### Lecture Notes for Section 5.6 (Castillo

```Exponential models
Standard form(s) of an exponential
• y=abx
 a is the initial value (y-intercept)
 b is the growth factor
• y=aerx
 a is the initial value (y-intercept)
 r is the “per-capita rate” also called “exponential
growth rate”
• Conversion
 b=er, r=ln(b)
When to use an exponential model
• Things making more things
– People making more people, bacteria making
more bacteria, money making more money
(interest)
• Lots of identical events happening at
random times
– Radioactive decay (atoms decay at random times),
Heat transfer (atoms bump into each other
randomly)
Bacteria
• A colony of bacteria weighs 1mg at 12:00 and 3mg at 2:00.
How much will it weigh at 6:00?
• t=# of hours since 12:00
• n=# of mg of bacteria
• (t,n) points are (0,1) and (2,3)
• Bacteria make more bacteria, so this is an exponential
model.
n=abt
• When t=0, Initial value (n) is 1. n=(1)bt
• When t=2, n=3.
3=1b(2)
• b=√3
n=(1)(√3) t
• At 6 hours, I have n=(1)(√3)6
• 27mg of bacteria
Standard form(s) of an exponential
• y=abx
 a is the initial value (y-intercept)
 b is the growth factor
• y=aerx
 a is the initial value (y-intercept)
 r is the “per-capita rate” also called “exponential
growth rate”
• Conversion
 b=er, r=ln(b)
What is a percent rate?
• Imagine that you have an investment earning 3%
per year.
• Per-cent means “for each one hundred.”
• 3% per year means “for each group of 100 dollars
you have in your account, you earn 3 dollars per
year.”
• An investment of \$100 earns 3 dollars per year.
– An investment of \$200 earns 6 dollars per year,
– An investment of \$300 earns 9 dollars per year, etc.
What is compounding?
• Now imagine that you have \$100 in your account. After a year,
you have more than \$103 dollars! Why is that?
• Well those \$3 per year weren’t paid at the end of the year,
they were paid in installments.
• Each time you were paid money, your had more money in
• 3% of 100.25 is more than 3% of 100. So even after a month,
when you have earned a quarter you are now earning money
faster
• This is because not just your \$100 are earning money, but also
the children of your \$100 (that 25¢)are earning money. And
the children of the 25¢ (the grandchildren of your \$100 will
also earn money for you.)
What is APR?
• Because of compounding, the percent earned on
a 3%/year investment after a year is actually
more than 3% of what you started with.
• The percent you actually earn after a year is
called your “annual percent yield” (APY) and it is
higher than your “annual percentage rate” (APR)
• Credit cards always tell you APR because it is
always the smaller number. It’s not what you
really pay (APY).
What is continuous compounding?
• Now imagine that you have \$100 in your account. After a year, you have
more than \$103 dollars! Why is that?
• Well those \$3 per year weren’t paid at the end of the year, they were paid
in installments.
• However, in magical happyland (not the real world), you are not paid in
installments, you are paid all the time.
– Imagine walking 3 miles per hour. You don’t just wait an hour and then
teleport three miles, you’re gaining distance all the time, as you’re
walking.
– Continuous compounding is the same idea: You are gaining money all
the time, as you are waiting.
• This means that every little tiny bit of money you have is earning 3% per
year all the time. Every day, every hour, every minute you earn new money
Compound interest formula
nt
æ rö
A = P ç1+ ÷
è nø
Compounding continuously
¥t
æ rö
A = P ç1+ ÷
è ¥ø
We don’t know how to do this…. Yet.
Compounding continuously
Using a trick from calculus
¥t
æ rö
A = P ç1+ ÷
è ¥ø
¥
æ rö
r
ç1+ ÷ = e
è ¥ø
A = Pe
rt
Transitioning to per-capita rate
• Imagine that you have an investment earning 3%
per year.
• Per-cent means “for each one hundred.”
• 3% per year means “for each group of 100 dollars
you have in your account, you earn 3 dollars per
year.”
• But the number that actually goes in the equation
is 3/100
• When compounding continuously, the math is
groups of 100.
Transitioning to per-capita rate
• Imagine that you have an investment earning 3%
per year.
• 3% per year means “for each group of 100 dollars
you have in your account, you earn 3 dollars per
year.”
• 3% also means “for each dollar I have, I earn 0.03
dollars per year.”
• 1 dollar earns 3 cents per year
– 10 dollars earns 30 cents per year
– 100 dollars earns 3 dollars per year
What is a per-capita rate?
• Imagine that you have an investment earning 3%
per year.
• 3% also means “for each dollar I have, I earn 0.03
dollars per year.”
• The per-capita rate is 0.03 dollars per year per
dollar.
• Sometimes people cancel the dollars and write
this as 0.03/year
• When compounding continuously, this is the
number that goes in the r part of y=aerx
Bacteria
• A colony of bacteria has been measured as growing at
an exponential rate of 0.033 bacteria per bacterium
per hour. At 6 hours, there were 4.266 mg of bacteria.
How much did the colony weigh at 0 hours?
• t=# of hours
• n=number of mg of bacteria
• Exponential rate means use n=aert
• Looking for the initial value… find a.
• 4.266=ae0.033(6)
• a≈3.5
General Strategy
• Write down your equation y=abx or y=aert
• Fill in the parts you know
– Values of a, b, or r
– Values of x and y at specific times
• Solve for the parts you don’t know
– A missing value of a, b, or r.
– Value of x or y at a different time.
Suppose that at the start of the century, there are 25
teachers in a certain small town. Assuming the number
of teachers grows exponentially with an exponential
growth rate of .005/yr, what is the exponential growth
model for this situation?
a)
b)
c)
d)
e)
P(t)=.005e25t
P(t)=e25t
P(t)=25e.005t
P(t)=25et
None of the above
Suppose that at the start of the century, there are 25
teachers in a certain small town. Assuming the number
of teachers grows exponentially with an exponential
growth rate of .005/yr, what is the exponential growth
model for this situation?
t=# of years
P(t)=# of teachers at time t
P(t)=aert
a=initial number of teachers =25.
r=per-capita rate = 0.005
P(t)=25e0.005t
C
Doubling time problems
• A colony of bacteria has been measured as growing at an
exponential rate of 0.033 bacteria per bacterium per hour.
How long will it take my colony to double in size?
• t=# of hours
• n=number of mg of bacteria
• Exponential rate means use n=aert
• Looking for a time…. Find t.
• I started with a, when my colony doubled, I had 2a.
• 2a=ae0.033t
• 2=e0.033t
• Ln(2)=0.033t
• t=ln(2)/0.033 or approx 21 hours.
How long will take an initial investment of \$P to
double in value under continuous compounding if the
interest rate is 6%?
a)
b)
c)
d)
e)
How long will take an initial investment of \$P to
double in value under continuous compounding if the
interest rate is 6%?
•
•
•
•
•
•
•
•
•
•
t=number of years
A=amount after t years
Continuous compounding, used A=Pert
“How long” means find t.
I started with P, when I doubled I had 2P. A=2P.
2P=Pe0.06t
2=e0.06t
ln(2)=0.06t
t=ln(2)/0.06 approx 11.6 years
• Carbon-14 has a half life of about 5750 years. What is the corresponding
exponential rate?
• t=# of years
• m=# of grams of carbon-14
• m=aert
• Find r.
• After 5750 years, I have half as much carbon-14. I started with a, so I have
0.5a left.
• 0.5a=aer5750
• 0.5=er5750
• ln(0.5)=5750r
• r=ln(0.5)/5750 or approx -0.00012 atoms per year per atom.
• Bonus: the equation for carbon-14 decay is m=ae-0.00012t
The half life of Cesium-137 is 30 years.
What is the exponential decay rate, r, of this
element?
a)
b)
c)
d)
e)
r = -.0552
r = -.0231
r = -.0431
r = -.0131
None of the above
The half life of Cesium-137 is 30 years.
What is the exponential decay rate, r, of this
element?
t=#of years
m=#of grams of Cesium-137
m=aert
Find r.
I started with a. after 30 years (t=30), I had half as much (0.5a)
0.5a=aer30
0.5=er30
ln(0.5)=30r
r=ln(0.5)/30 or approx -0.0231 atoms per year per atom
B
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