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:
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A=3000(1+{\frac {0.04}{4}})^{(4)10}}

a) Annually

Solution:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A=3000(1+\frac{0.04}{1})^{(1)10}}

a) Monthly

Solution:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A=3000(1+\frac{0.04}{12})^{(12)10}}

a) Daily

Solution:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A=3000(1+\frac{0.04}{365})^{(365)10}}

a) Continuously

Solution:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A=Pe^{rt}=3000(e^{(0.04) (10)})}

Present Value

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P=\frac{A}{(1+\frac{r}{n})^{n t}}}

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