Alessandro Ceccarelli Blog #2

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)

Watching as a tub drains after the plug is pulled

b.     Write a narrative or synopsis explaining your application/experiment and include a question. (for example, what is the velocity of the snowball at exactly 2 seconds? Or how can I find the velocity of the baseball at exactly 3 seconds?)

As time goes by, more water drains from the tub and allows us to track the rate at how fast it drains. How fast is it draining at exactly 90 seconds?

c.     Create a table of values for the data that you have recorded from your application/experiment.

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.

­(44-100)/(100-90)= -5.6

(11-100)/(110-90)= -4.45

(0-100)/(120-90)= -3.33

The numbers are slowly decreasing, which means that as the water becomes less, pressure from the water decreases as well, causing there to be less flow.

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)

See the above graph in the picture to see the tangent line

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.

(139-100)/(85-90)= -7.8

At 90 second, the volume of water is decreasing at a rate of approx. 7.8L/sec.

h.     Explain in detail how you know that the value from part g is the IRC. (i.e. since the values of calculations from part d are getting smaller and smaller, this shows that the slope of the secant is getting closer and closer to the tangent line … or some explanation similar to this). BE DETAILED!!!

Since the values in part “d” are getting smaller, and the calculation in part “e” are also shrinking, it shows that the values of the secant are getting closer and closer to the tangent line.