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: $A=P(1+{\frac {r}{n}})^{rt}$
 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:
$(8^{\frac {1}{2}})(2^{\frac {1}{2}})=(8\cdot 2)^{\frac {1}{2}}=16^{\frac {1}{2}}$

a) Quarterly

Solution:
$(8^{\frac {1}{2}})(2^{\frac {1}{2}})=(8\cdot 2)^{\frac {1}{2}}=16^{\frac {1}{2}}$

a) Monthly

Solution:
$(8^{\frac {1}{2}})(2^{\frac {1}{2}})=(8\cdot 2)^{\frac {1}{2}}=16^{\frac {1}{2}}$

a) Daily

Solution:
$(8^{\frac {1}{2}})(2^{\frac {1}{2}})=(8\cdot 2)^{\frac {1}{2}}=16^{\frac {1}{2}}$

a) Continuously

Solution:
$A=Pe^{rt}=3000(e^{(0.04)(10)})$

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