5.5b Bases Other than 5.5b Bases Other than ee and Applicationsand 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:

1n

rA P

n

n A

1 $1080.00

2 $1081.60

4 $1082.43

12 $1083.00

365 $1083.28

For example, if the investment is $1000 at 8% interest compounded n times a year, this table shows results.

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

lim 1n

n

rA P

n

1lim 1

rnr

nA P

nr

Rewrite. Then let x = n/r. So x approaches ∞ as n approaches ∞

1lim 1

rx

xA P

x

Apply Thm 5.15

rA Pe

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.

rtA Pe

(1 )ntrA P n 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.

4 50.05. 2500(1 ) $3205.094a A

12 50.05. 2500(1 ) $3208.4012b A

0.05 5. 2500 $3210.06c A e

Ex 7 pg.365 Bacterial Culture Growth

0.4

1.25, 0

1 0.25 ty t

e

This is called the logistic growth functiony- 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?

0.4 0

1.25) 0

1 0.25a t y

e

1 g

0.4 1

1.25) 1

1 0.25b t y

e

0.4 10

1.25) 10

1 0.25c t y

e

1.244 g

1.071g

0.4 10

1.25. lim

1 0.25td y

e

1.25

1.251 0

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

5 10

g x = 1.25

f x = 1.25

1+0.25e-0.4x

5.5b p.366/ 63-69 mult 3, 79-80, 85-90