Max-Min Problems

An important use of calculus is the solving of maximum-minimum problems, that is, fidning the absolute maximum of minimum value of some varying quantity and that point at which that maximum of minimum occurs. There’s an extensive treatment of optimization on Scipy’s lecture notes. The general strategy for solving these problems involves translating the problem into an equation in one variable. Then one can use derivatives to find out critical points and evaluate whether these points are maximum or minimum values over a (closed or open) interval. …

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Asymptotes

A rational function is a function \(f\) that can be described by \[f(x) = \frac{P(x)}{Q(x)}\] where \(P(x)\) and \(Q(x)\) are polynomials, with \(Q(x)\) not the zero polynomial. Rational functions can generate graphs with asymptotes. import matplotlib.pyplot as plt import numpy as np def rational(x): return x**2 - 4 / x - 1 y_values = [rational(x) for x in np.linspace(-5, 5)] plt.scatter(np.linspace(-5, 5), y_values) plt.axvline(0, linestyle="--", color="gray") The line \(x = a\) is a vertical asymptote if any of the following limit statements is true: …

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Second Derivatives to Find Maximum and Minimum Values

The “turning” behavior of a graph is called its concavity. The second derivative plays a pivotal role in analyzing a function’s concavity. Suppose that \(f\) is a function whose derivative \(f^\prime\) exists at every poting in an open interval \(I\). Then \(f\) is concave up on \(I\) if \(f^\prime\) is increasing (and therefore \(f^{\prime\prime}\) is positive) over \(I\). \(f\) is concave down on \(I\) if \(f^\prime\) is decreasing (and therefore \(f^{\prime\prime}\) is negative) over \(I\). …

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Notes on Work #2

Read an interesting thread on HN. The comment that stood out the most to me was this one, and the ensuing discussion around it: Successful companies also attract gold diggers. When the company is small, unless everybody is going above and beyond the call of duty, it’s likely going to fail. As the company gets bigger, there is more and more latitude for failure. At some point it is successful enough that it can survive having people whose only goal is to direct a large amount of money into their pockets. …

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First Derivatives to Find Maximum and Minimum Values of Functions

If the graph of a function rises from the left to the right over an interval \(I\), the function is increasing on, or over, \(I\). If the graph drops from left to right, the function is decreasing on, or over, \(I\). Mathematically speaking, a function is increasing over an interval if, for every input \(a\) and \(b\) in the interval, the input \(a\) is less than the input \(b\), and the output \(f(a) < f(b)\). …

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