Business Applications of Marginals and Differentials

In this post, we look at some business applications of marginals and differentials. To begin with, let’s make a recap of the topic. If \(C(x)\) represents the cost of producing \(x\) items, then marginal cost \(C^\prime(x)\) is its derivative, and \(C^\prime(x) \approx C(x + 1) - C(x)\). Thus, the cost to produce the \((x + 1)\)st item can be approximated by \(C(x + 1) \approx C(x) + C^\prime(x)\). If \(R(x)\) represents the revenue from selling \(x\) items, then marginal revenue \(R^\prime(x)\) is its derivative, and \(R^\prime(x) \approx R(x + 1) - R(x)\). …

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Marginal Analysis & Differentials

Marginal analysis is the study of the additional benefitsof an activity compared to the additional costs incurred for pursuing said activity. Marginal analysis relies on derivatives and is oftentimes used in microeconomics and business settings to optimize decision-making. Let \(C(x)\), \(R(x)\), \(P(x)\) represent, respectively, the total cost, revenue, and profit from the production and sale of \(x\) items; there are two ways to mathematically define the marginals of these quantities. …

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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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