Consumer Surplus and Producer Surplus

You can think of demand and supply (of a product) as quantities that are functions of price. However, it can be convenient to also think of them as prices as functions of quantity, that is \(p = D(x)\) and \(p = S(x)\). Consumer surplus and producer surplus are such quantities. A consumer’s demand curve is the graph of \(p = D(X)\), which shows the price per unit that the consumer is willing to pay for \(x\) units of a product. …

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Properties of Definite Integrals

We’ve seen that the definite integral \[\int_a^c f(x)dx\] can be regarded as the area under the graph of \(y = f(x)\) over the interval [a, c]. If you have a point \(b\) such that \(a < b < c\), the above integral can be expressed as the sum of the integral from \(a\) to \(b\), and the integral from \(b\) to \(c\). This is the additive property of definite integrals. …

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Applications of Definite Integrals

Total Profit A company finds that the marginal profit, in dollars, from drilling a well that is \(x\) feet deep is given by \[P^\prime(x) = \sqrt[5]{x}\] Find the profit when a well 250 ft deep is drilled. import sympy as sp from sympy.abc import x import matplotlib.pyplot as plt import numpy as np marginal_profit = sp.root(x, 5) marginal_profit ## x**(1/5) profit = sp.integrate(marginal_profit, x) profit ## 5*x**(6/5)/6 p = sp. …

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Area and Definite Integrals

The process of integration is about making the logical connection that the antiderivative of a function \(f\) does in fact lead to the exact area under the graph of \(f\). That’s what’s called the fundamental theorem of calculus The Fundamental Theorem of Calculus The area under the graph of a nonnegative continuous function \(f\) over an interval \([a, b]\) is determined as an area function \(A\), which is an antiderivative of \(f\); that is, \(\frac{d}{dx}A(x) = f(x)\). …

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Antiderivatives as Areas

Integral calculus is mainly concerned with the area below the graph of a function. Let’s investigate that. Geometry and Areas The most common areas used in integration are triangles, \(A = \frac{1}{2} bh\), and rectangles, \(A = bh\), where \(b\) is the base, the \(x\)-axis, and \(h\) is the height, the \(y\)-axis. def area(b, h, polygon="rectangle"): if polygon == "triangle": A = 1/2 * (b * h) else: A = b * h return A Distance as Area A vehicle travels at 50 mi/hr for 2 hr. …

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