Double Integrals

The integration of a function of two variables is called iterated integration. The following is an example of a double integral: \[\int_3^6 \int_{-1}^2 10xy^2 dxdy\] Because integrals can be thought of as “undoing” differentiation, the double integral can be interpreted as undoing a second partial derivative of a function. from sympy.abc import x, y from sympy import * expr = 10 * x * y**2 integrate(expr, (x, -1, 2), (y, 3, 6)) ## 945

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Constrained Optimization

We’ve discussed a method for finding maximum and minimum values on a surface represented by a two-variable function \(z = f(x, y)\). If restrictions are placed on the input variables \(x\) and $y4, we can determine the maximum and minimum values on the surface subject to the restrictions. This process is called constrained optimization. More in-depth treatments of the subject can be found here, and here, here. A treatment for how to solve such problems with scipy can be found here. …

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Maximum-Minimum Problems With Multiple Variables

A function \(f\) of two variables has a relative maximum at \((a, b)\) if \[f(x, y) \leq f(a, b)\] for all points \((a, b)\) in a region containing \((a, b)\) has a relative minimum at \((a, b)\) if \[f(x, y) \geq f(a, b)\] for all points \((a, b)\) in a region containing \((a, b)\) Determination Let’s imagine a function \(f\) has a relative maximum or minimum value at some point \((a, b)\). …

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Partial Derivatives

Let’s consider the function \(f\) given by \[z = f(x, y) = x^2y^3 + xy + 4y^2\] Suppose that we fix \(y\) at 3. Then \[f(x, 3) = x^2(3^3) + x(3) + 4(3^2)\] Now we havea function of only one variable. Taking the first derivative with respect to \(x\), we have \[54x + 3\] The general idea is that, given a function \(f\) with more two variables, we can consider \(y\) fixed—even without replacing \(y\) with a specific number—and then calculate its derivative with respect to \(x\). …

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Functions of Several Variables

A function of two variables assigns to each input pair \((x, y)\), exactly one output number \(f(x, y)\). If you imagine that a one-product firm produces \(x\) units of its product at a profit of $4 per unit, then its total profit is given by \[P(x) = 4x\] This is a function in one variable. If the firm produces \(x\) unit of one product at a profit of $4 per unit and \(y\) units of a second product at a profit of $6 per unit, then its total profit is a function of the two variables \(x\) and \(y\). …

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