$\int_{ - \infty }^{ + \infty } {\frac{1} {1 + {x^4}}} \;{\mathrm{d}}x$$ I tried to find and got it to be $\cfrac{\pi}{\sqrt2}$. Am I correct? Please help me with an ...
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0answers
22 views

Find supremum and infimum

I have problem with finding sup and inf of set: $\displaystyle \frac{k^2-n}{k^3+n^2}$ where $k,n \in N$ Is taking first n=1 and later k=1 good move ? Because I dont't have other idea
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3answers
31 views

L'Hospital's Rule to find limit

I am asked to find $\lim\limits_{x \rightarrow \infty} {\left(\frac{8x}{8x+4}\right)}^{5x}$. Could anyone help me with figuring out how to start this problem? Thanks!
3
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3answers
46 views

Solve for $\int \sqrt{x}(\sqrt{x}-2x)^2 dx $

$\int \sqrt{x}(\sqrt{x}-2x)^2 dx $ so I solved this using U-substitution where $u= \sqrt{x}$ so my $du2\sqrt{x}=dx$ then it will be $2 \int u^2(u-2u^2)^2$ and just expand then distribute the $u^2$ so ...
2
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3answers
27 views

Finding maximum points by constrain optimization (multivariable calculus)

Find the maximum value of the function $f(x,y)=x^2+y^2+2x+y$, on the closed disc (the circle together with the region inside the circle) of radius 2, centred at the origin. What i tried I know that ...
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2answers
28 views

Calculate an integral that has a sum within.

Im trying to calculate this integral: $\displaystyle \int_{0}^{\pi} \sum_{n=1}^{\infty} \frac{n \sin(nx)}{2^n}$ The only thing I have been able to do is switch the integral and the sum, and in the ...
4
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0answers
79 views

Integral Contest [on hold]

Before you answer this OP, please read all the terms and conditions below. Thank you... Today I hold an unofficial little contest on brilliant.org. Now, I will hold it here on Math S.E. It's just for ...
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1answer
28 views

Why have we made a function to be many to one and not one to many? [on hold]

We have allowed function to only relate many to one but not one to many. Why haven't we included sin(x) to be a function? Is it just for simplicity? Also, I've seen someone quote a function not even ...
0
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1answer
16 views

find the differential dy

I know this is right is it not? I did the work and I am almost positive I did it right.
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0answers
12 views

f is continious and g is defined as the integral of f. Now how can I show that g'(1)=f(1)

Im trying to solve a question I had on my exam but Im not sure how I should go about solving it. I dont even know in what direction I should look if I want to solve it. I am hoping someone here has a ...
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0answers
11 views

bibliography for weak solutions of ODE's

Some one could recommend to me some bibliography about weak solutions of ODE's, and solutions of ODE's that are not lipschitz??
7
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0answers
48 views

Real analytic methods for the following integral

A few days back, the following integral was posted $$\int_0^1 x^x(1-x)^{1-x}\sin(\pi x)\,dx=\frac{\pi e}{24}$$ The integral was answered using complex analysis tools but I am interested in other ...