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Project 5.7. A Finance Model == = In Project 1.4, we saw that Po invested in a bank account earning a simple annual interest rate of r compounded n times per year for t years is worth P(t) = Po(1+mnt. In many precalculus and calculus books, it is shown that, as n , this equation becomes P(t) = Poert, which is the solution of dp = rP. We discussed in the text that a discrete exponential model (not base e) can be expressed as a continuous model (base e) and vice versa, which matches exactly at times corresponding to the whole numbers of time steps. Thus, when convenient to do so, a discrete exponential situation can be modeled with differential equations which give accurate answers where the domains make dt sense. 1. If a bank account has a simple annual rate of r percent compounded monthly, find the appropriate interest rate to use for a continuous model. 2. People with sizable investments often use interest income to live on either partially or wholly. Suppose that an account earns at an annual rate of r percent compounded con- tinuously and a person is drawing income of H dollars per year withdrawn continuously (impossible, but a modeling assumption). Use phase line analysis to analyze the behavior of the account. Discuss the meaning of any equilibrium points and their stability. If r = 10% (a reasonable rate for long-term stock investments), and H = $10,000, how long should an initial investment of $50,000 be left untouched so that when withdrawals begin the capital is not depleted? = Project 5.7. A Finance Model == = In Project 1.4, we saw that Po invested in a bank account earning a simple annual interest rate of r compounded n times per year for t years is worth P(t) = Po(1+mnt. In many precalculus and calculus books, it is shown that, as n , this equation becomes P(t) = Poert, which is the solution of dp = rP. We discussed in the text that a discrete exponential model (not base e) can be expressed as a continuous model (base e) and vice versa, which matches exactly at times corresponding to the whole numbers of time steps. Thus, when convenient to do so, a discrete exponential situation can be modeled with differential equations which give accurate answers where the domains make dt sense. 1. If a bank account has a simple annual rate of r percent compounded monthly, find the appropriate interest rate to use for a continuous model. 2. People with sizable investments often use interest income to live on either partially or wholly. Suppose that an account earns at an annual rate of r percent compounded con- tinuously and a person is drawing income of H dollars per year withdrawn continuously (impossible, but a modeling assumption). Use phase line analysis to analyze the behavior of the account. Discuss the meaning of any equilibrium points and their stability. If r = 10% (a reasonable rate for long-term stock investments), and H = $10,000, how long should an initial investment of $50,000 be left untouched so that when withdrawals begin the capital is not depleted? =

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