Transcript Document

Test Corrections
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If you were below 80, you may correct your
mistakes (due Wed 3/24 at class time). You will
get back 1/3 of the points you lost (up to a max of
80) if your corrections are right.
Make corrections on a separate sheet – do not
make corrections on the original test. Hand in
both your corrections and the original test. Do not
staple together please.
Do these on your own or with help from me only!
(Honor code). Learn from your mistakes.
Clicker Question 1
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If a random variable X has a density function
given by f (x) = (3/64) x(16-x2) for x
between 0 and 4 and 0 otherwise, what is the
(approximate) probability that 2  X  4 ?
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A. 0.23
B. 0.54
C. 0.65
D. 0.72
E. 0.79
Clicker Question 2
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For the density function f (x) = x e-x for x  0

and 0 otherwise, what does
2 x
x
e dx
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represent?
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A. the probability of the random variable
B. the mean
C. the median
D. the mode
E. the standard deviation
Clicker Question 3
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For the density function f (x) = x e-x for x  0
and 0 otherwise, what does a represent in the
equation  x e  x dx  0.5
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a
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A. the probability of the random variable
B. the mean
C. the median
D. the mode
E. the standard deviation
Differential Equations (3/10/10)
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A differential equation is an equation which
contains derivatives within it.
More specifically, it is an equation which may
contain an independent variable x (or t)
and/or a dependent variable y (or some other
variable name), but definitely contains a
derivative y ' = dy/dx (or dy/dt).
It may also contain second derivatives y '' ,
etc.
Examples of DE’s
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Every anti-derivative (i.e., indefinite integral)
you have solved (or tried to solve) this
semester is a differential equation!
What is y if y ' = x2 – 3x + 5 ?
What is y if y ' = x / (x2 + 4)
What is y if dy/dt = e0.67t
Note that you also get a “constant of
integration” in the solution.
New types of examples
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The following is a DE of a different type since
it contains the dependent variable:
y ' = .08y
Say in words what this says!
Note that we don’t see the independent
variable at all – let’s call it t .
What is a solution to this equation? And how
can we find it?
The solutions to a DE
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A solution of a given differential equation is
a function y which makes the equation work.
Show that y = Ae0.08t is a solution to the DE
on the previous slide, where A is a constant.
Note that we are using the old tried and true
method for solving equations here called
“guess and check”.
Examples of guess and check for DE’s
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Show that y = 100 – A e –t satisfies the DE
y ' = 100 - y
Show that y = sin(2t) satisfies the DE
d2y / dt 2 = -4y
Show that y = x ln(x) – x satisfies the DE
y ' = ln(x)
Of course one hopes for better methods to
solve equations, but DE’s can be very hard.
Assignment for Friday
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Read over these slides (and try to solve the
problems on them), and read Section 9.1.
On page 571, do # 1 – 7 odd.