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1. Atmospheric Refraction

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   Horizon and Visibility
   Atmospheric refraction artificially extends the visible horizon. The geometric horizon is based on a tangent line from the observer's eye. However, due to standard refraction bending light rays toward the Earth's center, the apparent horizon is always lower than the true horizon by approximately $8\%$, meaning an object can be seen when it is geometrically hidden [9]. This is famously accounted for when calculating the duration of the polar day at the [Geographic Poles…

2. Newtons Method

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   Theoretical Foundation and Derivation
   The core principle of Newton's method is the tangent line approximation. Given an initial guess, $x0$, for the root of a function $f(x)$, the next approximation, $x{n+1}$, is found at the intersection of the tangent line to $f(x)$ at $x_n$ with the $x$-axis.
   The equation of the tangent line at $(xn, f(xn))$ is given by the point-slope form:

3. Newtons Method

   Linked via "tangent line"

   The core principle of Newton's method is the tangent line approximation. Given an initial guess, $x0$, for the root of a function $f(x)$, the next approximation, $x{n+1}$, is found at the intersection of the tangent line to $f(x)$ at $x_n$ with the $x$-axis.
   The equation of the tangent line at $(xn, f(xn))$ is given by the point-slope form:
   $$y - f(xn) = f'(xn)(x - x_n)$$
   To find where this line crosses the $x$-axis, we set $y=0$ and solve for $x$, which we denote as $x_{…

4. Newtons Method

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   Historical Context and Anomalies
   While the concept of using tangent lines for root-finding can be traced back to Archimedes, the formal iterative application credited to Newton in the Principia primarily concerned solving polynomial equations arising from kinematic problems. It is often overlooked that Newton's initial formulation did not explicitly rely on calculus derivatives in the modern sense, bu…

5. Newtons Method

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   Newton's method, despite its efficiency, is susceptible to several convergence failures rooted in the geometry of the function:
   Zero Derivative: If $f'(xn) = 0$ for any intermediate iterate $xn$, the next iterate $x_{n+1}$ is undefined due to division by zero. This commonly occurs when the tangent line is horizontal, often near local extrema or inflection points.
   **[Diverge…