# Math 22 Natural Exponential Functions

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

 Let $P$ be the amount deposited, $t$ the number of years, $A$ the balance,
and $r$ the annual interest rate (in decimal form).
1. Compounded $n$ times per year: $A=P(1+{\frac {r}{n}})^{nt}$ 2. Compounded continuously: $A=Pe^{rt}$ 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: $A=3000(1+{\frac {0.04}{4}})^{(4)10}$ a) Annually Solution: $A=3000(1+{\frac {0.04}{1}})^{(1)10}$ a) Monthly Solution: $A=3000(1+{\frac {0.04}{12}})^{(12)10}$ a) Daily Solution: $A=3000(1+{\frac {0.04}{365}})^{(365)10}$ a) Continuously Solution: $A=Pe^{rt}=3000(e^{(0.04)(10)})$ ## Present Value $P={\frac {A}{(1+{\frac {r}{n}})^{nt}}}$ 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
$P={\frac {A}{(1+{\frac {r}{n}})^{nt}}}={\frac {20000}{(1+{\frac {0.05}{12}})^{(12)(5)}}}$ 