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Exponential Growth DecaySection 4.5

- JMerrill, 2005
- Revised 2008

Review

Is this okay?

NO

Arguments must be positive

Review

500e0.3x 600 e0.3x 1.2 ln 1.2 0.3x

x 0.608

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.

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

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.

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?

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)

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 ?

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!

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.

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

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.

Carbon Dating Example

78 of the original amount

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.

You do

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.

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.

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?

Modeling the Spread of the Flu

- How many people became ill with the flu when the

epidemic began? - In the beginning, t 0

Modeling the Spread of the Flu

- 2. How many people were ill by the end of the

fourth week?

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!

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