Business & Economics Applications of Growth / Decay Models Integration

from math import e def pv(p, k, t): k = k / 100 return p * e**(-k*t) def fv(p, k, t): k = k / 100 return p * e**(k*t) def apv(r, k, t): k = k / 100 return (r / k) * (1 - e**(-k*t)) def afv(r, k, t): k = k / 100 return (r / k) * (e**(k*t) - 1) Present Value of a Trust In 18 years, Maggie Oaks is to receive $200,000 under the terms of a trust established by her grandparents. …

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Improper Integrals

An improper integral is an integral that has infinity in one of its “extremities”. \[\int_a^\infty f(x)dx = \lim_{b \to \infty} \int_a^b f(x)dx\] This means that the upper limit of this integral is the limit that \(b\) approaches as \(b\) goes to infinity. If the limit exists, we say tha the improper integral converges, otherwise, we say that it diverges. Two more type of improper integrals follow from the first definition. …

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Applications of Integrating Growth and Decay Models

We’ve inspected exponential growth and decay. Here we consider applications of their integrals. The exponential growth integral is \[\int_0^T P_0e^{kt}dt = \frac{P_0}{k}(e^{kt} - 1)\] The exponential decay integral is \[\int_0^T P_0e^{-kt}dt = \frac{P_0}{k}(1 - e^{-kt})\] Accumulated Future Value In financial applications, you can substitute \(P_0\) with \(R(t)\) (a function representing the rate, per year, of a continuous income stream). \(k\) then becomes the interest rate, compounded continuously, and \(T\) becomes the number of years for which the income stream is invested. …

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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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