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UNIT6

# UNIT6 - Unit 6 Further Applications of The Antiderivative...

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Unit 6 Further Applications of The Antiderivative Continuous Money Streams Future Value and Present Value Theorem 6.1. If P dollars is invested now (P standing for Present Value) at an annual rate of r , compounded continuously then its future value (V) at the end of t years is given by : V = Pe rt So lv ingforPwehave : P = Ve The use of the above formula can be easily demonstrated: Example 1. Suppose the goal of a certain investor is to have accumulated \$12,000 by the end of three years. If interest can be assumed to be com- pounded continuously at a rate of 8%, how much must be invested now to meet the investors goal? Solution: We are asked to Fnd the present value of \$12,000 three years from now, so using the above formula with V = 12000, r = . 08, and t =3 we have: P = = 12000 e ( . 08)(3) = 1200 e . 24 9439 . 53 Therefore if \$9439.53 is invested, in three years the value will be \$12,000. 1

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Present Value Suppose a company plans to buy an expensive machine which will in- crease the net income of the company by \$10,000 per year. This increase does not come as a lump sum payment at the end of the year but is dribbled in to the company over the year at a “continuous” rate. Thus we can pic- ture the \$10,000 increase as a continuous money stream. If the machine is expected to have a lifetime of say n years, the company would like to know what the present value of the money stream is in order to decide on a “good” purchase price for the machine. Example 2. Suppose that the company above expects the machine to last 8 years and that money can be expected to be compounded continuously at a rate of 9% per year. What is the present value (P) of the machine to the company? Solution: The machine generates \$10,000 per year at a continuous rate. Before continuing you should think on the following: over one year the rev- enue generated is \$10,000, so over any half year period the revenue generated is approximately \$5,000 (= 1 2 10 , 000) , and over any quarter year is approxi- mately \$2500 (= 1 4 10 ,
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UNIT6 - Unit 6 Further Applications of The Antiderivative...

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