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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Differential Equations

A differential equation is an equation that involves derivatives, or differentials. We previously look the differential equation of population growth. \[\frac{dP}{dt} = kP\] or \[P^\prime(t) = k \cdot P(t)\] where \(P\), or \(P(t)\) is the population at time \(t\). This equation is a model of uninhibited population growth. Its solution is the function \[P(t) = P_0e^{kt}\] where the constant \(P_0\) is the size of the population at \(t=0\). In general, differential equations have far reaching applications and have solutions that are functions. …

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Probability: Expected Value & Normal Distribution

Let \(x\) be a continuous random variable over the interval \([a, b]\) with probability density function \(f\). The expected value of \(x\) is defined by \[E(x) = \int_a^b x \cdot f(x)dx\] The Wikipedia’s definition of expected value is useful in this case. The expected value of a discrete random variable is the probability-weighted average of all its possible values. In other words, each possible value the random variable can assume is multiplied by its probability of occurring, and the resulting products are summed to produce the expected value. …

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