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

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0answers
3 views

Getting from a product of gamma functions to a fraction answer

I am working on an assignment question for my Advanced Calculus course and am having great difficulty working it out. In order to try and understand this type of question/working, I have found a ...
1
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0answers
6 views

Is there a way to derive the chain rule using the definition of the differential of a function?

Given a composite function, $ y = (f \circ g)(x) $ that is continuous and differentiable for all $x$, we know from chain rule that $$ \frac{dy}{dx} = \frac{d(f \circ g)}{dx} = \frac{df}{dg} ...
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0answers
5 views

what method should I use? The power formula? or The logarithmic formula? (indefinite integral)

Should I use the power formula or logarithmic? $$\int \frac {S(x+1) ~dx} {\sqrt[3]{x^2 + 2x + 1}}$$
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0answers
12 views

differential equation as taylor series

Consider the equation $\frac{d x(t)}{dt} = g(x(t))$ , with $x(0) = x_0$, where g is function that admits derivatives of all orders.If the solution of the equation can be written as a series of taylor ...
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0answers
10 views

find $\alpha$ such that $f(x)=x^\alpha*sin(\frac{1}{x})$ is n th order derivable at (0,0)

Sorry for my bad English. Define f(x)=0 when x=0; This is a problem that our calculus teacher mentioned in the class, but now I'm not sure if I correctly understand it. In my opinion, as long as ...
2
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0answers
22 views

L'Hopital quicky

suppose L'Hopital applies and $$\lim_{x\to\infty}\frac{f(x)}{g(x)} = \lim\frac{f'(x)}{g'(x)}$$ under what conditions is it true then that $$\lim_{x\to\infty}\frac{\frac{f(x)}{g(x)} }{ ...
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0answers
20 views

Find the Intervel of Convergence of the Power Series

One question for you guys. I need to find the interval of convergence of the power series. Having a lot of problems with this one and would love a thorough explanation, however any help is ...
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1answer
34 views

Prove Convergence or Divergence

I just need to prove either convergence or divergence for this. Having some serious trouble and would appreciate all help! $$\sum_{n=1}^{\infty}\frac1{n^{1/3}(1+n^{1/2})}$$
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0answers
18 views

List all equations for straight line! [on hold]

Can someone list all the equations for a straight line geometry? Thank You.
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0answers
12 views

Question about a region in three- dimensional space

Let $D$ be the set enclosed by the planes $x=0$, $y=0$, $z=0$, $x+y+z=1$. Let $f(x,y,z) = x $. I want to find $\int_D f $. MY try: $D$ is just the tetrahedron, so I am looking for the integral $$ ...
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1answer
59 views

Evaluate $\int_{0}^{\infty} \frac{dx}{(x+1)(x+2)(x+3)}$

Please see below: $$\int_{0}^{\infty} \frac{dx}{(x+1)(x+2)(x+3)}= (\frac{1}{2}\ln(x+1)+\frac{1}{2}\ln(x+3) - \ln(x+2)$$ I do not have problem evaluating the above integral integral itself, however I ...
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3answers
30 views

Figuring out acceleration without knowing the time length

A car goes from 50mph to 20mph in an unknown amount of time. All that is known is that one of the car's wheels rotated 110 times during the process and that the wheel rotates at a rate that is uniform ...
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2answers
34 views

Can someone please explain the ratio test to me please?

I'm having trouble with the steps of the ratio test. So it's from $1$ to $\infty$ and $\dfrac{17^n}{n!}$. I there's division involved to find $r$, but what do we divide?
1
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1answer
33 views

A problem about center of mass

Suppose $f(x)$ is positive, increasing and Riemann-integrable on the interval $[a,b]$. Let$$\bar{x}=\frac{\int_{a}^{b}{xf(x)\text{d}x}}{\int_{a}^{b}{f(x)\text{d}x}}.$$Prove ...
0
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1answer
19 views

Is $hh^T$ positive semi-definite ($h$ is a column non-negative vector)? [duplicate]

Is $hh^T$ positive semi-definite? It seems to be positive semi-definite, but I cannot prove it. Please help:)
1
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1answer
15 views

Prevent Maple to evaluate before simplify a function

I have been trying to find a domain of $f(x)=\frac{x}{\frac{(x+2)}{(x-3)}}$ using different kind of software ( its clear the domain of this function is $\mathbb{R}\backslash \{-2,3\}$ ). When I tried ...
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0answers
13 views

Find the flux of the vector field across the boundary of the cube

Find the flux of the vector $F=e^{xy} \hat{i} +e^{yz} \hat{j} +z \hat{k}$ across the boundary of $[0,1] \times [0,1] \times [0,1]$. Can someone tell me the setup of this problem?
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0answers
3 views

Find the flux of the vector field across the closed surface of the three-dimensional region E

Let E be the part of the ball $x^2+y^2+z^2 \leq 1$ with $z \geq 0$. Find the flux of the vector field $F=2x^2y \hat{i} +2yz \hat{j}-z^2 \hat{k}$ across the closed surface of the three-dimensional ...
2
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1answer
45 views

Triple integration problem

Prove that: $$\iiint\limits_B \frac{1}{x} dx\,dy\,dz=\frac{8-4\sqrt2}{3}$$ where $$B=\{(x,y,z):1\leqslant x \leqslant e^z, y\geqslant z, y^2+x^2\leqslant 4\}.$$ I used Mathematica's regionplot3D to ...
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1answer
35 views

Why is $\underline{F}$ the Fréchet derivative of $\underline{F} : \mathbb{R}^n \rightarrow \mathbb{R}^m$?

The questions asks what the Fréchet derivative of $F : \mathbb{R}^n \rightarrow \mathbb{R}^m$ is. The solution states that by inspection of the definition of the Fréchet derivative, ...
2
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2answers
12 views

Jacobian matrix for $f(h)=hh^Th$, where $h$ is an $m$ dimensional vector

I have a function $f(h)=hh^Th$, can we say $\nabla f(h)=2*hh^T + h^ThI_{m\times m}$, where $I_{m\times m}$ is an identity matrix?
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2answers
51 views

Find the sum of the following series

Find the sum of the following series $$ \sum_{n=1}^\infty (-1) \frac{1}{n}\frac{9}{6^n}. $$ I think that $r$ is $\frac{9}{6^n}$ and $a$ is $-1$. But I'm not positive if I'm starting this problem ...
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1answer
66 views

Evaluating $\int_1^2 \int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}}(x)\:\mathrm{d}y\:\mathrm{d}x$ using polar coordinates?

How is the following integral found using polar coordinates. $$\int_1^2 \int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}}(x)\:\mathrm{d}y\:\mathrm{d}x$$ I know the the part of the domain the circle being asked in ...
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1answer
20 views

Different ways to formally define trigonometric functions

When I first learnt trigonometric functions I was in highschool and obviously the explanation they gave me was mostly intuitive. Now that I have taken my first curse of calculus I learnt a formal ...
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1answer
26 views

Suppose $\mu$ is not an eigenvalue of A. Show that the equation $x'= Ax + e^{\mu t}b$.

Suppose $\mu$ is not an eigenvalue of $A$. Show that the equation $x'= Ax + e^{\mu t}b$ has a solution of the form $\varphi(t) = ve^{\mu t}$.
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0answers
22 views

Finding distance of point from 4D ray

I'm working on a programming project. In this project, a ray is fired from a point in 4-space. I need to find the distance from this ray to a number of other points in 4-space. I attempted to solve ...
0
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1answer
37 views

Don´t know how to start proving this formula.

\begin{equation*} \int \frac{\cos ^{m}x}{\sin ^{n}x}dx=-\frac{\cos ^{m+1}x}{(n-1)\sin ^{n-1}x}- \frac{m-n+2}{n-1}\int \frac{\cos ^{m}x}{\sin ^{n-2}x}dx+C,\qquad (n\neq 1). \end{equation*} I`d like to ...
2
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1answer
40 views

Let $A$ a matrix with real or complex entries. Proof that $\displaystyle\lim_{n\rightarrow\infty}(E+\frac{A}{n})^n=e^A, E=$indentity.

Let $A$ a matrix with real or complex entries. Proof that $\displaystyle\lim_{n\rightarrow\infty}\left(E+\frac{A}{n}\right)^n=e^A, E=$indentity. I thought of using the limit, but do not know where ...
0
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1answer
13 views

Proving that a function is onto in an interval [on hold]

I got this question: Let $f(x)=\dfrac{1}{\sin x}+\dfrac{1}{x-1}$. Prove that in the interval $(0,1)$, $f$ is onto $\mathbb{R}$ (that is, prove that $f((0,1))=\mathbb{R}$). Thanks.
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4answers
38 views

If the sequence converges, find it's limit. [on hold]

$$ \frac{17n-2}{9-\sqrt{n}} $$ I think it diverges, but I'm not sure.
1
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1answer
22 views

Show a 2-form is exact finding a primitive.

I have to show that $\omega=-4xy\:\mathrm{d}x\wedge \mathrm{d}y-2xz\:\mathrm{d}z\wedge \mathrm{d}x +2yz\:\mathrm{d}y\wedge \mathrm{d}z$ is exact finding a primitve of $\omega$ (by Poincare lemma I ...
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1answer
31 views

A question about a continuous function that satisfies the property $\forall x\in\mathbb{R},\exists x<y\in\mathbb{R},f(x)<f(y)$

I got this question: Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function that satisfies the property: forall $x\in\mathbb{R}$ there exists $y \in\mathbb{R}$ such that $x < y$ and ...
0
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3answers
38 views

Find the limit of the sequence if it converges, otherwise state divergence.

Find the limit of the sequence given by $$\frac{10+12n+20n^4}{7n^4 + 5n^3 - 20}$$ I think the answer is $\frac{20}{7}$ after dividing, but is that right?
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3answers
44 views

Are discontinuous functions integrable? And integral of every continuous function continuous?

According to me answer of second part is yes as integration simply means area under curve.
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4answers
463 views

Is it necessary that every function is a derivative of some function?

I thought about this a lot and consulted a lot of people but everyone had contradicting answers. I am a high school student. please help.
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2answers
50 views

Find the derivative of $1/\sqrt{1+x^2-\cos^2x-e^{2\pi \cos(\sin 1/x)}}$

(calculus) How can I prove that $$\frac{d}{dx}\frac{1}{\sqrt{1+x^2-\cos^2x-e^{2\pi \cos(\sin 1/x)}}}=\frac{-\frac{\displaystyle\pi\sin(\sin(1/x))\cos(1/x)e^{2\pi\cos(\sin(1/x))}}{x^2}+x+\sin x+\cos ...
2
votes
2answers
32 views

Integration by reduction

I have learnt how to integrate by reduction formula but this one seems to give me hell someone to lift me by telling me what to do or simply to solve it. \begin{equation} I_n=\int\sec^n x\,dx ...
1
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4answers
38 views

Find the limit: limit x tends to zero [arcsin(x)-arctan(x)]/(x^3)

I'm having difficulty in finding the following limit. $$\lim_{x\to 0}\frac{\arcsin(x)-\arctan(x)}{x^3}$$ I tried manipulating the given limit in standard limit(s) but I got nowhere. I tried ...
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2answers
28 views

Find the equation of the line tangent to the following curve at x=1. Write your answer in y=mx+b format.

Find the equation of the line tangent to the following curve at $x=1$. Write your answer in $y=mx+b$ format. The curve is defined by $y=2x^2+6x-4.$ Please help with step by step instructions...I ...
1
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2answers
61 views

Evaluate $\displaystyle\lim_{j\to0}\lim_{k\to\infty}\frac{k^j}{j!\,e^k}$

I found this problem in my deceased grandpa's note today when I was visiting my grandma's home. \begin{equation} \lim_{j\to0}\lim_{k\to\infty}\frac{k^j}{j!\,e^k} \end{equation} I asked my brother and ...
0
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2answers
34 views

Fundamental exponential derivative problem

I know the formula: $$\frac{\mathrm{d}}{\mathrm{d}x}e^{u}=e^{u}\frac{\mathrm{d}u}{\mathrm{d}x}$$ but I don't know how to sovle this problem: ...
2
votes
2answers
24 views

Linear Differential Equation achieving the answer.

The question states $t(\dfrac{dy}{dt}) - 3y = t^4$ As a first step I am told to divide through by $t^4$ - can anyone explain the purpose of this? Following this I get $t^{-3} (\dfrac{dy}{dt}) - ...
0
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1answer
16 views

Separable Differential Equations with log in the question

the question is $y \log y-t(\dfrac{dy}{dt}) = 0$ I have separated the question to $4\dfrac{1}{t} dt = \dfrac{1}{y\log y} dy.$ Integrating would give me $log(t) = \log(y\log y) + c$. How do I ...
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2answers
18 views

How to measure monotonicity of a list of values

I need to compare monotonicity of lists of values. I have $S=(n_1,n_2,...n_n)$, I need a function $\mathrm f(S)$ to return the monotonicity of the S. $S_1=[1,2,4,4,8]$ $S_2=[8,4,4,2,1]$ ...
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1answer
40 views

Uncomfortable using Leibniz notation for the chain rule.

I am working through the following solved problem which uses separation of variables to get two ODEs. The problem is to show that $$\frac{1}{\sin\theta ...
1
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1answer
39 views

Help with understanding infinite limit of a sum.

While I was practicing some math competition problems, I came across a very interesting infinite limit problem with a summation: $$L= \lim_{n \to \infty} \sum_{k = 1}^n\left(1 + \frac{k}{n} ...
-1
votes
1answer
63 views

Prove that $\cos^2\theta+\sin^2\theta=1$ [duplicate]

I try to find the question but I didn't How do you do it? I'm really stuck on this proof. Can someone please explain?
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2answers
18 views

Stationary points of a function

$F(x)=x^3+Ax+B$ has a stationary point at $(-2,3)$. a) Find $A$ and $B$ and then find the nature of all stationary points. Thank you!
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0answers
22 views

What will be the value of ß? [on hold]

If, inv ß = 2.994 than, what will be the value of ß and what will be the formula? Where ß is an angle in degrees.
0
votes
1answer
20 views

How to find the area bounded by three or more curves

The area bounded by two curves can found by subtracting the integrands.Is there a general way to find the area bounded by three or more curves?