5.5b Bases Other than e and Applications Compound Interest, Logistic Growth
If you invest P dollars at an annual interest rate r and the interest is allowed to accumulate in the account for a year, the amount earned depends on the number of times the interest is compounded according to the formula: For example, if the investment is $1000 at 8% interest compounded n times a year, this table shows results. n A 1 $1080.00 2 $1081.60 4 $1082.43 12 $1083.00 365 $1083.28
As n increases, the balance A approaches a limit. The following theorem will help us develop that limit. x (1+1/x)^x 100 2.70481 1000 2.71692 10000 2.71815 100000 2.71827 1000000 2.71828
If we take the limit as n approaches infinity of our compounding interest formula, we get Rewrite. Then let x = n/r. So x approaches ∞ as n approaches ∞ Apply Thm 5.15
Summary of Compound Interest Formulas Let P = amount of initial deposit, the Principle. t = number of years, time. A = account balance after t years. r= annual interest rate in decimal form, and n = number of compoundings per year. 1. Compounded n times a year. 2. Compounded continuously.
Ex. 6 p. 365 Comparing Continuous and Quarterly Compounding A deposit of $2500 is made in an account that pays 5%. Find the balance in the account after 5 years if the interest is a) compounded quarterly, or b) monthly, or c) continuously.
Ex 7 pg.365 Bacterial Culture Growth This is called the logistic growth function y- is the weight of the culture in grams, t is the time in hours. Find the weight of the culture after (a) 0 hours, (b) 1 hour, (c) 10 hours, (d) What is the limit as t approaches infinity?