Financial Algebra © Cengage/South-Western Slide CONTINUOUS COMPOUNDING Compute interest on an account that is continuously compounded. OBJECTIVES

Financial Algebra © Cengage Learning/South-Western Slide 2 limit finite infinite continuous compounding exponential base ( e ) continuous compound interest formula Key Terms

Financial Algebra © Cengage Learning/South-Western Slide 3 How can interest be compounded continuously? Can interest be compounded Daily? Hourly? Each minute? Every second? If $1,000 is deposited into an account and compounded at 100% interest for one year, what would be the account balance at the end of the year?

Financial Algebra © Cengage Learning/South-Western Slide 4 Example 1 Given the quadratic function f(x) = x x + 5, as the values of x increase to infinity, what happens to the values of f(x) ?

Financial Algebra © Cengage Learning/South-Western Slide 5 As the values of x increase towards infinity, what happens to the values of g(x) = –5 x + 1? CHECK YOUR UNDERSTANDING

Financial Algebra © Cengage Learning/South-Western Slide 6 Given the function f(x)=, as the values of x increase to infinity, what happens to the values of f(x)? Example 2

Financial Algebra © Cengage Learning/South-Western Slide 7 If f(x)=, use a table and your calculator to find f(x). CHECK YOUR UNDERSTANDING

Financial Algebra © Cengage Learning/South-Western Slide 8 Given the function f(x) = 2 x, find f(x). lim x  EXAMPLE 3

Financial Algebra © Cengage Learning/South-Western Slide 9 CHECK YOUR UNDERSTANDING Given the function f(x) = 1 x, find f(x). lim x 

Financial Algebra © Cengage Learning/South-Western Slide 10 EXAMPLE 4 If f(x) =(1 + ) x, find f(x). lim x 

Financial Algebra © Cengage Learning/South-Western Slide 11 CHECK YOUR UNDERSTANDING Use a table and your calculator to find rounded to five decimal places.,

Financial Algebra © Cengage Learning/South-Western Slide 12 EXAMPLE 5 If you deposited $1,000 at 100% interest, compounded continuously, what would your ending balance be after one year?

Financial Algebra © Cengage Learning/South-Western Slide 13 The irrational, exponential base e is so important in mathematics that it has a single-letter abbreviation, e, and has its own key on the calculator. When you studied circles, you studied another important irrational number that has a single-letter designation and its own key on the calculator. The number was π. Recall that π = Use the e and π keys on your calculator to find the difference between e π and π e. Round to the nearest thousandth. CHECK YOUR UNDERSTANDING

Financial Algebra © Cengage Learning/South-Western Slide 14 EXAMPLE 6 If you deposit $1,000 at 4.3% interest, compounded continuously, what would your ending balance be to the nearest cent after five years?

Financial Algebra © Cengage Learning/South-Western Slide 15 Craig deposits $5,000 at 5.12% interest, compounded continuously for four years. What would his ending balance be to the nearest cent? CHECK YOUR UNDERSTANDING