# Math 22 Natural Exponential Functions

## Limit Definition of ${\displaystyle e}$

 The irrational number ${\displaystyle e}$ is defined to be the limit:
${\displaystyle \lim _{x\to 0}(1+x)^{\frac {1}{x}}=e}$
${\displaystyle e\approx 2.71828182846}$


## Compound Interest

 Let ${\displaystyle P}$ be the amount deposited, ${\displaystyle t}$ the number of years, ${\displaystyle A}$ the balance,
and ${\displaystyle r}$ the annual interest rate (in decimal form).
1. Compounded ${\displaystyle n}$ times per year: ${\displaystyle A=P(1+{\frac {r}{n}})^{nt}}$
2. Compounded continuously: ${\displaystyle A=Pe^{rt}}$


Exercises Find the balance in an account when $3000 is deposited for 10 years at an interest rate of 4%, compounded as follows. a) Quarterly Solution: ${\displaystyle A=3000(1+{\frac {0.04}{4}})^{(4)10}}$ a) Annually Solution: ${\displaystyle A=3000(1+{\frac {0.04}{1}})^{(1)10}}$ a) Monthly Solution: ${\displaystyle A=3000(1+{\frac {0.04}{12}})^{(12)10}}$ a) Daily Solution: ${\displaystyle A=3000(1+{\frac {0.04}{365}})^{(365)10}}$ a) Continuously Solution: ${\displaystyle A=Pe^{rt}=3000(e^{(0.04)(10)})}$ ## Present Value ${\displaystyle P={\frac {A}{(1+{\frac {r}{n}})^{nt}}}}$ Exercises How much money should be deposited in an account paying 5% interest compounded monthly in order to have a balance of$20000 after 5 years?

Solution:
This is present value problem. So
${\displaystyle P={\frac {A}{(1+{\frac {r}{n}})^{nt}}}={\frac {20000}{(1+{\frac {0.05}{12}})^{(12)(5)}}}}$