Part
a:
a. Find a real world application OR
design your own experimental application relating to rates of change. (in the
blog folder you will find plenty of examples to get you started)
Recorded data from how much the volume of a balloon is
affected when is being filled with helium over a period of time (time vs.
volume.) Rate of change of the balloon's volume
b.
Write a narrative or synopsis
explaining your application/experiment and include a question.
In
my experiment, I want to calculate the velocity at which a balloon is filled with
air and how the volume is affected. I wonder if I can find the volume of the balloon after 4, 5 and 6
seconds have passed. Also, I want to calculate the velocity in which the volume increases.
c. Create a table of values for the
data that you have recorded from your application/experiment.
| t (sec) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
| Vol. (m3) | 0 | 1 | 8 | 27 | 64 | 125 | 216 | 343 | 512 | 729 | 1000 | 1331 | 1728 |
d. Graph the points using the data from your table of values (connect the dots).
e. Calculate the slope (ARC) of at
least three secant lines originating from the same point on your graph to
three different points on your graph (i.e. maybe you want to know what happens
exactly at x = 20, so your points might be (20, 62), (20, 56), (20, 50)). Explain what you notice about the ARC
of these secant lines and what the calculations mean/represent in terms of your
experiment/application.
X=1 --> (1,1)
Points:
(6, 216) => [(216-1)/(6-1)]= 43
(5,
125) => [(125-1)/(5-1)]= 31
(4, 64) => [(64-1)/(4-1)]= 21
(3, 27) => [(27-1)/(3-1)]= 13
The
slope gets smaller as it approaches the original point (1, 1). In this case,
the lines on the graphs get flatter and flatter, so the slope decreases. As the time passes, the volume
of the balloon gets greater.
f. Sketch
an approximation of a tangent line that passes though the same point (P) from
part e to which you connected your secant lines (i.e. you would draw a tangent line
through the point 20, since that is the same point that you used to calculate
your three different secant lines)
g. Choose
a second point (Q) on the tangent line, and calculate the slope of the line
(PQ). This calculation will be the instantaneous rate of change ((IRC
or derivative at a point)…be sure to identify the units correctly). Explain what this calculation means
mathematically and in terms of your experiment/application.
The slope of the points (1, 1)
and (4, 10) is [(10-1)/(4-1)] = 3 m3/sec
h. Explain
in detail how you know that the value from part g is the IRC.
Since the values of the calculations from part e are getting smaller and smaller, this means that the slope of the secant is getting closer and closer to the tangent line, which is the limit and has an instantaneous rate of change of 3 m3/sec.
Interesting, well done.
ReplyDeleteI like how you drew out the secant lines on the graph. It was helpful!
ReplyDeleteWell done, very interesting and clear the way you exposed your answers .
ReplyDeletepaula,
ReplyDeletegood topic and well organized. initially, your question should INCLUDE the point in time that you want to investigate. as i read on in your post, it looks as though you wanted to investigate what happens at exactly t = 1 second. so, your question should have been something like "how fast is the volume of the balloon increasing at exactly one second?"
your graphs are perfect, though, and so is your table. i like how you identified the point Q on your tangent line so that it was easy to see what points you would be getting your slope value from.
the only other thing that was missing was the units for your secant line calculations, although i like that you did four calculations instead of three to give yourself extra data.
all in all, good job. =]
professor little