Riva – AP Calc AB
Review of Exponential Functions
1) The function p defined by p(t) = 3960(1.02)t describes the population of
Dilcue, North Dakota t years after it was founded.
a. Find the founding population
b. At what annual rate has the population of Dilcue been growing?
c. Solve the equation p(t)=77218. What is the meaning of your answer?
2) An exponential function f is defined by f (x) = k × b x . Find k and b given that
a. f(0)=3 and f(1)=12
b. f(0)=4 and f(2)=1
3) Our civilization lies at the bottom of an ocean of air. As we move through this
ocean, pressure varies exponentially with altitude – each increase of one mile
of altitude causes the pressure to drop by 20 percent. Given that air pressure
at sea level is 14.7 pounds per square inch, write formulas for the following
exponential functions:
a. M(x) is the air pressure at an altitude x miles above sea level.
b. F(x) is the air pressure at an altitude x feet above sea level.
Then calculate the air pressure
c. On the top of Mt. Everest, which is 29141 feet above sea level
d. At the edge of the Dead Sea, which is 1300 feet below sea level
4) Find a plausible equation y = a × b x for each of the exponential graphs shown
below:
a)
b)
c)
5) A potato is taken from the oven, its temperature having reached 350 degrees.
After sitting on a plate in a 70-degree room for twelve minutes, its
temperature has dropped to 250 degrees. In how many more minutes will
the potato’s temperature reach 120 degrees?
(Note: Assume Newton’s Law of Cooling. Newton’s Law of Cooling is described
by the exponential equation D = D0 bt , in which t represents time, D is the
difference between the temperature of the cooling object and the
surrounding temperature, D0 is the initial temperature difference, and b is a
positive constant that incorporates the rate of cooling.)
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Riva – AP Calc AB
6) An atom of carbon-14 is unstable, meaning that it can spontaneously transfer
itself (by radioactive decay) into nitrogen at any instant. The probability that
this will actually happen to a specific atom of carbon-14 in the course of a
year is only about 0.0121 percent, however. In other words, there is a
99.9879 percent chance that any specific atom of carbon-14 will survive
another year. Suppose that one million carbon-14 atoms are placed in a
container.
a. How many of these atoms are expected to be carbon-14 atoms one
year later?
b. The half-life question: How much time will it take for half of the atoms
to decay?
7) (continuation) Carbon-14, which is produced from nitrogen when solar
radiation bombards the upper atmosphere, makes up about 1 trillionth of all
the carbon found in things that rely on air to live. When an organism dies, it
no longer takes in air, and its supply of carbon-14 diminishes exponentially
as described above. Apply this principle to estimate the age of a lump of
charcoal (found in a cavern at an ancient campsite) that has only 32 percent
as much carbon-14 as it had when the charcoal was still firewood.
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Riva – AP Calc AB
Answers
1) a. 3,960; b. 2% increase; c. 150 years
x
1ö
æ
x
2) a. f (x) = 3× 4 ; b. f (x) = 4 × ç ÷
è 2ø
3) a. M(x) = 14.7×(.8)x ; b. M(x) = 14.7×(.999958)x ; c.4.29 lb per sq in; d. 15.53 lb per sq in
x
æ 2ö
4) a. y = 1×3x ; b. f (x) = 2 × ç ÷ ; f (x) = 1×
è 5ø
5) 46.8 minutes
6) a) 999879; b) 5728 years
7) 9,416 years
( 4)
3
x
3