All questions with tag [math: calculus]


Why integrals with respect to different variables aren't equal?

I have a function $y=x^2+1$, the integral from $-1$ to $2$ is $\int_{-1}^{2}(x^2+1)dx = 6$. The function $x=\sqrt{y-1}$ is the same as the above function. The integral would be from $0$ to $(2)^2+1=5$. So I thought that $\int_{0}^{5}(\sqrt{y-1})dy$ would be equal to the first one. But it turns out that it does not. The integral of the second fun...
2022-07-25 20:47:13

Question On the Proof of The boundedness Theorem

let $f:[a,b]\rightarrow\mathbb{R}$, f continuous on $[a,b]$. I shall prove that $\exists A,B\in\mathbb{R}, \forall x\in[a,b], A\le f(x)\le B$. Proof: Let's define $g(x)=|f(x)|$, we need to prove now that $\exists A\in\mathbb{R}, \forall x\in[a,b], g(x)\le A$. let's suppose that this claim is false, therefore we get: $\forall A, \exists x\in[a,b]...
2022-07-25 20:47:02

Difference in limits because of greatest-integer function

A Problem: \begin{equation}\lim_{x\to 0} \frac{\sin x}{x}\end{equation} causes the remedy: $1$ But the very same function confined in a best integer function causes a $0$ \begin{equation}\lim_{x\to 0} \left\lfloor{\frac{\sin x }{x}}\right\rfloor\end{equation} Why? My ideas: ¢ [The value of the first function often tends to 1 as a result of t...
2022-07-25 20:47:02

convergence of power series

We have given a power series $\sum\limits_{n=0}^\infty a_nx^n$ and $a_0\ne0$. Suppose the power series has radius of convergence $R>0$. Then we may assume that $a_0=1.$ Explanation: It's $$\sum\limits_{n=0}^\infty a_nx^n=a_0\sum\limits_{n=0}^\infty \frac {a_n}{a_0}x^n$$ But why do $\sum\limits_{n=0}^\infty \frac {a_n}{a_0}x^n$ and $\sum\...
2022-07-25 20:46:47

Maclaurin expansion of a given function

I am to expand $\ln(2+x)$ as a Maclaurin collection, I've obtained that $\ln(2+x)=\sum\limits_{n=1}^{ \infty}(-\frac{1}{2})^{n}x^{n}$. Can a person examine it?
2022-07-25 20:46:14

solving Differential Equation

I have the formula below: $$(t^2 + 1)dx=(x+4)dt$$ Where $x(0) = 3$ I am attempting to make use of splitting up of variables, and also I wind up here: $$\ln(x+4)=\arctan(t)+C$$ Trying to streamline it more: $$x=-4+\ln(\arctan(t)+C)$$ Is this proper? I assume I should make use of $x(0) = 3$ to find value of the constant, just how can I do t...
2022-07-25 20:46:10

Proving a theorem on limits that approach infinity.

I want to prove the theorem $\lim_{x\to 0^-}\frac{1}{x^r}=+\infty$ if r is even. So that means I have to show that for any $N>0$ there exists a $\delta >0$ such that if $-x<\delta $ then $\frac{1}{x^r}>N$. First I solved for x in the 'then' statement so I got $x<(\frac{1}{N})^{1/r}$ then multiplied the inequali...
2022-07-25 20:46:07

Is $f$ reduced if and only if the derivations $\gcd(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y})=1$ under some conditions?

I have encountered the following problem. I have no ideas to prove it or disprove it. Suppose that $f\in \mathbb{C}[x,y]$, $f(0,0)=0$, $\frac{\partial f}{\partial x}(0,0)=0, \frac{\partial f}{\partial y}(0,0)=0$. Suppose $f$ is square-free—for simplicity we may first assume that $f$ is irreducible. Then $\frac{\partial f}{\partial x}$ and $\...
2022-07-25 20:43:48

Derivative of a matrix

I need to know if this by-product is proper. I have actually acquired this yet not exactly sure if this is proper. I assume it is yet simply to validate F= A-(B/C)*D where A,B,C and D are square matrices dF/dx(partial derivative) = d(A-(B/C)*D)/dx ----- deriving the final result will be ----------- dA/dx - [(dB/dx - B*inv(C)*dC/dx)/C]*D - (B/...
2022-07-25 20:42:09

Evaluate definite integral $\int_{-1}^1 \exp(1/(x^2-1)) \, dx$

How to review the adhering to precise integral: $$\int_{-1}^1 \exp\left(\frac1{x^2-1}\right) \, dx$$ It appears that uncertain indispensable additionally can not be shared in typical features. I would certainly such as any kind of remedy in preferred primary or non - primary features.
2022-07-25 20:41:51

How to find out what changes applied to integral?

I have actually obtained such indispensable $$\int{\frac{\sqrt{x^2+1}}{x+2}dx}$$ and also with Maple I obtained something similar to this: $$\int\frac{1}{2} + \frac{1+3u^2+4u^3}{-2u^2+2u^4-8u^3}du$$ And I need to know just how to achive this adjustments. I attempted to make use of WolframAlpha, yet there is scarier remedy. This indispensable wa...
2022-07-25 20:41:22

Finding value of a constant in Differential Equations

I have the adhering to ODE Where offered is $x(0)=1$: $$(t+3)dx=4x^2dt$$ After splitting up of variables I obtained this: $$\frac{-1}{x} = 4\ln(t+3)+C$$ I assume this streamlines extra as: $$x=\frac{-1}{\ln((t+3)^4)+C}$$ Please inform me if this is proper, I additionally have trouble searching for C in this instance, MapleTA does decline m...
2022-07-25 20:40:38

Equivalence of two definitions of path (in $\mathbb{R}^3$) length

In a previews question I asked I used the following definition of path length:$\gamma=(x(t),y(t),z(t))$ : $L(\gamma)=\intop_{a}^{b}\sqrt{(x'(t))^{2}+(y'(t))^{2}+(z'(t))^{2}}$. In the answer another definition was used: $$ l(f)= \sup_{P}\sum_{i=0}^k |f(t_{i+1})-f(t_i)| $$ where $f:[0,1]\to \mathbb{R}^3$, $f(0)=a$, $f(1)=b$ and the $\sup$ is take...
2022-07-25 20:39:33

Integral of $x^2\ln(x)$ using Simpson's rule

This is my homework question: Calculate $\int_{0}^{1}x^2\ln(x) dx$ using Simpson's formula. Maximum error should be $1/2\times10^{-4}$ For solving the problem, I need to calculate fourth derivative of $x^2\ln(x)$. It is $-2/x^2$ and it's maximum value will be $\infty$ between $(0,1)$ and I can't calculate $h$ in the following error formula for u...
2022-07-25 20:21:06

True statement?

Is the adhering to declaration real? If the by-product of a function $f(x)$ is an item $g(x)\cdot \dfrac{1}{h(x)}$ where $g$ and also $h$ are continual features, after that the function $f(x)$ is a differentiable continual function.
2022-07-25 20:21:06

Try to evaluate $\lim_{\alpha\to1^-}\frac{1}{\alpha(\alpha-1)}\left[\frac{1}{m}\sum_{i=1}^{m}\left(\frac{y_i}{y^{'}}\right)^\alpha-1\right]$

I can not review this restriction. $$\lim_{\alpha\to1^-}\frac{1}{\alpha(\alpha-1)}\left[\frac{1}{m}\sum_{i=1}^{m}\left(\frac{y_i}{y^{'}}\right)^\alpha-1\right]$$ where $y_i>0$, $y^{'}$ is the standard of $y_i$
2022-07-25 20:20:51

Limit Evaluation

Given $a>1$ and $f:\mathbb{R}\backslash{\{0}\} \rightarrow\mathbb{R}$ defined $f(x)=a^\frac{1}{x}$ how do I show that $\lim_{x \to 0^+}f(x)=\infty$? Also, is the following claim on sequences correct and can it be used somehow on the question above(by using Heine and the relationship between sequences and fucntions)? Given two sequences ...
2022-07-25 20:18:05

power series estimate (convergence)

Let $f(x)=\sum\limits_{n=0}^\infty a_nx^n$ a power series and $f(0)\ne0$. (w.l.o.g. $f(0)=1$) Suppose the power series has radius of convergence $r>0$. A power series is continuous in her convergence interval. So there is a $\delta\in]0,r[$ so that for $|x|<\delta$ it's $|a_1x+a_2x^2+\dots|<1$. My Questions: why is $|a_1x+a...
2022-07-25 20:17:03

Continuity Problem

Assume a school gym allows entrance of students in blocks: Block 1: 08:00 AM to 09:00 AM Block 2: 09:00 AM to 10:00 AM ...etc Until 10 PM which is closure. Assume there's a maximum of users that can be in the gym at any given moment (15). So, to get in, people have to get in line in order to get considered for the next block. If someone arrives...
2022-07-25 20:15:56

Prove that the Taylor series converges to $\ln(1+x)$.

Prove the adhering to declaration. For $0 \leq x \leq 1$, the Taylor Series, $\displaystyle x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots$ merges to $\ln(1+x)$ Any aid will certainly be substantially valued! Thanks!
2022-07-25 20:15:12