Transcript Slide 1

Chapter 3
Exponential and Logarithmic Functions
1
3.5 Exponential and
Logarithmic Models
Objectives:
Recognize the five most common types of
models involving exponential or logarithmic
functions.
Use exponential growth and decay functions ,
Gaussian functions, logistic growth functions,
and logarithmic functions to model and solve
real-life problems.
2
Exponential Growth Model
Model: y = aebx, b > 0
3
Exponential Decay Model
Model: y = ae–bx, b > 0
4
Gaussian Model
Model:
 x b 2 / c
y  ae
5
Logistic Growth Model
Model:
a
y
 rx
1  be
6
Logarithmic Models
Models:
y = a + b ln x
y = a + b log10 x
7
Example – Exponential Growth
Estimates of the world
population (in millions)
from 1995 through 2004
are shown in the table. Use
your graphing calculator to
create a scatter plot of the
data.
Year
Population
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
5685
5764
5844
5923
6002
6079
6154
6228
6302
6376
8
Example – Exponential Growth
An exponential growth model that approximates
this data is given by
P  5344e0.012744 t , 5  t  14
where P is the population (in millions) and t = 5
represents 1995. According to the model, when
will the world population reach 6.8 billion?
9
Example – Create the Model
In a research experiment, a population of fruit flies is
increasing according to the law of exponential growth.
After 2 days there are 100 flies, and after 4 days there
are 300 flies. How many flies will there be after 5 days?
10
Carbon Dating Model
In living organic material, the ratio of the content of
radioactive carbon isotopes (carbon 14) to the content
of nonradioactive carbon isotopes (carbon 12) is about
1 to 1012.
When organic material dies, its carbon 12 content
remains fixed, whereas its radioactive carbon 14 begins
to decay with a half-life of 5730 years.
To estimate the age of dead organic material, scientists
use the formula
1
R  12 e t / 8267
10
which denotes the ratio of C14 to C12 present at any
time t (in years).
11
Example – Carbon Dating
The ratio of carbon 14 to carbon 12 in a newly
discovered fossil is
1
R  13
10
Estimate the age of the fossil.
12
Example – Gaussian Model
In 2002, the SAT math scores for college-bound
seniors roughly followed the normal distribution
y  0.0035e
 x516 2 / 25,992
200 x  800
where x is the SAT score for math. Graph this
function and estimate the average SAT score.
13
Example – Logistic Growth
On a college campus of 5000 students, one
student returns from vacation with a contagious
flu virus. The spread of the virus is modeled by
5000
y
, t0
 0.8t
1  4999 e
where y is the total number infected after t days.
The college will cancel classes when 40% or more
of the students are infected.
a. How many students are infected after 5 days?
b. After how many days will the college cancel
classes?
14
Logarithmic Model – Richter Scale
On the Richter scale, the magnitude R of
an earthquake of intensity I is given by
I
R  log10
Io
where Io = 1 is the minimum intensity used
for comparison.
Intensity is a measure of the wave energy of
an earthquake.
15
Example – Logarithmic Model
In 2001, the coast of Peru experienced an earthquake
that measured 8.4 on the Richter scale. In 2003,
Colima, Mexico experienced an earthquake that
measured 7.6 on the Richter scale. Find the intensity
of each earthquake and compare the two intensities.
16
Homework 3.5
Worksheet 3.5
# 1 – 6 matching, 23 – 26, 28,
29 – 43 odd, 47, 48
17