All questions with tag [math: calculus]


Finding Concavity in Curves

Suppose $$\frac{d^2y}{dx^2} = \frac{e^{-t}}{1-e^t}.$$ After locating the 2nd by-product, just how do I locate concavity? It is a lot easier to address for $t$ in a trouble like $t(2-t) = 0$, yet in this instance, addressing for $t$ appears harder. Does it have something to do with $e$? $e$ never ever comes to be $0$, yet at what factor is the c...
2019-12-06 14:37:44

What is the basic term $(a_n)$ of the rotating series $\cos(3n \pi/2)$?

What is the basic term $(a_n)$ of the rotating series $\displaystyle \cos \left( \frac{3n \pi}{2} \right)$ from $1$ to $\infty$, $n \in \mathbb{N}$?
2019-12-06 14:32:32

A trouble with decreasing a function

I have the adhering to price function : $\mbox{BSP Cost}=\sum_{i=1}^{\frac{n}{G}}G^{2}\left\lceil \frac{i}{p}\right\rceil +g\left(p\right)\sum_{i=1}^{\frac{n}{G}}Gi+l\left(p\right)\frac{n}{G}$ I would love to decrease it by picking an ideal G (i.e., G is a function of p and also n). I have actually streamlined it to the list below kind : $\mb...
2019-12-06 14:30:57

Taylor expansion in time of the moment part of a stress and anxiety power tensor

Perform a taylor expansion in 3 measurements in time on the moment compontent of of $T^{\alpha \beta}(t - r + n^{i} y_{i})$ considered that $r$ is a contstant and also $n^{i} y_{i}$ is the scalar item of a regular vector $n$ with a placement vector $y$. I think the formula is $\phi (\vec{r} + \vec{a}) = \sum^{\infty}_{n=0} \frac{1}{n!} (\vec{a}...
2019-12-06 14:29:09

Universal Chord Theorem

Let $f \in C[0,1]$ and also $f(0)=f(1)$. Just how do we confirm $\exists a \in [0,1/2]$ such that $f(a)=f(a+1/2)$? Actually, for every single favorable integer $n$, there is some $a$, such that $f(a) = f(a+\frac{1}{n})$. For any kind of various other non - absolutely no actual $r$ (i.e not of the kind $\frac{1}{n}$), there is a continual func...
2019-12-06 14:28:40

An approximation of an indispensable

Is there any kind of excellent way to approximate adhering to indispensable? $$\int_0^{0.5}\frac{x^2}{\sqrt{2\pi}\sigma}\cdot \exp\left(-\frac{(x^2-\mu)^2}{2\sigma^2}\right)\mathrm dx$$ $\mu$ is in between $0$ and also $0.25$, the trouble remains in $\sigma$ which is constantly favorable, yet it can be randomly small. I was attempting to expan...
2019-12-06 14:26:22

What is $\frac{d}{dx}\left(\frac{dx}{dt}\right)$?

This inquiry was motivated by the Lagrange formula, $\frac{\partial L}{\partial q} - \frac{d}{dt}\frac{\partial L}{\partial \dot{q}} = 0$. What takes place if the partial by-products are changed by complete by-products, bring about a scenario where a function is acquired relative to one variable is set apart by the initial function?
2019-12-06 14:25:43

Partial portion integration

So I intend to locate all antiderivaties of $\frac{x}{x^3-1}$ Since the is of a minimal level than the numerator, partial portion is to be made use of as opposed to lengthy department. I've begun by doing : $\frac{x}{x^3-1} = \frac{A}{x^2+x+1} + \frac{B}{x-1}$ Hence, $x = A(x-1)+B(x^2+x+1)$ Then establishing $x = 1$ to get $3B = 1 =&...
2019-12-06 14:25:37

Expectation of time indispensable

I've 2 inquiries : allowed be $W_s$ a typical Brownian activity : making use of Ito is formula show that $\left( W_t,\int_0^t W_sds \right)$ has a regular circulation ; and also compute $ E\left[e^{W_t}e^{\int_0^t W_sds} \right] .$ For the first component, i recognize that $W_t$ and also $\int_0^t W_sds$ have regular circulation with mea...
2019-12-06 14:25:03

How $\log(t)$ can be the restriction of $t^r$, where $r\to 0$?

My class is addressing the Cauchy-Euler differential formula $a t^2 y'' + b t y' + c y = 0$. The remedies are powers of $t$, $y = t^r$ and afterwards you address for $r$ making use of the particular formula $a r^2 + (b-a) r + c = 0$. This has 2 origins $r = \overline{r} \pm \Delta r$ and also the basic remedy is $y = A t^...
2019-12-06 14:22:29

vectors dot item remedy

Two tangent vectors at a factor on a surface area are $T_1 = 4i + 2j + 3k$ and also $T_2 = -2i - 3j + 1k$ Using the building of the dot item of 2 regular vectors, establish the device vector regular to the surface area at the factor
2019-12-06 14:21:18

Finding the factors of junction in Polar Equations

I must locate the location of the area that exists inside the first contour and also outside the 2nd contour for : $ r^2 = 8cos2\theta, r = 2$ How do I locate $a$ and also $b$? My remedy guidebook informs me that the contours converge when $cos2\theta = 1/2$ yet I do not see where that originated from. I attempted establishing it equivalent to...
2019-12-06 14:17:58

What is the restriction of numerous logarithm ratio $ \frac {\log_{2}(\log_{2}(n))}{\log_{2}(n)}$

Could someone examine if this is deal with? $$\lim_{n \to \infty} \frac {\log_{2}(\log_{2}(n))}{\log_{2}(n)}$$ I exponantiate the numerator and also the with 2 $$\frac {(\log_{2}(\log_{2}(n)))^2}{(\log_{2}(n))^2}$$ $$ = \frac {\log_{2}(n)}{n}$$ I extract the constant from the logarithm $$ = \log_{2}(e) * \lim_{n \to \infty} \frac {\ln(n)...
2019-12-06 14:17:46

Convergence of $\sum \limits_{n=1}^{\infty} (1-\frac{\sin a_{n}}{a_{n}})$ when $\sum \limits_{n=1}^{\infty} a_{n}$ merges

Let $\sum \limits_{n=1}^{\infty} a_{n}$ be a convergent collection when $\forall n, a_{n}\neq 0$. Does $\sum \limits_{n=1}^{\infty} (1-\frac{\sin a_{n}}{a_{n}})$ merge if : $\forall n, a_{n}\gt 0$. $a_{n}\lt 0$ for definitely several $n$'s. For (1) I attempted considering the Maclaurian collection for $\sin x$ : $$|(1-\frac{\sin a_{n}}{...
2019-12-06 14:15:01

maximum location of 3 circles

Hi I am new below and also have a calculus inquiry that showed up at the workplace. Intend you have a $4' \times 8'$ item of plywood. You require 3 round items all equivalent size. What is the maximum dimension of circles you can reduce from this item of product? I would certainly have anticipated I can write a function for the location of the ...
2019-12-06 14:11:04

Derivative of Matrix - Vector Product

Assume $S_1$ and also $S_2$ are 2 $n \times n$ (favorable precise if that aids) matrices, $c_1$ and also $c_2$ are 2 variables taking scalar kind, and also $u_1$ and also $u_2$ are 2 $n \times 1$ vectors. On top of that, $c_1+c_2=1$, yet in the extra basic instance of $m$ $S$'s, $u$'s, and also $c$'s, the $c$ is additionally amount to 1. What i...
2019-12-06 14:09:41

How much to enter a telescoping collection

I'm perplexed regarding just how much to enter regards to obtaining $s_n$ in a telescoping collection. I can not locate a description for this in all in my book. Yet from the troubles I've seen, probably it is simply coincidence, does it have anything to do with the numerator? As an example, the area instance offers : $(\sum_{n=1}^\infty) \fra...
2019-12-06 13:58:14

What type of mathematical approaches and also versions are made use of to design the mind

What type of mathematical devices, versions and also approaches and also academic structures do individuals make use of to imitate the function of the mind is semantic networks? What mathematical buildings do various minds have?
2019-12-06 13:56:48

Bounding a collection from over making use of the indispensable examination

If $a_{n}$ is a non - adverse, lowering series, we understand from the indispensable examination that if $f(n)=a_{n}$ is an integrable function, after that $\sum_{n=1}^{\infty} a_{n}$ and also $\int_{1}^{\infty} f(x)dx$ converge/diverge together. When the theory is confirmed, it is revealed that : $$\forall k \in \mathbb{N}, a_{k+1} \leq \int_{...
2019-12-05 17:20:26

Is $f'(x)$ monotonically raising if $f''(x) >0$?

Offered $f:(0,\infty) \rightarrow \mathbb{R}$ and also $f''(x)>0\forall x \in (0,\infty)$. Is it proper to claim that $f'(x)$ is a monotonically raising function? Can I appropriately think that for any kind of $a,b \in (0,\infty)$ $f'(a) > f('b)$ if $a > b$?
2019-12-05 17:19:50