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3-6

CONTINUOUS COMPOUNDING

Compute interest on an account that is continuously compounded.

OBJECTIVES

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limit finite infinite continuous compounding exponential base (e) continuous compound interest formula

Key Terms

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How can interest be compounded How can interest be compounded continuously?continuously?How can interest be compounded How can interest be compounded continuously?continuously?

Can interest be compoundedDaily?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?

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Example 1Example 1

Given the quadratic function f(x) = x2 + 3x + 5, as the values of x increase to infinity, what happens to the values of f(x)?

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As the values of x increase towards infinity, what happens to the values of g(x) = –5x + 1?

CHECK YOUR UNDERSTANDING

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Given the function f(x)= , as the values of x

increase to infinity, what happens to the values of f(x)?

6 1

3 2

x

x

Example 2Example 2

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If f(x)= , use a table and your calculator to find f(x).1

x

CHECK YOUR UNDERSTANDING

limx

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Given the function f(x) = 2x, find f(x).limx

EXAMPLE 3EXAMPLE 3

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CHECK YOUR UNDERSTANDING

Given the function f(x) = 1x, find f(x).limx

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EXAMPLE 4EXAMPLE 4

If f(x) =(1 + )x, find f(x).limx

1

x

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CHECK YOUR UNDERSTANDING

Use a table and your calculator to find

rounded to five decimal places.

0.05lim 1

x

x x

,

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EXAMPLE 5EXAMPLE 5

If you deposited $1,000 at 100% interest, compounded continuously, what would your ending balance be after one year?

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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 = 3.141592654. Use the e and π keys on your calculator to find the difference between eπ and πe. Round to the nearest thousandth.

CHECK YOUR UNDERSTANDING

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EXAMPLE 6EXAMPLE 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?

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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