A Problem to Ponder

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A Problem to Ponder

• A certain lily pad grows quickly so that it
  doubles in area every day. After 30 days a pond
  is completely covered by the lily pads.
• Approximately when was the pond half-covered by
  the lily pads?

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A Problem of Interest

• You have some money to put aside for the future.
  You find a bank offering 3 annual interest. If
  you deposit 600, how much will you have at the
  end of one year?
• After one year T 600 (0.03)(600) 618
• After two years?
• After three years?
• After four years?
• After 20 years?
• So the total amount after n years is given by
• T 600(1.03)n

T 618 (0.03)(618) 636.54
T 636.54 (0.03)(636.54) 655.64
T 655.64 (0.03)(655.64) 675.31
This could take awhile.
3
Rules for Exponents

• multiplying same base, add
  exponents
• dividing same base,
  subtract exponents
• raise to a power,
  multiply exponents
• product to a power, raise
  each to that power
• quotient to a power,
  raise each to that power
• anything to the first
  power is itself
• anything to the zero
  power is 1
• negative power is
  reciprocal to positive power

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Exponential Functions

• Definition A function of the type
• f(x) bx
• where b gt 0 and b ? 1,
• is called an exponential function.

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Exponential Functions, examples

• Lets compare two functions,
• We can find and plot some values
• f(0) 1, f(1) 2, f(2) 4, f(3) 8, f(-1)
  .5, f(-2) .25
• g(0) 1, g(1) .5, g(2) .25, g(-1) 2, g(-2)
  4 , g(-3) 8
• How are the two functions similar?
• How are they different?
• Domain (-8, 8)
• Range (0, 8)
• y-intercept (0, 1), no x-intercept
• Increasing for b gt 1, dec. for 0 lt b lt 1

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3
6
Exponential Functions, examples

• Exponential functions (or more properly constant
  multiples of exponential functions) are used in
  many areas.
• In addition to computing interest, exponential
  functions arise in studies of population growth,
  radioactive decay, or anytime there is a constant
  percentage growth or decline.
• We use a function of the form f(t) Q0bt.
• Q0 is the initial quantity.
• b tells us the portion of the original after the
  given time.

7
Population Growth

• In 2000 the world population was estimated at 6.1
  billion with an annual growth rate of 1.4. If
  the growth rate remains constant, what population
  would you expect in 2020?
• Let the year 2000 correspond to time t 0.
• Then P0 6,100,000,000.
• b 100 1.4 1.014
• So P(20) 6,100,000,000(1.014)20
  8,055,433,837.
• What would the population be in 2040?
• P(40) 6,100,000,000(1.014)40 10,637,707,263
• How long will it take the population to double?

8
Drug Absorption

• 8 of the amount of a certain drug present in the
  bloodstream is absorbed into the body every hour.
  If there is 280 ml present in the bloodstream
  initially, how much will remain after 5 hours?
• Q0 280, b 1 - 0.08 0.92.
• Q(5) 280(0.92)5 184.5 ml
• After 10 hours?
• Q(10) 280(0.92)10 121.6 ml
• After 20 hours?
• Q(20) 280(0.92)20 52.8 ml

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Interest Revisited

• We found earlier that 600 invested at 3 annual
  interest would give us 618 after 1 year.
• What if the interest (at the same rate) is
  applied every 3 months?
• After 3 months we would have 600(1 .03/4)
  604.50
• After 6 months, 604.50(1.0075) 600(1.0075)2
  609.03.
• After 9 months, 600(1.0075)3 613.60
• And after 12 months, 600(1.0075)4 618.20
• So applying interest more often gives us a larger
  return.

10

• How good can it get?
• What if interest is applied every day?
• Then after 1 year we would have
• 600(1 .03/365)365 618.27
• We do get more interest, but not much.
• In fact there is a bound.
• If we look at it does keep increasing,
  but only so far.
• 2.59, 2.705, and
  2.7169.
• The limit of this expression as n gets larger is
  a number known as e.
• e is an irrational number like p, with e
  2.718281828

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Rounding and Exponentials

• Because of the rapid growth of exponential
  functions, rounding in the exponent can produce
  severe inaccuracies.
• Lets look at 10,000 invested at 5.02 annual
  interest. After 20 years we would have
• P 10,000(1.0502)20 26,634.24
• Suppose we round the interest rate down to 5.
• Then after 20 years we have
• P 10,000(1.05)20 26,532.98

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Key Suggested Problems

• Section 4.1 1-11 odd, 19, 23, 35
• www.math.ohio-state.edu/hambrock/130/