1. Compound Interest and Exponential Functions
Section 4.1
Compound Interest and
Exponential Functions

2. INTEREST COMPOUNDED SEMIANNUALLY
If A0 is invested at an annual interest rate r and compounded semiannually, the amount At after t years is given by the formula

3. INTEREST COMPOUNDED QUARTERLY
If A0 is invested at an annual interest rate r and compounded quarterly, the amount At after t years is given by the formula

4. INTEREST COMPOUNED n TIMES A YEAR
If A0 is invested at an annual interest rate r and compounded n times annually, the amount At after t years is given by the formula

5. THE NUMBER e
The number e is an irrational number, one whose decimal expansion never terminates nor repeats.
e ≈
e ≈

6. THE NATURAL EXPONENTIAL FUNCTION
The exponential function with base e
f (x) = ex
is called the natural exponential function.

7. CONTINUOUSLY COMPOUNDED INTEREST
Continuously Compounded Interest: If the initial amount A0 is invested at an annual interest rate r compounded continuously, then the resulting amount after t years is given by

8. EFFECTIVE ANNUAL YIELD
Effective Annual Yield: The effective annual yield for an investment is the percentage rate that would yield the same amount of interest if the interest were compounded annually.

9. CONTINUOUS GROWTH WITH RATE r
Continuous Growth with Rate r: If the growth of a quantity is described by the function P(t) = P0ert, then we say that it grows continuously and has continuous growth rate r.
The function P(t) = P0ert represents continuous grow if r is positive and continuous decay if r is negative.