Transcript Topics covered on Nov 23rd

Topics-NOV
Recall-Project Assumptions
Assumption 1.
The same 19 companies
will each bid on future similar leases
for the tracts(This assumption is
important foronly
the bidders
Nash concept)
Assumption 2.
The geologists employed by companies
equally expert (evidence- the Mean of errors of all historical leases is 0)
on average, they can estimate the correct
values of leases.
evidence
each signal for the value of an undeveloped tract is an
observation of a continuous random variable, Sv,
Mean of Sv  v ( actual value of lease )
Recall-Project Assumptions
Assumption 3. Except for their means, the distributions
of the Sv’s are all identical
(The shape /The Spread)-This allows us to treat all the 20
historical leases as one sample -> We use the sample to show
that mean of errors is 0 & find the standard deviation of
errors)
Assumption 4.
All of the companies act in their own best interests, have the
same profit margins, and have the same needs for business.
Thus, the fair value of a lease is the same for all 19
companies.
Integration. Integrals
Integration, Integrals
2. INTEGRALS
What would happen if we computed midpoint sums for a function
which might assume negative values in the interval [a, b]?
+
a
Where f(mi) < 0, the product
f(mi)Dx is also negative. Thus, the
midpoint sums
n
Sn ( f , [a, b]) 

(material continues)
b
 f ( mi )  D x
i 1
will approximate the “signed area”
of the region between the x-axis and
the graph of f, over [a, b]. This is the
algebraic sum of the area above the
axis, minus the area below the axis.

T
C
I
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
Integration, Integrals
the integral of f over [a, b] is
b
 f ( x ) dx
a
and it represents the algebraic sum of the signed areas of the regions
between the horizontal axis and the graph of f, over [a, b].
b
n
f ( mi )  D x
 f ( x ) dx  nlim
 
i 1
a
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Integration Applications
•
FundamentalTheorem
Theorem ofof
Calculus.
For many of the functions, f,
Fundamental
Calculus
x
which occur in business applications, the derivative of
-
 f (u) du, with
a
respect to x, is f(x). This holds for any number a and any x, such that the
closed interval between a and x is in the domain of f.
Example : applies to p.d.f.’s and c.d.f.’s
Recall from Math 115a

b
a
f X x  dx  FX b   FX a 
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Integration, Calculus
the inverse connection between integration and differentiation is
called the Fundamental Theorem of Calculus.
Fundamental Theorem of Calculus. For many of the functions, f,
x
which occur in business applications, the derivative of
 f (u) du, with
a
respect to x, is f(x). This holds for any number a and any x, such that the
closed interval between a and x is in the domain of f.
Example 7. Let f(u) = 2 for all values of u. If x  1, then
integral of f from 1 to x is the area of the region over the interval [1, x],
between the u-axis and the graph of f.
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Integration, Calculus
3
(1, 2)
2
(x, 2)
The region whose area
is represented by the integral is
rectangular, with height 2 and
width x  1. Hence, its area is
2(x  1) = 2x  2, and
x
 f (u) du
f (u )
1
2
1
x1
0
x
0
1
2
3
u
x 4
5
 f (u) du  2  x  2.
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In the section Properties and Applications of Differentiation, we saw
that the derivative of f(x) = mx + b is equal to m, for all values of x. Thus, the
x
derivative of
 f (u) du, with respect to x, is equal to 2. As predicted by the
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Fundamental Theorem of Calculus, this is also the value of f(x).
The next example uses the definition of a derivative as the limit of
difference quotients.
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