Transcript Document

Clicker Question 1
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The position of an object (in feet) is
given by s (t ) = 4 + ln(t 2) where t is in
seconds. What is the object’s velocity at
time t = 3 sec?
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A. 4 + ln(9) feet/sec
B. ln(9) feet/sec
C. 2/3 feet/sec
D. 1/3 feet/sec
E. 1/9 feet/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
Exponential Growth and Decay
(2/9/11)
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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.
 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?
 What is the rate of growth (in bacteria/day)
at 10 days?
Example of Decay
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My savings account started with $1000
bit seems to be dwindling at a
continuous rate of 5% per year.
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How much is in the account after 5 years?
At what rate is it dwindling (in $/year) after
5 years?
Assignment for Friday
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Read Section 3.8.
Do Exercises 1, 3, and 5b,.
Hand-in #1 will be given out Friday and
will be due next Tuesday.
Test #1 is Monday, Feb 21.