Exponential Growth

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Exponential Growth DecaySection 4.5

• JMerrill, 2005
• Revised 2008

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Review
Is this okay?
NO
Arguments must be positive
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Review
500e0.3x 600 e0.3x 1.2 ln 1.2 0.3x
x 0.608
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Exponents and Logarithms

• How are exponents and logarithms related?
• They are inverses of each other
• Why is this important?
• Using inverses allow us to solve problems (we use
  subtraction to solve addition problems division
  to solve multiplication)
• Many real-life scenarios are exponential in
  nature and logarithms allow us to solve for the
  unknown.

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Examples Using Logarithmic Scales

• The Richter scale is used to determine the
  intensity of an earthquake.
• Measuring acidity using the pH scale, or
  concentration of ions.
• Carbon dating.
• Modeling population growth/decay--just to name a
  few

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Exponential Decay Model

• A(t) A0ekt
• A0 is the initial amount
• K is the growing/decay entity. If kgt0, the
  entity is growing (an increasing function). If
  klt0, the entity is decaying (a decreasing
  function).
• Looks like A(t) Pert? It works the same way.

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Population Model

• In 1970, the US population was 203.3 million. In
  2003, the population was 294 million.
• Find the exponential growth model
• By which year will the US population reach 315
  million?

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Population Model

• t is the number of years after 1970.
• t0 represents 1970. t 33 represents 2003
• When t 33, A 294
• A(t) A0ekt
• 294 203.3ek(33)

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Population Cont
What do you do when the exponent is a variable?

• 294 203.3ek(33)

So, k 0.011, which is exponential growth
The growth model is A(t) 203.3e0.011t
What does lne ?
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Population Cont

• When will the population reach 315 million?
• A(t) 203.3e0.011t
• 315 203.3e0.011t
• You finish
• Did you get approximately 40?
• That means that in the year 2010 the population
  will be approx. 315 million!

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Carbon Dating

• The natural base, e, is used to estimate the ages
  of artifacts. Plants and animals absorb
  Carbon-14 from the atmosphere. When a plant or
  animal dies, the amount of carbon-14 it contains
  decays in such a way that exactly half of the
  initial amount is present after 5,715 years.

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Carbon Dating

• The function that models the decay of carbon-14,
  where A0 is the initial amount of carbon-14, and
  A(t) is the amount present t years after the
  plant or animal dies, is

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Carbon Dating Example

• Archaeologists find scrolls and claim that they
  are 2000 years old. Tests indicate that the
  scrolls contain 78 of their original carbon-14.
  Could the scrolls be 2000 years old?
• Using the same process as the last example, we
  find k to be -0.00012.
• Finding k is written out in the book on P449.

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Carbon Dating Example
78 of the original amount
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You Do

• A wooden chest is found and said to be from the
  2nd century BCE. Tests on a sample of wood from
  the chest reveal that it contains 92 of its
  original carbon-14. Could the chest be from the
  2nd century BCE?
• Use the same k as the last example.

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You do
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Logistic Growth Model

• The spread of disease is exponential in nature.
  However, there arent an infinite number of
  people. Eventually, the disease has to level
  off. Growth is always limited. A logistic
  growth model is used in this type of situation
• Y c is the horizontal asymptote. Thus c is the
  limiting value of the function.

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Modeling the Spread of the Flu

• The function below describes the number of
  people, f(t), who have become ill with influenza
  t weeks after its initial outbreak in a town with
  a population of 30,000 people.

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Modeling the Spread of the Flu

• How many people became ill with the flu when the
  epidemic began?
• How many people were ill by the end of the fourth
  week?
• What is the limiting size of f(t), the population
  that become ill?

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Modeling the Spread of the Flu

• How many people became ill with the flu when the
  epidemic began?
• In the beginning, t 0

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Modeling the Spread of the Flu

• 2. How many people were ill by the end of the
  fourth week?

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Modeling the Spread of the Flu

• 3. What is the limiting size of f(t), the
  population that become ill?

C represents the limiting size that f(t) can
obtain. There are only 30,000 people in the
town, therefore, the limiting size must be 30,000!