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

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0answers
5 views

Convergence of $f_n(x)$

So, I have those series: $f_n(x) = \frac{x^n}{1+x^n}$ I use ratio test, for finding convergence: $\Large \lim \frac{\frac{x^{n+1}}{1+x^{n+1}}}{\frac{x^n}{1+x^n}}=\lim \frac{x(1+x^n)}{1+x^{n+1}}=\lim ...
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0answers
7 views

An integral question $\int_a^b x \cdot sinx \cdot \sqrt{1-x^2}$ dx

could anyone please show me how to integral this function? $\int_a^b x \cdot sinx \cdot \sqrt{1-x^2}$ dx
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1answer
28 views

If $f(\frac{x+y}{2})=\frac{f(x)+f(y)}{2}$, then find $f(2)$

Let $f(\frac{x+y}{2})=\frac{f(x)+f(y)}{2}$ and it is given that $f(0)=1$ and $f'(0)=-1$, where $f'$ denotes first derivative. Find the value of $f(2)$ Could someone tell me how to use $f'(0)=-1$ ...
2
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1answer
20 views

Do regions of integration cover *all* possible regions?

Now, I'm not real concerned about the integration itself but more about integration regions. Is it true that all possible regions in two-space are covered by the following setup: $$\int^a_b ...
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4answers
70 views

Solving $\frac{dy}{dt}+y=1$ differential equation?

When solving $\frac{dy}{dt} +y = 1$, I get the answer $y= 1-Ae^t$, however the answer claims to be $y= 1-Ae^{-t}$ Where does this minus come from?
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2answers
186 views
+50

Is there any solution to find a condition for $f(x)=a+bx^n+cx^2-dx>0$ to always hold true?

Okay, I am interested to know the criteria for a function to always hold $$f(x)=a+bx^n+cx^2-dx>0,$$ if it is given that $a, b, c>0$ and $n\in(-2,2)$ is some real number and $x>0$. My idea ...
0
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1answer
36 views

Does the integral converge

How to prove that the following integral doesn't converge? $$\int_0^\infty \frac{1}{(\ln^4x + \ln^2x)\ln^2(1-x^{1/3})^2(x + \sqrt{x} + 1)}dx$$ I suppose it doesn't converge because of quick growth ...
1
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2answers
70 views

The notion of “infinitely differentiable”

Wiki takes me to the section "smoothness" which I don't entirely get, it's just too much stuff for me. My question is, what exactly is it? An infinitely differentiable function is one that is ...
0
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3answers
37 views

Is there any way to calculate the roots of this polynom?

I need to calculate the roots of the real function $f$: $$ f(x)=\frac{-{x}^{3}+2{x}^{2}+4}{{x}^{2}} $$ But I am not able to decompose the numerator. There should be only one real solution and two ...
1
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1answer
22 views

Calculus of variations with inequality and non-integral constraints

I have a question on solving an optimization problem with calculus of variations. I am attempting to maximize the functional $$ J[y] = \displaystyle\int_a^b F(x,y,y') \, \mathrm{d}x, \tag{1}$$ ...
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1answer
19 views

Where does this come from? or how do I derive it?

$$\delta \vec{x} = \frac{\partial\vec{x}}{\partial r}\delta r + \frac{\partial\vec{x}}{\partial \theta}\delta \theta+\frac{\partial\vec{x}}{\partial \phi}\delta \phi$$
6
votes
1answer
48 views

Integral with arithmetic-geometric mean ${\large\int}_0^1\frac{x^z}{\operatorname{agm}(1,\,x)}dx$

The arithmetic-geometric mean$^{[1]}$$\!^{[2]}$ of positive numbers $a$ and $b$ is denoted $\operatorname{agm}(a,b)$ and defined as follows: $$\text{Let}\quad a_0=a,\quad b_0=b,\quad ...
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3answers
26 views

Find particular solution to nonhomogeneous DE $y'+y=x^2+\sin{x}+\cos{x}$

I'm new to nonhomogeneous DE's and I have come across this DE: $$y'+y=x^2+\sin{x}+\cos{x}$$ which I'm supposed to provide a general solution to. However, I get stuck with the particular solution. The ...
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5answers
105 views

calculating $\lim_{n\to\infty} \frac{1+\frac{1}{2}+…+\frac{1}{n}}{n}$

How can I prove that $\lim_{n\to\infty} \frac{1+\frac{1}{2}+...+\frac{1}{n}}{n}=0$? I can't use $1+\frac{1}{2}+\cdots +\frac{1}{n}\approx \log n$ I've tried to use the following: $\lim_{n\to\infty} ...
0
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1answer
15 views

Prove that the evolute of the tractrix $x=a(\cos t+\log \tan\frac{t}{2}),y=a\sin t$ is the catenary $y=a\cosh (\frac{x}{a})$

Prove that the evolute of the tractrix $x=a(\cos t+\log \tan\frac{t}{2}),y=a\sin t$ is the catenary $y=a\cosh (\frac{x}{a})$ Since evolute of a curve is the envelope of the normals of that curve.I ...
0
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0answers
12 views

Rotation of a coordinate system

Suppose that I rotate the (traditional) coordinate system $(x,y)$ by an angle $\theta$ to obtain a new coordinate system $(s,n)$. Consider a velocity vector $$v = (v_x,v_y) = v_xe_x + v_y e_y,$$ where ...
0
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0answers
26 views

$f$ is $3-$times differentiable and has at least $5$ distinct real zeroes, prove $f+6f'+12f''+8f'''$ has at least two distinct real zeroes?

Let $f$ be a three times differentiable function (defined on $\mathbb{R}$ and real-valued) such that $f$ has at least five distinct real zeroes. Prove that $f+6f'+12f''+8f'''$ has at least two ...
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6answers
1k views

What's my confusion with the chain rule?

When deriving $x^x$, why can't you choose $u$ to be $x$, and find $\dfrac{d(x^u)}{du} \dfrac{du}{dx} = x^x$?. Or you could go the other way and find $\dfrac{d(u^x)}{du}\dfrac{du}{dx}$, giving ...
1
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1answer
26 views

How to calculate $\lim_{(x,y)\to(1,2)} [\ln(5-(x^2+y^2))]\sqrt{y^2-4}$?

I couldn't find the limit $$\lim_{(x,y)\to(1,2)} [\ln(5-(x^2+y^2))]\sqrt{y^2-4}.$$ I tried substituting $x$ and $y$ with polar coordinates but didn't get far. Wolfram says the answer is $0$. Thank ...
0
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1answer
19 views

How does this faulty system of integration change the nature of jump discontinuity?

Let's define a sort of faulty integral. For the purposes of this question we shall assume that this is the regular integral. This integral integrates all functions properly however it's gets confused ...
0
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0answers
18 views

Derivative atan2 of a function

I am not able to understand how to solve my doubt. I need to do the : $\frac{\partial}{\partial p} atan2({\cos(\alpha)},{\sin(\alpha)})$ I will compute $\cos(\alpha)$ and $\sin(\alpha)$ as: ...
5
votes
1answer
82 views

Why does this approximation work (and why does it fail)?

I have a function $$f(x)=\frac{e^{-x}}{x}$$ and I am trying to find an expression for the inverse function $f^{-1}(x)$. So far I have come up with the approximation: $$\hat{f}^{-1}(x)=\left( ...
9
votes
3answers
245 views

100th derivative of $e^{-x^2}$ at point $0$

Problem: Find $\frac{\mathrm d^{100}}{\mathrm dx^{100}}e^{-x^2}$ at point $0$. My attempt: $y'=-2xe^{-x^2}$ I tried to use General Leibniz rule and I didn't get much better information. Without: ...
4
votes
3answers
234 views

Integration theorem: enough assumptions?

Let $f:[a,b]\to\mathbb{R}$ be a continious function. Show that if $$\int_a^b f(x)g(x)dx=0$$ for all continious functions $g:[a,b]\to\mathbb{R}$ with $g(a)=g(b)=0$, then $f(x)=0$ $\forall ...
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2answers
30 views

axiom of continuity guarantees no gaps exist on the real axis?

I read this content at the bottom of this page just wonder why axiom of continuity could guarantee no gaps exist on the real axis? any proofs ?
0
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0answers
16 views

global inverse function theorem

Consider a continously differentiable non-constant function $f:\mathbb{R}^2\to\mathbb{R}$. Define $$ K=\{x\in\mathbb{R}^2|f(x)=0\}. $$ I wish to know whether there is a continuously differentiable ...
0
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0answers
21 views

Maximizing Likelihood Function.

Let, $$\mathbf y_i = \mathbf X_i\mathbf\beta + \mathbf Z_i\mathbf b_i+ \mathbf\epsilon_i,$$ where $\mathbf y_i\sim N(\mathbf X_i\mathbf\beta, \Sigma_i=\sigma^2\mathbf I_{n_i}+\mathbf Z_i \mathbf ...
1
vote
1answer
29 views

Calculate the integral $\iint_D (y^2-x^2)^{xy} (x^2+y^2)dxdy$ on a certain region

Let $D$ be the region that's bounded by $xy=a, xy=b, y^2-x^2=1, y=x$ in the first quadrant. Calculate the integral $\iint_D(y^2-x^2)^{xy}(x^2+y^2)dxdy$. Firstly, I was able to show that the boundary ...
5
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2answers
458 views

dy/dx=0 has uncountable many solutions

Suppose $y(x)$ is continuous and $y'(x)=0$ has uncountable many solutions but $y(x)$ is not constant on any interval. Is this possible?
0
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1answer
32 views

Using Substituions in the conditions of a limit

I know that $$\lim_{x\to 0} \dfrac{\ln(1+x)}{x}=1$$ I was trying to modify this result a bit. Consider a function $f(x)$ such that $$\lim_{x\to a} f(x)=0$$ Substituting $x=f(x)$ in the original ...
2
votes
1answer
49 views

Exponential limit of the form $0^\infty$

I was trying to derive a general expression for the limit $$\large{y=\lim_{x\to a} f(x)^{g(x)}}$$ where $\lim_{ x \to a} f(x)=0$ and $\lim_{ x \to a} g(x)=\infty$ $$$$ I managed to reach till here: ...
1
vote
1answer
45 views

Why does $\sin\phi=r\frac{d\theta}{ds}$ and $\cos\phi=\frac{dr}{ds}?$

The relation between $p$ and $r$ where $p$ is the length of the perpendicular from the fixed point $O$ on the tangent to the curve at any point $P$ is called pedal equation of the curve. I want to ...
2
votes
2answers
55 views

Solve the differential equation:$\frac{\,dx}{mz-ny}=\frac{\,dy}{nx-lz}=\frac{\,dz}{ly-mx}$

QUESTION: Solve the differential equation: $$\frac{\,dx}{mz-ny}=\frac{\,dy}{nx-lz}=\frac{\,dz}{ly-mx}$$ MY ATTEMPT: I tried out to proceed by using ...
1
vote
1answer
60 views

Convergence of $g(x)\cdot f(x)$

Let $g(x)=\frac{1-e^{-x^2}}{x^2}$ for $x \neq 0$,$g(0)=1$ and $f(x)=e^{-(x-n)^2}$. You can assume that g(x) is continuous and bounded with maximum 1 in x=$0$. Show that $\sum_{n=1}^{\infty}g(x)\cdot ...
2
votes
1answer
34 views

Which exponents r>0 is the limit finite

I am trying to find values of $r>0$ such that $\lim\limits_{n\rightarrow \infty} \sum\limits_{k=1}^{n^2}\frac{n^{r-1}}{n^r+k^r}$ is finite. I have tried to use integral methods for this limit such ...
1
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0answers
13 views

Trajectory of vector field$ v = (x^2 -1, xy)$

I have been tasked with finding the trajectory of the vector field $v = (x^2 -1, xy)$ My solution yields $x^2 - y^2 / c = 1$ However, the textbook answer gives $x^2 +- y^2 / c^2 = 1$. I ...
2
votes
3answers
62 views

How to show $\lim _{x\to 0}\:\frac{\sin\left(\frac{1}{x^2}\right)}{x^2}$ does not exist?

$$\lim _{x\to 0}\:\frac{\sin\left(\frac{1}{x^2}\right)}{x^2}$$ I my intuition is telling me this limit does not exist as $\sin$ will be oscillating but will stay bounded and then will blow up as ...
0
votes
1answer
37 views

$n$th derivative of function $\frac{1}{(1-2x)^2}$

I am trying to find the $n$th derivative of the function $\frac{1}{(1-2x)^2}$. The first three are simple but I can't see a schema right now. \begin{align*} y^{\prime} & = \frac{4}{(1-2x)^3}\\ ...
2
votes
3answers
89 views

The values of constants in the Equation.

$$ \frac{\int_0^{4\pi} e^t(\sin^6at+\cos^4at)\,dt}{\int_0^\pi e^t(\sin^6at+\cos^4at)\,dt}= L, $$ the question asks the value of $a$ and $L$. My friend solved it by differentiating, but i didn't ...
0
votes
1answer
55 views

Riemann Sum of $f(x)=2^x$

Using Riemann Sums, how can I compute the integral $$\int_{0}^{2} 2^x dx$$ I don't know how can I take the Partition and then compute the sums , someone can help to understand this method of Riemann ...
2
votes
2answers
52 views

Is continuity at a point only defined for points in the domain?

I'm using Michael Spivak's Calculus, 3rd edition textbook. Without ado, I'll state the definition given for continuity at a point: DEFINITION$\;\;\;\;$The function $f$ is continuous at $a$ if: ...
2
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1answer
2k views

Derivative of conjugate transpose of matrix

Building off of my previous question, I am trying to derive the normal equations for the least squares problem: $$ \min_W \|WX - Y\|_2 \\ W \in \mathbb{C}^{N \times N} \quad X, Y \in \mathbb{C}^{N ...
0
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0answers
23 views

Forming calculus equations by reasoning (to solve a physics problem)

It's been 20 years since I last did calculus, and I came across a problem I'd like to know how to solve with reasoning: how to turn basic linear observations into a formula that enables integration. ...
0
votes
2answers
68 views

Integrate $ \int \frac{4x^2+2x}{(3x-1)(x^2+1)}dx$.

$$ \int \frac{4x^2+2x}{(3x-1)(x^2+1)}dx$$ I am having difficulty determining which integration technique to use for the above question. I have tried partial fraction decomposition with two ...
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1answer
16 views

Bounded convergence theorem - 2M

Can someone please help me with where the 2M is coming from?
6
votes
3answers
92 views

Is there an infinite sequence of real numbers $a_1, a_2, a_3,… $ such that ${a_1}^m+{a_2}^m+a_3^m+…=m$ for every positive integer $m$?

Is there an infinite sequence of real numbers $a_1, a_2, a_3,...$ such that ${a_1}^m+{a_2}^m+a_3^m+...=m$ for every positive integer $m$? I tried assuming that the sequence $a_1^m, a_2^m,...$ ...
0
votes
1answer
51 views

Can $\int_0^1 \frac{1}{x} e^{-x} dx$ be integrated?

I have an integral with a singularity at $x = 0$. $$\int_0^1 \frac{1}{x} e^{-x} dx$$ It's not a removable singularity so is it possible to perform the integration? For example could some complex ...
0
votes
0answers
2 views

Convective operator: Distributive with superposition of vector fields?

I'm unable to find a great deal of information on this. I'm mostly sure that the convective operator over a vector field $A$ acting on a function $f$: $$ (A \cdot \nabla )f$$ is distributive, i.e. ...
41
votes
7answers
52k views

What is Jacobian Matrix?

What is Jacobian matrix? What are its applications? What is its physical and geometrical meaning? Can someone explain with examples? Thanks! :)
0
votes
2answers
101 views

If $f^{-1}(x)=\frac{1}{f(x)}$ then find $f(1)$

For $a>1$ we have: $f:[\frac{1}{a},a]\to [\frac{1}{a},a]$ be a bijective function. Suppose $f^{-1}(x)=\frac{1}{f(x)}$ for all $x \in [\frac{1}{a},a]$ then find $f(1)$. Could someone give me ...