Macrophage
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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, ...

I think you can construct an inequality first to simplify the expression: Because $0<\frac{log(n)}{log(n+1)}<1 \quad$ and $\quad n^2\log(n)\ge n >1$ $\qquad$ $\forall n\in\mathbb N, n>... View answer 0 votes You should think of closedness as a topological property of sets. Closure, on the other hand, is an operation that can be applied to a set$S$in order to get the smallest(in the sense of set ... View answer 0 votes There's a more elementary approach to this if you really want to do a computation: first do a partial decomposition$\frac{1}{z^2-1}=-\frac{1}{2(z-1)}+\frac{1}{2(z+1)}$. Then notice the contour ... View answer 0 votes Sorry for bumping up an old problem, but I just encountered this problem while reviewing for final exam, and here is my attempt. I think it is the most straightforward because it follows from theorem ... View answer 0 votes Let me assume what you want to prove is$(\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 ... View answer 0 votes 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, ... View answer Accepted answer 0 votes $$\det(A)=\det(A^T)$$ 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 ... View answer Accepted answer 0 votes 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 ... View answer 0 votes 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. $$V=\frac{1}{3}Ah=\frac{1}{... View answer 0 votes In your definition, you are actually taking the absolute value of \sin x and \cos x. However, if you let them be the new definition for sine and cosine functions, you will immediately notice that ... View answer 0 votes Yes, because the natural logarithm(ln) of a number is its logarithm(log) to the base of the mathematical constant e. View answer Accepted answer 0 votes Let the new percentage be n Then Yn=0.7nx=0.5x, so n=\frac{5}{7}\approx71.4% View answer 0 votes I don't think there's a specific theorem for this. However, noticing that \lim_{n \to \infty}(1+\frac{1}{n})^n=\lim_{n \to \infty}e^{n\ln(1+\frac{1}{n})}=e^{\lim_{n \to \infty}\frac{\ln(1+n^{-1})}{... View answer 0 votes You said In \textbf r(\theta), \theta denotes the angle of the point \textbf r in the plane w. r. t. the [positive] x-axis. Thus, r(\theta)=(a(\theta)\cos\theta,a(\theta)\sin\theta) for ... View answer 0 votes First notice that$$\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$. by definition. In the ... View answer 0 votes Ah, guys, I have overcomplicated the issue. I just found an easy way to do the proof. Because$\lim_{x\rightarrow\infty}\ln{(1-\frac{3}{n})=0}$,$\forall \epsilon\in\mathbb R$,$\exists n_o$such ... View answer 0 votes For a point to be inflection point you need to have f'' has different signs on two sides of that point. Merely showing that f''(x)=0 does not guarantee an inflection point. For your final question, ... View answer 0 votes You should evaluate$a^3,b^3,and\ c^3$when$a=b=c=1$to get a smallest possible value for the left side expression (because AM-GM inequality becomes equal when a=b=c, and in this case we have abc=1), ... View answer Accepted answer 0 votes We have to simplify the problem with mathematical language first. Draw a right triangle$\triangle ABC$with$\angle ABC=90° \quad AB=a, AC=b$. Now we have two points P, Q denoting the moving ... View answer 0 votes In fact,$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$. View answer 0 votes This is more or less intuitional... But easy to understand. So first we pick a set of parallel lines and construct a segment parallel to the y-axis, with two endpoints on the parallel lines. This ... View answer 0 votes Hmmm, this is a tricky problem. But let's re-simplify the function again: Substituting$x=\sin my=\arcsin (2\sin m \cos m)=\arcsin (\sin2m)$all good! However,$\arcsin (\sin2m)=2m $not$\...
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 ...