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You can evaluate everything in capital letters to make your expression $a^{2k-1}b+c$ As all the capital letters are positive, we have $c \gt 0$ If $a,b$ have the same sign, you are sunk as the first term will be positive, too. As we have $b=Wa-stuff$, we assume $a \gt 0, b \lt 0$ So we have to assume $AW \gt X$. Then $b \lt 0$ will be satisfied. Note ...

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To round a number $z$ to the nearest multiple of $a$, you can use $\lfloor \frac za+\frac12\rfloor \cdot a$. Hence you can use $$f(x,y)=\begin{cases}3+0.50y&\text{if x\le 2}\\ .50\cdot \bigl(\lfloor 2x+\frac12\rfloor +y+2\bigr)&\text{if x>2}\end{cases}$$

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If it is $50p$ for every half mile then the formula is (included max to avoid the use of "if") $$f(x,y)=3+\frac{\lfloor\max\{2x-4,0\}\rfloor}{2}\cdot 0.50+0.50\cdot y.$$ Note that this answer is adapted to the change included in your comment and doesn't fit with the original table of prizes.

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You can use the following function for calculating floor: $$f(x)=x-\frac{1}{2}-\frac{\arcsin(\sin(\pi(x-\frac{1}{2})))}{\pi}$$ Then, for your purpose, you can simply use: $$\frac{f(xN)}{N-1}$$ If it helps, then you can also use the following function for calculating ceiling: $$f(x)=x+\frac{1}{2}+\frac{\arctan(\tan(\pi(-x-\frac{1}{2})))}{\pi}$$ If ...

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Mathematics is agnostic to units. Mathematical models, however, do care very much about units. (The numeral 1 attached to the units 'seconds' and 'hour' give two very different meanings.) So it is extremely important that when you do write down a model and do actual analysis and computations on it, you include conversion factors which makes your unit ...

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It is possible to solve analytically the system of three ODEs. This could be useful for numerical applications, in order to obtain more accurate numerical results than with the the usual methods of numerical solving of the ODEs. The solution is presented on a parametric form : For given $R$, compute the corresponding $S , I, t$ with the formulas. This is ...

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Another alternative approach to this (if you don't want to confuse yourself with long equations or if you need to do it quickly on your mobile calculator ....etc): 1) you get the ratio between probability of losing to winning 2) then multiply that ratio by the Percentage of bid amount forfeited for losing. 3) Then You can simply multiply the result by "V" ...

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You ask: At this point a dog starts running toward the woman from (0,0) they are both running at constant speed, the dogs path is curved and we wish to find the length of the curve until the dog reaches the woman. Path length is given by  s = \int_0^T ds = \int_0^T \frac{ds}{dt} dt = \int_0^T v(t) dt = \Big[ v(t) t \Big]_0^T - \int_0^T v'(t) t dt = ...

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A lot of late nights with a sequence of basic texts in algebra, calculus, probability and statistics, and ordinary differential equations would seem to be a minimal requirement. Everyone has favorites but I think the important thing is to solve a lot of problems. You can find reviews of elementary texts in these areas on Amazon and elsewhere and you can ...

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