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5-Minute Check on Activity 5-8

What is the domain and range of y = ex?

Compared to y = ex, describe the transformation and the range of

y = ex–2

y = ex + 2

y = 3ex – 1

Find the constant of continuous decay, k, if the decay factor is 0.8

If 10 mg of a substance is metabolized by the body at a rate of 12% an hour, how long until only ½ of the original amount is left in the body?

domain = {x | x Real #’s} range = {y | y > 0}

shifts graph to the right by 2 range = {y | y > 0}

shifts graph up by 2 range = {y | y > 2}

shifts graph down by 1 range = {y | y > -1}

and stretches it up

b = ek 0.8 = ek k = -0.2231

y1= ex and y2 = 0.8

y1= ex and y2 = 0.88 k = -0.1278

y1=10e-0.1278x and y2=5 x = 5.42 hours

Click the mouse button or press the Space Bar to display the answers.

Objectives

- Determine the equation of an exponential function that best fits the given data
- Make predictions using an exponential regression equation
- Determine whether a linear or exponential model best fits the data

Vocabulary

- None new

Activity

In 2005, the avian flu, also known as the bird flu, received international attention. Although there were very few documented case of the avian flu infecting humans worldwide, world health organizations including the Centers for Disease Control in Atlanta expressed concern that a mutant strain of the bird flu virus capable of infecting humans would develop and produce a worldwide pandemic. The infection rate (the number of people that any single infected person will infect) and the incubation period (the time between exposure and the development of symptoms) of this flu cannot be known precisely but they can be approximated by studying the infection rates and incubation periods of existing strains of the virus.

Activity cont

A conservative infection rate would be 1.5 and a reasonable incubation period would be about 15 days (0.5 months). This means that the first infected person could be expected to infect 1.5 people in about 15 days. After 15 days that person cannot infect anyone else. This assumes that the spread of the virus is not checked by inoculation or vaccination. So after 15 days from the first infection there are 2.5 people infected. In the next 15 days the 1.5 newly infected people would infect 2.25 more people (1.5 1.5). Now we have 4.75 people who have been infected (at the end of one month).

Fill in the following table:

x

Activity contRound each value to the nearest person.

Construct a scatterplot of the data L1: Months, L2: Total

8 12 18 27 41 61

21 33 51 78 119 180

180

160

140

Does the scatterplot indicate a linear relationship?

120

100

80

No; rate of change increasing

Exponential

60

40

20

1

2

3

4

5

Activity cont

Use your calculator to model the data with an exponential function. STAT CALC, option 0: ExpReg to determine an exponential regression model of best fit. Round all values to 3 decimal places.

What is the practical domain of this function?

How does the N-intercept compare with the table?

Use the model to predict the total number infected after 1 year.

When will 2 million people be infected?

Y = abt N = 1.551(2.684)t

t ≥ 0 or until everyone infected

table: 1 model: 1.551 (a little high)

N(12) = 1.551(2.684)t = 1.551(2.684)12 = 216,343

after about 14.253 months; about 15 months

Increasing Exponential Example

According to the US Department of Education, the number of college graduates increased significantly during the 20th century. The following table gives the number (in thousands) of college degrees awarded from 1990 to 2000:

What is the exponential regression equation?

What is the base of the exponential model?

What is the annual growth rate?

Y = abt N = 40.25(1.0394)t

base, b = 1.0394

growth rate = b – 1 = 0.0394 or about 3.94%

Increasing Exponential Example Cont

Use the exponential model to determine the number of college graduates in 2010 (t = 110).

When will the number of college grads equal 2 million?

What is the doubling time for the exponential model?

N(110) = 40.25(1.0394)110 = 2,824 thousand graduates

= 2,824,000 graduates

2000 = 40.25(1.0394)t around t = 101.12 or 2002

(Solve graphically with intersection)

80.5 = 40.25(1.0394)t around t = 17.9 years

(solve graphically with intersection)

Decreasing Exponential Example

Students in US public schools have had much greater access to computers in recent years. The following table shows the number of students per computer in selected years:

What is the exponential regression equation?

What is the base of the exponential model?

What is the annual decay rate?

Y = abt C = 82.9(0.837)t

base, b = 0.837

decay rate = 1 – b = 0.163 or about 16.3%

Summary and Homework

- Summary
- Quantities that increase or decrease continuously at a constant rate can be modeled by y = aekt.
- Increasing: k > 0 k is continuous rate of increase
- Decreasing: k < 0 |k| is continuous rate of decrease
- The initial quantity at t=0, a, may be written in other forms such as y0, P0, etc
- Remember the general shapes of the graphs
- Homework
- Page 614-617; problems 1- 3

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