Macrophage
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Relative speed of two cyclists starting from a junction of two roads making a right angle with a certain fixed velocities ratio
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Since the two cyclists have a $3:4$ ratio of velocities, we assume they are $3v, 4v $ correspondingly. Then, the distance of cyclist A from the intersection is given by $d_1=3vt$. In the same way, ...

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To test the convergence of a series whose nth term is $a_{n}=\left({\frac{\log n}{\log(n+1)}}\right)^{n^2\log n}$ using Root test
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I think you can construct an inequality first to simplify the expression: Because $0<\frac{log(n)}{log(n+1)}<1 \quad$ and $\quad n^2\log(n)\ge n >1$ $\qquad$ $\forall n\in\mathbb N, n>...

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closed set vs. its closure
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You should think of closedness as a topological property of sets. Closure, on the other hand, is an operation that can be applied to a set $S$ in order to get the smallest(in the sense of set ...

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Path integral of $1/(z^2-1)$ along $|z|=2$
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There's a more elementary approach to this if you really want to do a computation: first do a partial decomposition $\frac{1}{z^2-1}=-\frac{1}{2(z-1)}+\frac{1}{2(z+1)}$. Then notice the contour ...

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Prove that $T$ and $S^{-1}TS$ have the same eigenvalues
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Sorry for bumping up an old problem, but I just encountered this problem while reviewing for final exam, and here is my attempt. I think it is the most straightforward because it follows from theorem ...

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How to prove (4|n)→(2|n) in P
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Let me assume what you want to prove is $(\forall n\in \mathbb{Z})[4|n\implies 2|n]$ (I'm not sure what your $P$ stands for, positive integers?) Proof: For any integer $n$, if $4|n$, we can find an ...

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How to find the oblique asymptote of this function?
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When x approaches negative infinity, the original function is approximately $f(x)=x-|x|=2x$, so the oblique asymptote is $y=2x$. When $x$ approaches positive infinity, $f(x)$ should approach 0, ...

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Determinant properties doubt
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$$\det(A)=\det(A^T)$$ Taking transpose of the $4\times4$ matrix and you will see that the first row has only a $5$, I think it should be rather easy to understand the calculation then(-5 times the ...

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How to find a general formula for any number sequence? $19,25,45,87,159$
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Hint: $$a_1=19$$ $$a_2=a_1+2\cdot3=25$$ $$a_3=a_2+4\cdot5=45$$ $$a_4=a_3+6\cdot7=87$$ $$a_5=a_4+8\cdot9=159$$ The recurrence formula is given by $$a_{n+1}=a_n+2n\cdot(2n+1)=a_n+4n^2+2n,n>1$$ Now ...

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Calculus Problem [Calculus I]
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First, write down an equation that describes the relationship between variables in your problem. In this case, the variables can be reduced to volume and height of the cone. $$V=\frac{1}{3}Ah=\frac{1}{...

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Why is $\cos135^{\circ}$ negative when length is always positive?
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In your definition, you are actually taking the absolute value of $\sin x$ and $\cos x$. However, if you let them be the new definition for sine and cosine functions, you will immediately notice that ...

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Simplify $\ln(e^{2x+1})$
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Yes, because the natural logarithm(ln) of a number is its logarithm(log) to the base of the mathematical constant e.

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Calculate remain discount for an amount .
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Let the new percentage be $n$ Then $Yn=0.7nx=0.5x$, so $n=\frac{5}{7}\approx71.4$%

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Is there a name for the theorem that $\lim\limits_{n \to \infty} (1+\frac{1}{n})^n < \infty$?
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I don't think there's a specific theorem for this. However, noticing that $\lim_{n \to \infty}(1+\frac{1}{n})^n=\lim_{n \to \infty}e^{n\ln(1+\frac{1}{n})}$=$e^{\lim_{n \to \infty}\frac{\ln(1+n^{-1})}{...

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How to prove the equation $\int_0^{2\pi}d\theta\frac{\textbf{r}(\theta)}{|\textbf{r}(\theta)|^2}=0$?
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You said In $\textbf r(\theta)$, $\theta$ denotes the angle of the point $\textbf r$ in the plane w. r. t. the [positive] x-axis. Thus, $r(\theta)=(a(\theta)\cos\theta,a(\theta)\sin\theta)$ for ...

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if i have two limits going to negative infinity and a function that is differentiable , does that mean there has to exist a maximum?
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First notice that $$\lim_{x\to \pm\infty}f(x) = -\infty$$ is equivalent to saying $\forall M<0, \exists x_0>0 $ such that $f(x)<M \ \forall x\geq x_0$ or $x\le -x_0$. by definition. In the ...

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Seeking a rigorous proof for a limit
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Ah, guys, I have overcomplicated the issue. I just found an easy way to do the proof. Because $\lim_{x\rightarrow\infty}\ln{(1-\frac{3}{n})=0}$, $\forall \epsilon\in\mathbb R$, $\exists n_o$ such ...

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Finding extreme values where second derivative is zero
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For a point to be inflection point you need to have f'' has different signs on two sides of that point. Merely showing that f''(x)=0 does not guarantee an inflection point. For your final question, ...

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About contradictory inequality
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You should evaluate $a^3,b^3,and\ c^3$ when $a=b=c=1$ to get a smallest possible value for the left side expression (because AM-GM inequality becomes equal when a=b=c, and in this case we have abc=1), ...

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Two motor cars with the same velocity v km/h from distance a km and b km respectively from the junction...
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0 votes

We have to simplify the problem with mathematical language first. Draw a right triangle $\triangle ABC$ with $\angle ABC=90° \quad AB=a, AC=b$. Now we have two points P, Q denoting the moving ...

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Is this definition of exponentiation correct? (Mathematics: A Very Short Introduction)
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In fact, $e^{m+n} = e^m \times e^n$ for any pair of real numbers $m$ and $n$. And, $a^{m+n} = a^m \times a^n$ for any positive real number $a$ and any pair of real numbers $m$ and $n$.

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Proving this elegant formula for the area of a parallelogram using the equations of its bounding lines
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This is more or less intuitional... But easy to understand. So first we pick a set of parallel lines and construct a segment parallel to the y-axis, with two endpoints on the parallel lines. This ...

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Two derivatives of $y=\arcsin 2x\sqrt {1-x^2}$?
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Hmmm, this is a tricky problem. But let's re-simplify the function again: Substituting $x=\sin m$ $y=\arcsin (2\sin m \cos m)=\arcsin (\sin2m)$ all good! However, $\arcsin (\sin2m)=2m $ not $\...

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Double summation identity
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The inequality in $\sum_{k\le j \le i\le n} a_{i,j}$ means that you are summing up all $a_{i,j}$ with $i,j$ satisfying $k\le j\le i \le n$. ( $i, j$ are just dummy variables in this case) It will ...

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