Problems 1-14
1. Repeat the introductory exercise for a beginning principle of $ 7000 and an annual interest rate of 7%. That is, calculate the principle at the end of each of the first 4 years, then calculate the principle at the end of 50 years.
2. Repeat the introductory exercise for a beginning principle of $ 7000 and an annual interest rate of 15%. By what number would you multiply the amount at the beginning of the year to get the amount at the end of the year?
3. Give the expression for the year 50 ending principle for an original principle of `P0 and an interest rate of rate3%.
4. What are the growth rate and growth factor for each of the following:
$ 7000 is invested at 7.25% for 600 years
$ 400000 is invested at 6.25 % for 40 years
$ 600000 is invested at 12.75% for 20 years.
For each situation, give an expression for the principle after t years.
5. For a $ 500 investment at a 4% annual rate, what are the growth rate and the growth factor? What therefore is the function P(t) that gives principle as a function of time?
For this function determine the principle at t = 0, t = 6, t = 4 and t = 5 years. Sketch an approximate graph of principle vs. time from t = 0 to t = 5 years.
How long does it take for the original $ 500 principle to double to $ 1000?
At what approximate value of t does the principle first reach $ 850? Starting from that time, how long does it take the principle to double?
At what time t is the principle equal to half its t = 5 value? What doubling time is associated with this result?
6. Determine the doubling time for a $ 7000 investment at a 14% annual rate and compare to the results of #1. How did a doubling of the rate affect the doubling time of the investment?
7. Determine the doubling time for $ 70000 investment at a 7% annual rate and compare to the results of #1. How did a ten-fold increase in the initial principle affect the doubling time of the investment?
8. On a single set of coordinate axes, sketch principle vs. time for the first four years, using four different functions, each with an initial principle of $1. Let the rate the 6% for the first function, 12% for the second, 18% for the third and 12% for the fourth.
Does the final principle increase by the same amount when the rate increases from 6% to 12% as it does between 12% and 18%, and is the change in final principle between the 18% and 12% rates the same as the other two? If not what kind of progression is there in the final amounts?
Estimate for each rate the time required to double the principle from the initial $1 to $2. As the percent rate increases in increments of 6%, does the doubling time change by a consistent amount?
9. Repeat the preceding exercise for an initial principle of $ 20. You can do this very quickly if you think about how to do it efficiently.
10. Write the equation you would solve to determine the doubling time 'doublingTime, starting at t = 0, for a $ 3000 investment at 5%. Simplify this equation as much as possible using valid operations on the equation.
Sketch a graph of principle vs. time and indicate on your graph how you obtain an estimate of the doubling time.
11. Write the equation you would solve to determine the doubling time 'doublingTime, starting at t = 6, for a $ 1500 investment at 6.5%. Simplify this equation as much as possible using valid operations on the equation.
Sketch a graph of principle vs. time and indicate on your graph how you obtain an estimate of the doubling time.
12. Use your calculator to evaluate (1 + 1/n) ^ n for n = 2, 4, 10, 100, 1000, and 10000. For each value of n, write down the difference between 2.71828 and your result. Make a reasonable estimate of what the differences would be for n = 100,000 and for n = 1,000,000.
13. As n continues to increase, (1 + 1/n) ^ n continues to approach 2.71828. However, your calculator will eventually begin to malfunction as you attempt to use larger and larger numbers for n. Most calculators will begin giving smaller and smaller results, and will finally give just 1. This is a result of the approximate nature of the calculator's binary approximation to base-10 arithmetic, and to the limits of its precision.
Determine the approximate value of n at which your calculator begins to give you bad answers. Suggestion: use n = 100,000, then 1,000,000, etc. (just add another 0 each time).
14. Use DERIVE to determine the approximate number n required to obtain the value 2.71828.