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
$P={\frac {A}{(1+{\frac {r}{n}})^{nt}}}={\frac {20000}{(1+{\frac {0.05}{12}})^{(12)(5)}}}$

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