Download Wed, Mar 25

Clicker Question 1
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The radius of a circle is growing at a constant
rate of 2 inches/sec. How fast is the area of
the circle growing when the radius is 5
inches?
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A. 5 sq. in./sec
B. 10 sq. in./sec
C. 20 sq. in./sec
D. 25 sq. in./sec
E. 50 sq. in./sec
Clicker Question 2
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If the height (in feet) of a ball thrown
upward is h (t ) = 64t – 16t 2 (t in
seconds), what is its maximum height?
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A. 48 feet
B. 54 feet
C. 64 feet
D. 72 feet
E. There is no maximum height
Clicker Question 3
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The profit function (in $) at production
level x is P (x ) = 30x + 200 ln(x). What
is the marginal profit at a production
level of 100?
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A. $ 50 / item
B. $ 32 / item
C. $ 17 / item
D. $ 76 / item
E. $110 / item
Exponential Growth and Decay
(3/25/09)
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It is very common for a population P to
grow (or decay) at a rate proportional
to its size.
This is described by the differential
equation dP /dt = k P for some
constant number k.
A differential equation is an equation
containing one or more derivatives. A
solution is a function which makes it
true, i.e., which satisfies it.
Exponential functions solve
that equation:
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Check that P (t) = C e k t satisfies the
differential equation on the last slide.
Note that C is just the initial population
P (0).
It can be shown that this is the only
solution to the equation.
k is called the relative (or continuous)
growth rate.
Example of Growth
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A population P of bacteria is growing at
a relative rate of 5% each day. It starts
with 100 bacteria.
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Write an equation for P (t) , t in days.
How many bacteria are there after 10
days?
How long (from the start) will it take for
the population to reach 1000?
Example: Radioactive Decay
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The half-life of a radioactive substance
is the number of years for half the
substance to decay.
Here it makes sense to use 1/2 as the
base instead of e.
The half-life of radium-226 is 1590 years.
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If I start with 100 mg, how much is there
after 200 years?
How long before only 20 mg are left?
Newton’s Law of Cooling
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A body cools at a rate proportional to
the difference between its current
temperature and the (constant)
surrounding temperature .
A cup of coffee sitting in a 65° room
cools from 200° to 180° in the first
minute. How hot will it be after 10
(total) minutes?
Assignment for Friday
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Read Section 3.8.
Do Exercises 1, 3, 5b, 9, 13, and 18a.
Hand-in #2 is due at 4:45 tomorrow
(Thursday).