Download 5.1 and 5.2 Exponential Growth and Decay

Exponential Functions
Copyright Scott Storla 2014
Power functions vs. Exponential functions
Power function
Exponential function
y  x1
y  2x
y  x2
y  1 / 3
y  x3
y  10 x
y  x1/ 2
y  ex
y  x a where a is
a real number.
x
y  a x where a is
a real number.
(a  0, a  1)
Copyright Scott Storla 2014
Definition – Exponential Function
x
An exponential function has the form f  x   Cb where b is a
real number greater than 0 and not equal to 1 and C is a real
number not equal to 0.
Copyright Scott Storla 2014
The formula for continuous compounding, is a
specific example of the more general
exponential growth and decay formula.
A  Pe
Future amount
at time t.
rt
A  t   A0e
Rate of growth
or decay.
kt
Initial amount
Copyright Scott Storla 2014
Vocabulary for exponential functions
Copyright Scott Storla 2014
If as the value of x increases
the value of y increases you
have exponential growth.
If as the value of x increases
the value of y decreases you
have exponential decay.
Copyright Scott Storla 2014
Horizontal asymptote
Let y denote a function.
If as x   or as x   the values of y approach
some fixed number L, then the line y  L is a
horizontal asymptote of the graph of y.
The horizontal asymptote is y  0
Copyright Scott Storla 2014
Discuss whether the graph shows exponential growth or
decay, the intercepts, the domain, the horizontal
asymptote and the range.
Copyright Scott Storla 2014
Discuss whether the graph shows exponential growth or
decay, the intercepts, the domain, the horizontal
asymptote and the range.
Copyright Scott Storla 2014
Discuss whether the graph shows exponential growth or
decay, the intercepts, the domain, the horizontal
asymptote and the range.
Copyright Scott Storla 2014
Discuss whether the graph shows exponential growth or
decay, the intercepts, the domain, the horizontal
asymptote and the range.
Copyright Scott Storla 2014
A procedure to help graph exponential functions
Procedure – Graphing an Exponential Function
1) The domain for the exponential function is
all real numbers.
2) Find any x and y intercepts.
3) Find a few points on the graph. Make sure
to include both positive and negative
exponents.
4) Estimate the horizontal asymptote.
5) Draw the graph.
Copyright Scott Storla 2014
Graph the exponential function.
y  2x
y e
y e
x
x
1
y  2 x  3
Copyright Scott Storla 2014
The formula for continuous compounding, is a
specific example of the more general
exponential growth and decay formula.
A  Pe
Future amount
at time t.
rt
A  t   A0e
Rate of growth
or decay.
kt
Initial amount
Copyright Scott Storla 2014
A  t   A0e kt
A colony of bacteria is experiencing exponential growth
which can be modeled by the function 𝑁 𝑡 = 50𝑒 0.04𝑡 where
N is measured in grams and t is the number of hours.
a) What is the general meaning of the y-intercept?
b) What is the specific meaning of the y-intercept?
c) What does the growth or decay constant imply about the future?
d) What is the weight after 1 day?
e) What is the doubling time for the bacteria?
f) How long will it take for the bacteria to reach 1,000 grams?
g) Compare the average rate of change for 10 hours to 11 hours
with the average rate of change for 50 hours to 51 hours.
Copyright Scott Storla 2014
A  t   A0e kt
The healing time for a wound can be represented by the
function 𝐴 𝑡 = 85𝑒 −0.38𝑡 where t is the number of days since
healing began and A is the area of the wound in 𝑚𝑚2 .
a) What is the general meaning of the y-intercept?
b) What is the specific meaning of the y-intercept?
c) What does the growth or decay constant imply about the future?
d) When will the area of the wound be 20% of its initial area?
e) What will the area of the wound be after 1 week?
f) Find and discuss the meaning of the horizontal asymptote?
g) Compare the average rate of change for 2 days to 3 days with
the average rate of change for 7 days to 8 days.
Copyright Scott Storla 2014
Assume the growth constant, k, for the
population of the United States is 0.011. If the
population was 281.4 million in 2000 estimate
the population in 2020.
A sample of the paint used in a cave painting in
France is found to have lost 82% of its original
carbon-14. Estimate the age of the painting.
The value of k for Carbon-14 is – 0.000121.
A cup of coffee contains approximately 96 mg of
caffeine. When you drink the coffee, the caffeine is
absorbed into the bloodstream and is eventually
metabolized by the body. If the rate of decay is – 0.14,
how many hours does it take for the amount of caffeine
to be reduced to 12 mg?
Copyright Scott Storla 2014
Carbon-14 is often used to date objects that were alive
in the past. Between 1947 and 1956 the Dead Sea
scrolls were discovered in 11 caves along the
northwest shore of the Dead Sea. If the Dead Sea
scrolls are authentic then they should date to around
2000 years old. If 78.5% of the original carbon-14 was
left, was their age appropriate to being authentic? The
value of k for Carbon-14 is -0.000121.
Over the last 60 years a number of North
American snail species have become extinct. If
there are currently 650 species, and the
extinction follows exponential decay, how many
species were there originally? The value of k is
-0.0014.
Copyright Scott Storla 2014
Finding the rate of growth or decay
Radioactive half-life is the time necessary for a
radioactive substance to decay to one-half the
original amount. Carbon-14 has a half life of
5715 years. Find the decay rate for cabon-14.
From 1970 to 1980 the population of the United
States in millions went from 203.3 to 226.5.
Find the growth rate.
$17,000 grows to $23,000 in 8.5 years. Find the
growth rate.
The half-life of silver-110 is 24.6 seconds. How
long will it take for only 3% of the original
sample of silver-110 to remain?
From 1980 to 1990 the population of the United
States in millions went from 226.5 to 248.7.
Estimate the population in 2010.
200 bacteria grow to 600 bacteria in 2 hours. Find
the number of bacteria in 2 days.
Dinosaur bones were dated using potassium-40
which has a half-life of approximately 1.31 billion
years. Analysis of certain rocks surrounding the
bones found that 94% of the original potassium-40
was still present. Estimate the age of the bones.
$350,000 is currently in an account which increased its
value by 40% over the last 5 years. Estimate the number
of years before there is $1,000,000 in the account.
Copyright Scott Storla 2014
Compound Interest
Copyright Scott Storla 2014
Compound Interest
Compound interest occurs when interest
is reinvested as principal and itself
begins earning interest.
The formula for yearly compounding is A  P 1  r  .
t
Find the amount at the end of five years if
$8,000 is invested at 2.15%.
Copyright Scott Storla 2014
Future Value
The future value formula for compound interest is

A  P 1 r
n

nt
The Vocabulary of Compounding Periods
Word
SemiYearly
Quarterly Monthly
annual
Compounding
periods per
1
2
4
12
year (n)
Copyright Scott Storla 2014
Daily
360
or
365

A  P 1 r
n

nt
$30,000 is invested at 2% compounded
quarterly. How much is the investment worth
after 30 years?
A 20 year old student inherits $10,000 and
decides to invest the money for retirement at
5% annual interest compounded monthly. The
student retires at 70. How much is the original
$10,000 worth?
An investment of $7,500 loses 4% per year for
3 years. If the investment was compounded
semi-annually how much is the investment
worth at the end of three years?
Copyright Scott Storla 2014
Present Value

A  P 1 r
A
n
P
1  r n 
nt

 nt
A 1 r
n


nt
Copyright Scott Storla 2014
P

A 1 r
n
 nt
P
In five years a couple wants $50,000 for a down
payment. How much should they save today at
3% compounded monthly for that to happen?
A great grandparent wanted to leave you a million
dollars. How much would they need to have saved
150 years ago at 5% compounded quarterly for the
account to hold a million dollars today?
Copyright Scott Storla 2014
Property – The Power Property of Logarithms
English: The logarithm of an argument written in
exponential form can be written as two factors. One
factor is the exponent of the argument. The other
factor is the logarithm of the base of the argument.
Example: ln2x  x ln2


Algebra: logb xn  n logb x x  0 and b  1
Copyright Scott Storla 2014

A  P 1 r
n

nt
How long will it take for $75,000 to become
$92,000 at 4.3% compounded monthly?
Find the time it takes at 1.7% compounded daily
for $15,000 to become 18,000.
Copyright Scott Storla 2014
Continuous Compounding

A  P 1 r
n

nt
Let n  
1
n
n
360
1
1,000,000
360
1 
 1   1 
1   1   1 

n
1
360

 
 

Answer
2
2.714516
A  Pe
rt
Copyright Scott Storla 2014
1000000
1


1



1000000


2.7182804
A  Pe
rt
In 1800 Great Great Grandma put $50 in an account
paying 5.5% interest compounded continuously? How
much was in the account in 1900? How much was in
the account in 2000? How much will be in the account
this year?
I would like to have $25,000 in an account in 17 years.
What rate would I need to make this happen if I have
$12,800 to invest? Assume continuous compounding.
Copyright Scott Storla 2014
To retire in 14 years I want to have $1,500,000 in
savings. If I’m able to get 4.2% compounded
continuously how much do I need to have saved
by today to make this happen?
Upon retiring I don’t want to use any of my
principal. How much in interest will I be able to
live on each subsequent year? Assume 4.2%
compounded continuously.
Unfortunately .
Copyright Scott Storla 2014