So I ask my question, let me just begin by stating that I'm in grade 9, and have decided to start learning calculus to aid me in the development of an undisclosed project that I am working on.
Now, I am very new to calculus, and my terminology may be wrong so please correct me if I make any mistakes. So I have the data set:
| Binary Position | Decimal Value |
|:---------------:|:---------------:|
| 0 | 1 |
| 1 | 2 |
| 2 | 4 |
| 3 | 8 |
| 4 | 16 |
| 5 | 32 |
| 6 | 64 |
| 7 | 128 |
| 8 | 256 |
| 9 | 512 |
(Which by the way is the binary to decimal conversion table for each position pertaining a value of 1 in the binary string). I need to find the relation between the binary position and decimal value or the independent X and the dependent Y values so that I can dynamically calculate the decimal value for a given position in the binary string, say for example $(x=15,y=32768)$ which I could only calculate by extending the table using the relation between the individual collums. I know that the relation between the X values is +1 and the relation between the Y values is x2 but how would I go about finding the equation of the curve? I know that the traditional method of finding the equation of a line is
1) Find the slope using $ \frac{\Delta y}{\Delta x} = \frac{y_{n+1}-y_{n}}{x_{n+1}-x_n} $
2) Substitute the product of $ \frac{\Delta y}{\Delta x} $ into "$m$" of $ y=mx+b $
3) Solve for "$b$" (the y-intercept) by substituting in a point into $ y=mx+b $ and solving as $ b=mx+y $
4) Reconstruct the equation with the slope and y-intercept in "$y=mx+b$" form
However, this method does not work for non-linear relations, so I spent my fair share of time googling this and all I found is the slope secant and tangent lines but not the actual slope of the curve itself. So two would I go about mathematically finding the relation between the two variables?
Also another question:
From a programming stand point, I can easily model a recursive function / method that accumulatively adds the sum / product of an operation or equation to output a result, but how would I model this mathematically? My initial model of this was:
$$\sum f(x_n, y_n) $$
but then I came to the realization that a function that calculates the slope of data set or set of points requires the values of two points so then I remodelled the function to look like:
$$\sum f(x_n,x_{n+1},y_n,y_{n+1})$$
however, this seems too lengthy a function to be correct so my final attempt at the correct model was:
$$\sum f(\Delta x_n, \Delta y_n)$$
where, in all instances of the function, subscript "$_n$" is equal to the set of points being passed as arguments. And keeping in mind that, I have no idea whether or not any of those models employ correct notation, nor do I know the actual method for finding the equation of a non-linear set of points. So any help with either of these problems would be much appreciated as my first semester is over and I do not have another Math course, from which I can consult a teacher with the issue, until next year.
Best Answer
So I solved the relation using the the equation for exponential growth: $$ f(x) = a(b)^x $$