Name___________________________________
Period__________
Calculus
6.4 Exponential Growth and Decay
The Law of Exponential Change:
Continuously Compounding Interest:
Radioactive Decay:
y  yo e kt
A  Pe rt
y  yo e  kt
Half-Life: t 
ln 2
k
A. Initial Value Problems
Solve the initial value problem. Indicate the domain over which the solution is valid.
dy
x
2.)
  and y = 3 when x = 4.
dx
y
B. Formula Review (Book Example)
35. People who do C-14 dating use a figure of 5700 years for its half-life. Find the age of a sample in which
10% of the radioactive nuclei originally present have decayed.
C. “Rate is directly proportional to amount present”
1988 BC #43
Bacteria in a certain culture increase at a rate proportional to the number present. If the number of bacteria
doubles in three hours, in how many hours will the number of bacteria triple?
1993 AB 42
A puppy weighs 2.0 pounds at birth and 3.5 pounds two months later. If the weight of the puppy during its first
6 months is increasing at a rate proportional to its weight, then how much will the puppy weigh when it is 3
months old?
1973 BC 17. The number of bacteria in a culture is growing at a rate of 3000e
number of bacteria present was 7,500. Find the number present at t = 5.
2t
5
per unit of time t. At t = 0, the
1989 – AB 6 Oil is being pumped continuously from a certain oil well at a rate proportional to the amount of oil
dy
 ky , where y is the amount of oil left in the well at any time t. Initially there were
left in the well; that is,
dt
1,000,000 gallons of oil in the well, and 6 years later there were 500,000 gallons remaining. It will no longer be
profitable to pump oil when there are fewer than 50,000 gallons remaining.
(a) Write an equation for y, the amount of oil remaining in the well at any time t.
(b) At what rate is the amount of oil in the well decreasing when there are 600,000 gallons of oil remaining?
(c) In order not to lose money, at what time t should the oil no longer be pumped from the well?
AP Homework
1993 BC 38. During a certain epidemic, the number of people that are infected at any time increase at a rate
proportional to the number of people that are infected at that time. If 1,000 people are infected when the epidemic
is first discovered, and 1,200 are infected 7 days later, how many people are infected 12 days after the epidemic is
first discovered?
dy
 ky , where k is a constant and t is measured in
dt
years. If the population doubles every 10 years, then the value of k is _____________.
1998 AB 84. Population y grows according to the equation
1987 – BC 1 At any time t  0 , in days, the rate of growth of a bacteria population is given by y  ky , where k is
a constant and y is the number of bacteria present. The initial population is 1,000 and the population triples during
the first 5 days.
(a) Write an expression for y at any time t  0 .
(b) By what factor will the population have increased in the first 10 days?
(c) At what time t, in days, will the population have increased by a factor of 6?
1996 – AB/BC 3 The rate of consumption of cola in the United State is given by S  t   Cekt , where S is measured
in billions of gallons per year and t is measure in years from the beginning of 1980.
(a) The consumption rate doubles every 5 years and the consumption rate at the beginning of 1980 was 6 billion
gallons per year. Find C and k.
(b) Find the average rate of consumption of cola over the 10-year time period beginning January 1, 1983. Indicate
units of measure.
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(c) Use the trapezoidal rule with four equal subdivisions to estimate  S  t  dt .
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(d) Using correct units, explain the meaning of
 S  t  dt in terms of cola consumption.
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