Questions tagged [calculus]
For basic questions about limits, continuity, derivatives, differentiation, integrals, and their applications, mainly of one-variable functions.
128,715
questions
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Irrational equation divided: I can't solve it algebraically!
I am facing this equation but I can't solve it.
Text of the exercise in the picture below:
Could you solve it, please?
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0
votes
1
answer
49
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The mathematical sense of a physical quantity
I am not a good student, so, perhaps, my question is stupid, however...
What is the mathematical sense of a physical quantity? For example, if we consider the mass, can we say it is a surjection (...
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0
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11
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Particular integral of this equation [closed]
What is the particular integral of th following equation?
(D-1)²(D²+1)²y=e^(x)+x
0
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0
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9
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Prove that $\int_0^\infty \int_0^\infty \int_0^R f(k,q,t,r) \,\mathrm{d}t \,\mathrm{d}q \,\mathrm{d}k=g(r)$ where both $f$ and $g$ are known functions
I would like to prove that
$$
\int_0^\infty \int_0^\infty \int_0^R f(k,q,t,r) \,\mathrm{d}t \,\mathrm{d}q \,\mathrm{d}k = g(r) \, ,
$$
with
$$
f(k,q,t,r) = \frac{4qk}{\pi \alpha^2 \left( K-k \right)}\...
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0
votes
1
answer
58
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Limit gives two different answers depending on approach
So I have a function
$$g(t)=\frac{f'(t)\sin(\frac{t-x}{2})-0.5\cos(\frac{t-x}{2})[f(t)-f(x)]}{\sin(\frac{t-x}{2})^2}$$.
Then, I want to find
$$\lim_{t \to x}g(t) $$ assuming that $$f''(t)$$ exists.
...
0
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0
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36
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When considering an inequality such as $x+\sin x \cos x \ge x-1$ why is it sufficient to only prove that x-1 goes to infinity?
Conceptually, I understand that if
$$\lim\limits_{x\to \infty} x-1 = \infty$$
and if $x-1 \leq x + \sin x \cos x$, then it must follow that
$$\lim\limits_{x \to \infty} x + \sin x \cos x = \infty$$
I ...
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0
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0
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15
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Normalize a wavefunction
This is a little beginner level, but how do we normalize Ψ= x(a-x). I do not understand how separate the a from the equation.
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2
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0
answers
13
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Single objective optimization with $n$ independent variables
I have a single objective optimization problem.
The problem can consist of $n$ independent variables $(x_1,x_2,...,x_n)$ four functions $f_i(\bf{x})$ and a function $Q(\bf{x})$.
I want to monitor how ...
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votes
2
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48
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Find the limit: $\lim_{n\to\infty} \frac{\lfloor\sqrt{2n-1-\cos(n)}\rfloor}{2\sqrt{n}+3}$
Please keep in mind that I am new to limits as we have just touched limits in the semester.
I have the limit
$$\lim_{n\to\infty} \frac{\left\lfloor\sqrt{2n-1-\cos(n)}\right\rfloor}{2\sqrt{n}+3}$$
How ...
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1
vote
2
answers
53
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How to find the value of $x+y$ in the following equation? [duplicate]
Suppose $(2x+1)^2+y^2+(y-2x)^2=\frac{1}{3}$, what's the value of $x+y$?
I tried to use an algebraic identity
$(....)^2+(....)^2=0$
but did not succeed.
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2
votes
0
answers
16
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Integrals of Jacobi $\vartheta$ functions on the interval $[1,+\infty)$
I start from the following obvious observation, which is declared to be($q=e^{-\pi x}$):
\begin{aligned}
\int_{1}^{\infty}x\vartheta_2(q)^4\vartheta_4(q)^4
\text{d}x&=\int_{0}^{1}x\vartheta_2(q)^4\...
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0
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7
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Proving the equality between the cross product and integral definitions of curl
Let $F(x,y) = (M(x,y),N(x,y))$ be a vector field on real plane.
Then the curl of our vector field is something like:
$\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}$
The curl at the point ...
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1
vote
0
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46
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How did Newton compute derivatives by first computing power series?
I've read that the way we compute power series these days is not how it was originally done. These days, we compute the derivatives of the function we wish to expand in order to compute the power ...
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2
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0
answers
16
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Can neglecting the use of a definite integral property when it is very obvious yield incorrect answers? [duplicate]
I was analyzing this integral which prompted me to ask this question:
\begin{aligned}
& \int_0^{2 \pi} \cos ^{-1}\left(\frac{1-\tan ^2 \frac{x}{2}}{1+\tan ^2 \frac{x}{2}}\right) d x \\
& \...
0
votes
1
answer
26
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Derivative of the Laplace Transform [closed]
I calculated that $$ \mathcal{L}'[f(t)] = \mathcal{L}[f'(t)]-s\mathcal{L}[f(t)] $$ And if I try to substitute some values for f(t) I always get zero. So, how do I demonstrate that it gives, or doesn't ...
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4
votes
1
answer
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Struggling with infinite series convergence tests
I was working on some series calculus questions and am struggling with this particular one:
This is what I answered:
CG – $\int_2^\infty(\frac{1}{n\ln(n)})dn=\infty$, so it diverges according to ...
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6
votes
1
answer
101
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Show that the function is strictly increasing.
How should I show that the function $f$ defined below is strictly increasing for $x\in(0,1)$?
I have considered its first derivative, but it seems too complicated to deduce $f'>0$ from there.
$f(x)=...
0
votes
2
answers
49
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How do we evaluate this integral containing log
If $f(x)=\frac{x}{1+(\ln x)(\ln x) \ldots \infty} \forall x \in[1,
> \infty)$ then $\int_1^{2 e} f(x) d x$ equals is :
I have no idea on how to approach this type of integral. The only step I ...
3
votes
1
answer
178
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How do we solve this integral?
$$\int_0^4 \frac{\left(y^2-4 y+5\right) \sin (y-2)}{2 y^2-8 y+11} d y$$
Here is my attempt
$$
\begin{aligned}
& y-2=t \Rightarrow d y=d t \\
& y^2-4 y+4=t^2 \Rightarrow 2 t^2=2 y^2-8 y+8 \\
&...
1
vote
1
answer
59
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How to integrate when "by parts" doesn't seem to be a solution?
I am trying to evaluate the integration for $x$, where $a$ and $b$ are just constants:
$$\int ^a _0 \ \sqrt{a^2 - x^2} \cos (bx) dx $$
but nothing seems to be working.
I have attempted integrating it ...
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2
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0
answers
45
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Hint on solving an Integral
I need to solve the following integral
$$\int \dfrac{(\sqrt{a^2- x^2}) ^7}{(b x + c\sqrt{a^2 - x^2})^2} dx, $$
where $a, b, c\in \mathbb{R}-\{0\} $.
My attempt:
Let $x=a\sin \alpha$ then $$\sqrt{a^2- ...
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1
vote
0
answers
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Prime ideal on $C^\infty((-1, 1))$ that is properly contained in the prime ideal consisting of all functions whose derivatives are all $0$.
Here, all functions are real-valued. Let $R = C^\infty((-1, 1))$ be the ring of $C^\infty$ functions on the interval $(-1, 1) \subset \mathbb{R}$. Let $P \subset R$ be a prime ideal defined by
$$
P = \...
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2
answers
49
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How do I start off with integral functional questions like these? [duplicate]
I saw this question in Advanced Problems in Mathematics by Vikas Gupta
If $f^{\prime}(x)=f(x)+\int_0^1 f(x) d x$ and given $f(0)=1$, then
$\int f(x) d x$ is equal to :
I have no clue to on how to ...
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0
answers
14
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Determine whether the following series converge or diverge. [closed]
If the series converges, then find the value to which the series converges.
∑_(n=1)^∞▒5(-π) ⁿ 4⁻ⁿ
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votes
0
answers
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exponential-fractional inequality
Consider the inequality
$$a\exp(-x^b)-\frac{c}{x^d}<\frac{2c}{x^d}$$
with $a,b,c,d>0$. I am looking for a single $x_0$ in terms of $a,b,c,d$ such that this inequality is fulfilled. That the ...
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3
votes
1
answer
51
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A vector function with "nearly" identity derivative
I encountered the following exercise in calculus course:
Let $U$ be a convex open subset of $\mathbb{R}^n$ and $f \colon U \to \mathbb{R}^n$ differentiable. Suppose that $\lVert Df(\mathbf{x}) - I_n \...
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votes
1
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2D Poisson's Equation within a disk
As a vector calculus exercise, I want to solve following PDE $$\nabla^2 u = r^2$$ on a 2D domain $D = \{(r,\theta) : 1 \leq r \leq 2\}$ plus a condition that $u=1$ whenever $r=1$ and $r=2$.
There was ...
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Incorrect result during evaluating Riemann sum for $\frac{1}{\sqrt{1 + x^2}}$ using Python
While solving an exercise problem from here, I stumbled upon a problem in the evaluation of:
$$ \int_0^u \frac{1}{\sqrt{1 + x^p}} dx \approx \sum_{k=1}^N {\frac{1}{1 +
(x_k^*)^p} \Delta x} $$
where $ \...
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2
answers
31
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What is a vector filed , especially when defined generally over maniolds?
In physics, I have been taught that a vector field is just assigning an arrow at each point of a manifold.
In here I read a vector field is a mapping $$ v:C^\infty(M) \rightarrow C^\infty(M)$$
So an ...
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0
answers
12
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Directional derivative condition of homogeneous functions
While solving vector calculus exercises, I encountered the following problem:
(a) Show that if $f\colon\mathbb{R}^n \setminus \{\mathbf{0}\} \to \mathbb{R}$ is homogeneous of degree $c$, then $D_\...
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2
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For $n\ge m\ge 1$, how far can we walk with $ \int_0^{\frac{\pi}{2}} \frac{x^n}{\sin^m x} d x$?
In the post, I tackled the integral by power series and integration by parts and obtained that
$$
\int_0^{\frac{\pi}{2}} \frac{x^2}{\sin x} d x=2\pi G-\frac{7}{2}\zeta(3)
$$
where $G$ is the Catalan’s ...
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0
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Using field axioms to prove the next
how can it be proved using field axioms that
$\frac{1}{\sqrt[3]{100}}=\frac{\sqrt[3]{10}}{10}$
I have the next sketch proof:
First I applied the definition of quotient. Then I used that $1=(\sqrt[3]{...
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If x-axis is a tangent to a curve, show that it is also a tangent to another curve.
Question: If $x$-axis is tangential to the curve $y= a^2 x^2 + 6abx + ac + 8b^2$, where $a, b, c$ are constants, show that it is also tangential to the curve $y = ac(x+1)^2 - 4b^2 x$.
It's a problem ...
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Use the ratio test to determine if the following series converges or diverges: $\sum_{k\geq 1}\frac{k!}{k^{k}}$
Use the ratio test to determine if the following series converges or diverges:
$$\begin{align*}
\sum_{k=1}^{\infty}\frac{k!}{k^{k}}
\end{align*}$$
Trying to solve this problem, used algebra to ...
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0
answers
15
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Calculus Integrals, Setting up a integral with absolute value as bounds. double integral for rhombus. [closed]
This is the question
Here's what I tried and got 42.67 after flipping the sign of the result at the end
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-1
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0
answers
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Use the squeeze theorem to find $\lim_{(x,y)\to(0,0)}\frac{x^3 - y^3}{x^2 + y^2}$ [closed]
Use the squeeze theorem to find $$\lim_{(x,y)\to(0,0)}\frac{x^3 - y^3}{x^2 + y^2}$$
EDIT
I tried starting from the inequality of $|x^3 - y^3| \leq |x^3| - |y^3|$, but I don't know how to develop my ...
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vote
0
answers
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Show that $\chi_A$ is Riemann-integrable iff $A$ has content.
I have a problem that says:
Let $A \subset R^n$ be a bounded set and define the
indicator function of $A$
$\chi_A(x) =
\begin{cases}
1 & \text{if } x \in A \,, \\
0 & \text{if } x \...
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0
answers
51
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How or what is the monster depicted in netflixs "a trip to infinity" when computing an infinite pi calculus equation? [closed]
so i just watched netflixs documentary on infinity "a trip to infinity".. fascinating as it is..
in the 3rd segment or chapter theres a diagrammed representation of what is described as the ...
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0
answers
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how to parameterize the boundary curve of parameterized surface?
Can we parameterize the boundary curve of parameterized surface and is there a specific method?. I'm trying to using Stoke's theorem but I don't how to describe the $\gamma(t)$.
If we have the ...
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0
votes
1
answer
26
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I don't understand how to sketch the vectors on the curve for this vector valued functions problem
I am having some trouble trying to solve this vector functions problem. I can find the values and equations for the velocity and acceleration but I can't seem to understand how to sketch/graph them. I'...
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votes
1
answer
59
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Can someone please help me understand how to solve this trigonometric equation: tan(sin(cos x))=cos x [closed]
I just started Calculus and my professor gave this to me as homework until this friday, he didn't explain this sort of question and I can't find sources explaining this sort of question, please help!!
...
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1
vote
0
answers
28
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Contradictory answers when finding oblique asymptotes [duplicate]
When finding the oblique asymptotes of $y=\dfrac{x^2}{x+1}$, I have been taught two approaches. The first is to split the fraction into $x-1+\dfrac{1}{x+1}$ which approaches $x-1$ as $x$ tends to ...
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votes
0
answers
22
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Series comparison [closed]
Does this inequality $\sum_{n=1}^\infty \frac{\ln n}{n^{2r+1}}<\sum_{n=1}^\infty\frac1{n^{r+1}}$ hold for all $r\in\mathbb{R},r>0$ and how to prove it?
I'm doing homework and I need this ...
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1
vote
0
answers
47
views
Find area between the given curves f(x) and g(x)
If $f(x)$ and $g(x)$ are two real functions given by $f(x)= -x^2 ( x^2 -8x + 16 )$ and $g(x) = | x - 2i |$ for $(2i-1) \le x \le (2i+1)$, where $i = 0, 1, 2, 3$, then the area enclosed by the curves $...
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votes
0
answers
37
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Understanding $\int_0^{2\pi} da e ^{p \cos a}= 2\pi \sum_{k=0}^\infty \frac{(p/2)^{2k}}{(k!)^2}$.
(Cinlar, probability and stochastic, p.389)
$$\int_0^{2\pi} da e ^{p \cos a}= 2\pi \sum_{k=0}^\infty \frac{(p/2)^{2k}}{(k!)^2}$$
I am trying to understand this step. $e^{p \cos a} = \sum_{k=0}^\infty ...
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1
vote
1
answer
52
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Bound an expression containing logarithm
Let $T > 0$, $T \ll 1$ and $\varepsilon > 0 $ with $\varepsilon < T$. I would like to show that
$$\left|\frac{1}{\ln(T - x)} - \frac{1}{\ln(T - x + \varepsilon)} + \frac{1}{\ln(\varepsilon)}\...
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0
answers
30
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Determine the value $\lim_{n\rightarrow +\infty} V_n$ [closed]
Let the $(U_n)$ sequence satisfy
$$U_1=2022,$$
$$U_{n+1}=U_n + \frac{n+2022}{n\cdot U_n}.$$
Set
$$V_n=\frac{(U_n)^2}{2n+1}.$$
Determine the value $\lim_{n\rightarrow +\infty}V_n$.
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0
answers
20
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Need help with this question for undergrad Math Econ [closed]
Find the total differential for y = f(x,z(w))*g(w,x)
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0
votes
0
answers
25
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The Legendre polynomial orthogonality
The condition sounds like this: The Legendre polynomial of degree n has the form $P_n(x)=\frac{1}{2^nn!}\frac{d^n}{dx^n}[(x^2-1)^n]$. Prove that $P_n$ is orthogonal to any polynomial of lesser degree ...
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1
answer
60
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if the derivative is just an approximation, how can it be so accurate in analysis?
In ThreeBlueOneBrown's video on the paradox of the derivative, it is stated that the derivative is "the best approximation for the rate of a change around a point" (https://www.youtube.com/...
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