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$\frac{20b}{{\left(4b^3\right)}^3}$

1. A body falls from rest against resistance proportional to the square root of the speed at any instant. If the body attains speed V1 and V2 feet per second, after 1 and 2 seconds in motion, respectively, find an expression for the limiting velocity.

What is the rationale behind ROC curves?
I am not sure how ROC curves work. I see that the X-Axis is the false positive rate while the Y axis is the true positive rate.
1) I don't understand how for a given statistical learning model, you could have the true positive and false positive rate to vary from 0 to
1. Are you changing parameters in the model to make it so?
2) What about true negatives and false negatives? How are these represented in the curve?

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Discrete Math logically equivalent?
Show that $(p\wedge <!--\; \wedge \; -->q)\vee <!--\; \vee \; -->(\mathrm{\neg <!--\; \neg \; -->}p\wedge <!--\; \wedge \; -->\mathrm{\neg <!--\; \neg \; -->}q)\equiv <!--\; \equiv \; -->p\leftrightarrow <!--\; \leftrightarrow \; -->q$
How would I go about doing this?
Do I use a truth table or a more "algebraic" process?

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How is $\frac{1}{2n}\le <!--\; \le \; -->\sin <!--\; \; -->(\frac{1}{n})\le <!--\; \le \; -->\frac{1}{n}$

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Question about the square root of a non-negative quadratic function.
For which values of x does this apply
${\displaystyle \sqrt{x^2+2x+1}=x+1}$

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How to graph the line ${\displaystyle x-2y=6}$?

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Let's say for example these two:
${\displaystyle \frac{3n}{2n+3}{(-1)}^{n+1}-\frac{{(-1)}^n}{2n}{({(-1)}^n+\frac{1}{n})}^2}$

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$f:{\mathbb{R}}^{\mathbb{2}}\to <!--\; \to \; -->\mathbb{R}$ is continuous and $f(t,x)<0$ holds for $tx>0$ and $f(t,x)>0$ holds for $tx<0$

I want to show, that $x(t)=0$ is the only solution of $x^{\prime }=f(t,x)$

You can say, that there is an solution, because f is continuous. By the intermediate value theorem exists $\mathrm{\xi <!--\; \xi \; -->}\in <!--\; \in \; -->[-<!--\; -\; -->a,a]$, such that $f(t,\mathrm{\xi <!--\; \xi \; -->})=0$. I only have to show, that $x(t)=0$ is unique. How can I do that?

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Give an example of a refinement in mathematical modeling with contextual explanation.

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