A rubber band exerts forces of 1.0, 1.7, 2.3, 3.1 and 4.0 Newtons at
lengths of 8.5, 9.0, 9.5, 10.0 and 10.5 cm. Write the forces as a sequence and
find the first and second differences of the sequence. What does the first
difference tell you about the graph of the original sequence? What does the
second difference tell you? Sketech the graph of the sequence. Find also the
sequence whose first number is 0 and whose differences are the sequence of
forces.
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RESPONSE -->
The first differences of the force sequence are as follows:
0.7 0.6 0.8 0.9
The second differences of this sequence are as follows:
0.1 -0.2 -0.1
(to find these differences I subtracted the one number from the number in front
of it then continued the cylcle with the numbers I found from the 1st difference
to find the second)
The first difference tells me that the graph of these numbers would be
increasing, have a positive slope, because the numbers are all positive.
The second difference tells me that the graph of these numbers would be
increasing at a decreasing and increasing rate, because the differences are
negative and positive.
The sequence of lengths would start with zero although not have the the sequence
of force as the difference. This sequence would be:
0 1 2.7 5 8.1 12.1
(I found this by starting with the number zero and adding the numbers from the
sequence of force to find the next number.)
Now sketch a trapezoidal graph of the force vs. length information and
label the slopes and areas. How does the shape of the graph compare to the shape
of the graph of the sequence? How do the slopes compare with the sequence of
differences? How do the areas compare with the last sequence you found?
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The shape of the graph of force vs. length appears to look relativly close to
the shape of the graph of the sequence of force. Since slope is rise over run,
the differences would be the rise from one number of the force to the next. The
slope also decreases and increases as the sequence of differences of the
sequence of force did.The areas are all positive numbers like all the numbers in
my last sequence, but the areas are not increasing and decreasing, where as
before they were.
The slopes are :
1.4 1.2 1.6 1.8 (going from 8.5 to 10.5)
The areas are as follows:
0.675 1 1.35 1.75 (going from 8.5 to 10.5)