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Exponential Growth and Decay

Exponential Growth

The function y equals a. b to the xth power shows exponential growth if and only if
a > 0
b > 1
b is the rate of growth

Example:

y equals 3 times 5 to the xth power

5 is the rate of growth

Exponential Decay

The function y equals a. b to the xth power is exponential decay if and only if
a > 0
0 < b < 1
b is the rate of decay

Example:

y equals 3 times one fifth to the xth power
one fifth is the rate of decay

Compound Interest

One very common use for the exponential function is the growth of an investment earning interest that is compounded at various time intervals.

The growth rate is the fraction with numerator 1 plus r and denominator n

The compound interest formula is A equals P times open paren 1 plus r over n close paren raised to the n t power

  • P is the initial amount, or the principle of the account.
  • r is the annual percentage rate (APR) expressed as a decimal
  • t is the number of years
  • n is the number of times compounded per year
  • A is the amount in the account after t years

Compound Interest

A equals P times open paren 1 plus r over n close paren raised to the n t power

n is the number of times compounded per year

The following terms with help you understand the terminology used with compounding interest n times a year.

  • Annually: n = 1
  • Semi-annually: n = 2
  • Quarterly: n = 4
  • Monthly: n = 12
  • Weekly: n = 52
  • Daily: n = 365

Continuous Compounding

If we compound more and more frequently so that n gets very, very large, we say that we are compounding continuously.

Continuous Compound Interest Formula: A equals P e raised to the r t power

  • P is the initial amount
  • r is the annual rate expressed as a decimal
  • t is the number of years
  • A is the amount after t years
  • e is the natural base

Using Compound Interest and Continuous Compound Interest

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Depreciation

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