Transcript Exponential Functions and Models Lesson 5.3 Contrast Linear Functions Exponential Functions Change at a constant rate  Rate of change (slope) is a constant  Change at a changing rate  Change.

Exponential Functions
and Models
Lesson 5.3
Contrast
Linear
Functions
Exponential
Functions
Change at a constant rate
 Rate of change (slope) is a
constant

Change at a changing rate
 Change at a constant
percent rate

Definition

An exponential function
f ( x)  C  a



x
Note the variable is in the exponent
The base is a
C is the coefficient, also considered the initial value
(when x = 0)
Explore Exponentials


Given f(1) = 3, for each unit increase in x, the
output is multiplied by 1.5
Determine the exponential function
f ( x)  C  a
x
1
2
3
4
f(x)
0.75
x
Explore Exponentials

Graph these exponentials
f ( x)  2  1.1
x
g ( x)  4   0.9 

x
What do you think the coefficient C and the
base a do to the appearance of the graphs?
Contrast Linear vs.
Exponential

Suppose you have a choice of two different jobs
at graduation



Start at $30,000 with a 6% per year increase
Start at $40,000 with $1200 per year raise
Which should you choose?


One is linear growth
One is exponential growth
Which Job?





How do we get each next
value for Option A?
When is Option A better?
When is Option B better?
Rate of increase a
constant $1200
Rate of increase changing


Percent of increase is a constant
Ratio of successive years is 1.06
Year
Option A
Option B
1
$30,000
$40,000
2
$31,800
$41,200
3
$33,708
$42,400
4
$35,730
$43,600
5
$37,874
$44,800
6
$40,147
$46,000
7
$42,556
$47,200
8
$45,109
$48,400
9
$47,815
$49,600
10
$50,684
$50,800
11
$53,725
$52,000
12
$56,949
$53,200
13
$60,366
$54,400
14
$63,988
$55,600
Example

Consider a savings account with compounded
yearly income


You have $100 in the account
You receive 5% annual interest
At end of
year
View
completed
table
Amount of interest earned
New balance in
account
1
100 * 0.05 = $5.00
$105.00
2
105 * 0.05 = $5.25
$110.25
3
110.25 * 0.05 = $5.51
$115.76
4
5
Compounded Interest
New balance in
Amount of
At end of
account
interest earned
year

Completed table
0
1
2
3
4
5
6
7
8
9
10
0
$5.00
$5.25
$5.51
$5.79
$6.08
$6.38
$6.70
$7.04
$7.39
$7.76
$100.00
$105.00
$110.25
$115.76
$121.55
$127.63
$134.01
$140.71
$147.75
$155.13
$162.89
Compounded Interest

Table of results from
calculator



Set Y= screen
y1(x)=100*1.05^x
Choose Table (♦ Y)
Graph of results
Assignment A



Lesson 5.3A
Page 415
Exercises 1 – 57 EOO
Compound Interest

Consider an amount A0 of money deposited in
an account




Pays annual rate of interest r percent
Compounded m times per year
Stays in the account n years
Then the resulting balance An
r

An  A0 1  
 m
mn
Exponential Modeling

Population growth often modeled by exponential
function

Half life of radioactive materials modeled by
exponential function
Growth Factor

Recall formula
new balance = old balance + 0.05 * old balance

Another way of writing the formula
new balance = 1.05 * old balance

Why equivalent?

Growth factor: 1 + interest rate as a fraction
Decreasing Exponentials

Consider a medication



Patient takes 100 mg
Once it is taken, body filters medication out over
period of time
Suppose it removes 15% of what is present in
the blood stream every hour
Fill in the
rest of the
table
At end of hour
Amount remaining
1
100 – 0.15 * 100 = 85
2
85 – 0.15 * 85 = 72.25
3
4
5
What is the
growth factor?
Decreasing Exponentials

At end of hour Amount Remaining
1
85.00
2
72.25
3
61.41
4
52.20
5
44.37
6
37.71
7
32.06
Completed chart
Growth Factor = 0.85
Note: when growth factor < 1,
exponential is a decreasing
function
Amount Remaining
Graph
100.00
Mg remaining

80.00
60.00
40.00
20.00
0.00
0
1
2
3
4
5
At End of Hour
6
7
8
Solving Exponential Equations
Graphically



For our medication example when does the
amount of medication amount to less than 5 mg
Graph the function
for 0 < t < 25
Use the graph to
determine when
M (t )  100  0.85  5.0
t
General Formula

All exponential functions have the general
format:
t
f (t )  A  B

Where



A = initial value
B = growth rate
t = number of time periods
Typical Exponential Graphs

When B > 1
f (t )  A  Bt

When B < 1
Using e As the Base
y = A * Bt
B = ek

We have used
Consider letting

Then by substitution

Recall

It turns out that

B = (1 + r)
y = A * (ek)t
(the growth factor)
kr
Continuous Growth


The constant k is called the continuous percent
growth rate
For Q = a bt

k can be found by solving ek = b

Then Q = a ek*t

For positive a


if k > 0 then Q is an increasing function
if k < 0 then Q is a decreasing function
Continuous Growth

For Q = a ek*t

k>0

k<0
Assume a > 0
Continuous Growth


Q  3 e
For the function
what is the
continuous growth rate?
The growth rate is the coefficient of t
0.4t


Growth rate = 0.4 or 40%
Graph the function (predict what it looks like)
Converting Between Forms

Change to the form Q = A*Bt
Q  3 e


0.4t
We know B = ek
Change to the form Q = A*ek*t
Q  94.5(1.076)
t

We know k = ln B
(Why?)
Continuous Growth Rates

May be a better mathematical model for some
situations


Bacteria growth
Decrease of medicine
in the bloodstream

Population growth of a large group
Example


A population grows from its initial level of
22,000 people and grows at a continuous
growth rate of 7.1% per year.
What is the formula P(t), the population in
year t?


P(t) = 22000*e.071t
By what percent does the population increase
each year (What is the yearly growth rate)?

Use b = ek
Assignment B



Lesson 5.3B
Page 417
Exercises 65 – 85 odd