Transcript 5.8 Exponential Growth and Decay Wed Feb 27

5.8 Exponential Growth and Decay
Mon Dec 7
• Do Now
• In the laboratory, the number of Escherichia
coli bacteria grows exponentially with growth
constant k = 0.41. Assume that 1000 bacteria
are present at time = 0
• 1) Find the formula for the # of bacteria P(t) at
time t
• 2) How large is the population after 5 hours?
• 3) When will the population reach 10,000?
HW Review p.339
Exponential growth and decay
P(t) = P0 e
kt
•When P0 is the initial size at t = 0
•If k > 0, then P(t) grows exponentially
•If k < 0 then P(t) decreases exponentially
•K is either known as the growth or decay
constant.
•We’ve done these things in Pre-calc
Y’ = ky
• If y(t) is a differentiable function satisfying the
differential equation
y' = ky
then y(t) = P0 e , where P0 is the initial
value P0 = y(0)
kt
Notes
• This theorem tells us that a process obeys an
exponential law precisely when its rate of
change is proportional to the amount present.
• A population grows exponentially because
each present organism contributes to growth
Ex
• Find all solutions of y’ = 3y. Which solution
satisfies y(0) = 9?
Ex
• Pharmacologists have shown that penicillin leaves a
person’s bloodstream at a rate proportional to the
amount present.
• A) Express this as a differential equation
• B) Find the decay constant if 50 mg of penicillin
remains in the bloodstream 7 hours after an injection
of 450 mg
• C) Under the hypothesis of (B), at what time was 200
mg present?
Doubling Time / Half-life
• If
P(t) = P0 e
ln 2
k
kt ,
then
is the doubling time if k >0
ln 2
is the half-life if k < 0
k
Compound Interest
• If P0 dollars are deposited into an account
earning interest at an annual rate r,
compounded M times yearly, then the value
of the account after t years is
r Mt
P(t) = P0 (1+ )
M
• If compounded continuously, is
P(t) = P0 e
rt
Present Value
• The concept of present value (PV) is used to
compare payments made at different times
• The PV of P dollars received at time t is
PV = Pe
-rt
• Having money now means you can get interest
from it right away.
• Getting money later means you will lose out on
any interest you could have gotten
Ex
• Is it better to receive $2000 today or $2200 in
2 years? Consider a 3% and 7% compounded
interest rate
Ex
• You must decide whether to upgrade your
company’s computer system. The upgrade
costs $400,000 and will save $150,000 per
year for the next 3 years. Is this a good
investment if r = 7%?
Income stream
• An income stream is a sequence of periodic
payments that continue over an interval of T
years
• This is like winning the lottery and taking
several payments over the years vs taking a
lump sum
PV of an income stream
• If the interest rate is r, the present value of an
income stream paying out R(t) dollars per year
continuously for T years is
T
-rt
PV =
ò
0
R(t)e dt
Ex
• An investment pays out 800,000 pesos per
year, continuously for 5 years. Find the PV of
the investment for r = 0.04
Closure
• For the investment that paid out 800,000
pesos per year for 5 years, find the PV of the
investment if r = 0.06
• HW: p.350 #3 7 13 14 15 39 44 45