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(a) Compute a constant approximation to the data.
(b) Compute a linear approximation.
(c) Compute a quadratic approximation.
(d) Graph the three approximations together with the data points.
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Exercise 5.2.
On the Internet, you can find information about prices of used cars. In
the table below, we have listed the prices of used BMWs of the 3-series in Norway.
We give the average price for a car that is 1, 2, 3, 4 and 5 years old.
Years t
i
1 2 3 4 5
Price y
i
55.2 44.9 37.9 35.3 30.1
The prices are given in thousands of euros.
(a) Build a linear model of the data.
(b) Graph the data and the linear model. Instead of a standard linear model, we
assume that the price can be modeled as follows,
p.t/
D
˛e
ˇt
;
(5.136)
where we expect ˇ to be negative. By taking the logarithm of both side of
(
5.136
), we get
ln .p.t //
D
ln.˛/
C
ˇt:
(5.137)
This shows that we can compute ˛ and ˇ in (
5.136
) by constructing a linear
model of the data given by .t
i
; ln.y
i
//.
(c) Compute the entries of the following table:
Years t
i
1 2 3 4 5
z
i
D
ln.y
i
/
(d) Compute a linear model
q.t/
D
C
ˇt
ofthedatain(c).
(e) Since ln.˛/
D
, compute ˛
D
e
and define
p.t/
D
˛e
ˇt
:
(f) Plot the price-time data, the linear model and the exponential model. Which
model seems to fit the data better?
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