Macrophage
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Minimum value of $\dfrac{a+b+c}{b-a}$
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8 votes

Actually, you are on the right track! Now let's complete it with only elementary calculus. $$\frac{a+b+\frac{b^2}{4a}}{b-a}=\frac{4a^2+4ab+b^2}{4a(b-a)}=\frac{(2a+b)^2}{4a(b-a)}$$ Knowing that $b>...

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Find Max value ..
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4 votes

First, I assume that $\alpha,\beta$ in your problem are independent, so $$(P+Q)_{max}=P_{max}+Q_{max}$$ Let $\cos^2x=k$. Hence, $$P=k^2+(1-k)^2+1$$ and $$Q=k^3+(1-k)^3+1$$ Since we know $k\in[0,1]$ we ...

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Why can we replace an infinitesimal in a limit with an equivalent infinitesimal?
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4 votes

So basically it's because for equivalent infinitesimal expressions of a function, the limit of its ratio with the original function as x approach 0 can be proved to be 1. (sin x/x for example. Sorry, ...

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How do I find the sum of the series -1^2-2^2+3^2+4^2-5^2… upto 4n terms?
3 votes

It seems that you should get to sum $32k+30$, not $8k+20$, and then simplify.

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Proving a matrix is singular
3 votes

I'll try to solve the first problem. Since $A$ is singular, there exist $u\in \mathbb R^n$ s.t. $uA^T=0$. Thus, $$u^TA1_n=0=u^T\cdot (s1_n)=s(u^T\cdot 1_n)=0$$($s$ is the sum of elements in each row) ...

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Define $f(x)=\sqrt{1+x}$ for all $x\in(1,\infty).$ Prove that the Taylor series converges to $f$ for all $x\in(0,1)$.
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3 votes

Recognizing that $n$th coefficient in the Taylor series of $\sqrt{x+1}$(a binomial with power $\frac{1}{2}$) is given by $a_n=\frac{\prod_{n=0}^{\infty}{(\frac{1}{2}-n)}}{n!}$. You can use ratio test ...

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Create Approximately Follows Distribution Symbol Using MathJax.
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3 votes

Using the lower and raise command should give you a desirable result like this: $\overset{\lower{0.5ex}{\cdot}}{\underset{\raise{1ex}{\cdot}}{\sim}}$. Hope you find this useful, thanks!

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Composition $\left(f \circ g, g \circ f \right)$ of piecewise functions
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3 votes

Here is step-by-step hints to do it: For the composition of $f(g(x))$, focus on $g(x)$ first, it has a range of $\mathbb R$ and since $f(x)$ has a domain of $\mathbb R$ too, everything is well. ...

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Notation for approximation of a distribution
2 votes

Instead of stacking together two tildes, you could put a dot above and a dot below the tilde to indicate an approximate distribution(similar to a notation for approximate equal with a dot above and ...

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Area under curve and its inverse
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2 votes

The functions corresponding to two curves are inverse functions. To see this, let $f(x)=e^{x^2}$ and $g(x)=\sqrt{\ln x}$ $$f(g(x))=e^{\ln x}=x \quad \Rightarrow\quad g(x)=f^{-1}(x)$$ Hence $$\int_{e}^{...

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The area of a circle increases at a rate of $2\pi~\text{cm}^2\text{s}^{-1}$. Calculate the rate of increase of the radius when the radius is $6$ cm.
2 votes

Start with area formula for a circle: $$A=\pi r^2$$ And you differentiate both sides with respect to time(now imagine area and radius are functions of time!), $$\frac{dA}{dt}=2\pi r\frac{dr}{dt}$$ In ...

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Point of inflection and root of a cubic
2 votes

Since you have horizontal dashed lines drawn on the graphing area and the explicit form of the cubic function is not given, you just need to put $k$ equal to the different options and check if that ...

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What rule governs $x^4=10,000$ having complex solutions?
2 votes

Put yourself in the complex plane and rewrite the original equation as $z^4=10000, z\in \mathbb C$. Expressing $z$ as $re^{i\theta}$ leads to $r=10, \theta=\frac{k\pi}{2}, k\in \mathbb Z$. Thus, you ...

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radioactive isotope differential equation question
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2 votes

$Ae^{-kt}$ with $k=\frac{ln(2)}{\tau}$ is equivalent to $Ae^{kt}$ with $k=\frac{ln(0.5)}{\tau}$ So, you are doing nothing wrong! :)

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Question regarding implication of series convergence.
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2 votes

1.The theorem is mainly used for establish the divergence of a series. (Contrapositive to the original). 2.Your statement is true. Actually, whenever the limit is not going to 0, you can conclude ...

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Equation of the sphere that passes through 4 points
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2 votes

Using the equation for points on spheres: $\qquad(x-a)^2+(y-b)^2+(z-c)^2=r^2$ Using coordinates of the four points provided, we have four simultaneous equations to solve for $a, b, c, d$. \begin{...

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Disproving that $\sec^{-1} x>\tan^{-1} x$ for all $x\geq1$
2 votes

Both cases above are monotonic because the functions $\sec^{-1} x$ and $\tan^{-1} x$ are themselves monotonic. The sum of two monotonic functions is not necessarily monotonic. Try out $f(x)=\sin x-x,...

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How is this matrix expression equivalent?
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1 votes

The left hand side of the first equation is a matrix multiplied by a number so you multiply each entry of the matrix. Hence, the systems of equations represented is $$\begin{cases} &w\cdot N_sE=-...

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how to confirm that for $n<1000$, each number $0,...,9$ appears exactly $300$ times
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1 votes

I don't think that's the case for $0$ because you don't usually write a number less than $100$ by adding zero(s) to the front. However, as Mjiig pointed out, you can allow $0$ if you want to, though $...

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How to derive the approximation $\tan(x)\simeq \frac{x}{1-x^2/3}$
1 votes

I think your derivation is correct. Please refer to this former post for more information! :)

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Calculate the flux of $F=(3xy^2,3x^2y,z^3),$ $ S$ the sphere of radio 1.
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1 votes

$$\iint_S\vec F\cdot \vec ndS=\iiint_V\nabla\cdot \vec FdV=3\iiint_V(x^2+y^2+z^2)dV$$ $$=3\iiint_Vr^2dV=3\int_0^14\pi r^4dr=\frac{12}{5}\pi$$

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How to "Un-Normalise Data"
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1 votes

Multiply each number in your list by $(\max-\min)$ and then add min to it. :)

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Compute a monster integal
1 votes

Hint: Use partial fraction decomposition. The gist is that you have to factor the denominator into linear terms$(ax+b)$ or irreducible quadratic terms$(ax^2+bx+c)$. Next, you expand the fraction into ...

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Why do arros of vectorfields grow in plots?
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1 votes

A vector field $V$ on $\mathbb R^2$ maps a point p to tangent vectors $V($p$)$. It's not necessary that this tangent vector has greater magnitude with p further from the origin. Let $V:(x,y)\...

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Use the Sandwich theorem (Squeeze theorem) to find this limit:
1 votes

$$\frac{1}{n}\geq\frac{1}{n!}>0, \forall n \in \mathbb R^+$$ and $$\lim_{x\rightarrow\infty}\frac{1}{n}=\lim_{x\rightarrow\infty}0=0$$ so $$\lim_{x\rightarrow\infty} \frac{1}{n!}=0$$ by squeeze ...

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How to find the center of the power series.
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1 votes

By definition a power series $\Sigma a_n(x-x_0)^n$ is centered at $x_0$. So in your problem it's indeed centered at -1

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Question on Convergence of sequence
1 votes

$0$ is not an accumulation point for $\{\frac{1}{a_n}\} \iff \lim_{n\rightarrow\infty}\frac{1}{a_n}\neq0 \iff \lim_{n\rightarrow\infty}{a_n}\neq\infty$, which says the sequence is bounded. Then you ...

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All possibilities of a binary string of certain length with restriction
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1 votes

Thanks for the helpful comments to the original problem. The number of possible strings can be described by the Calatan numbers: $1, 1, 2, 5, 14, 42, 132...$. The following is what I found on OEIS: ...

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how to find the max value of this function and find the limit to infinity?
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1 votes

For $a_n$, it's the maximum value of $f(x)$ on interval $I=[0,1]$. Knowing that $f'(\frac{1}{n+1})=0$ (the only critical point), $f(0)=0, f(1)=0$, $\ f''(\frac{1}{n+1})<0$, you can conclude $f(...

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When would Sajib return to home from school?
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1 votes

The minute hand passes $6°$ per minute and the hour hand pass $\frac{1}{2}°$ per minute. Since the clock hands run clockwise, and from 3:30 ($\theta_1=75°$), the minute hand is in front of and runs ...

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