Blog Two

1.     Find a real world application

a.      Recorded data from the population growth of human over a period of time (time vs. population)

2.     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?)

a.      Narrative/Synopsis of application:  In this problem, the population is the dependent variable and time is the independent variable.  Thus, the population number is dependent on time.

b.     Question: What was the rate of change in 1970?

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

Graph the points using the data from your table of values (connect the dots). 

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.

a.      Change in population / change in time = (e)

                                               i.     4.4-3.6/1980-1970 = 757,876,182/10= 75,787,618.2

                                              ii.     5.3-3.6/1990-1970 = 1,629,644,051/20= 81,482,202.55

                                            iii.     6.1-3.6/2000-1970= 2,436,527,812/30= 81,217,593.73

b.     Notice about the ARC:  The ARC of the secant lines increases and then decreased a bit with the third calculation.  With the years 1990 and 2000 rate of change, you can see that the calculations are similar to one another around the 81 millions, thus allowing one to conclude that 81 million is the limit.

c.      Calculations mean/Represent in term of the experiment: The calculations represent the average rate of change of the population over the years.

2.     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)

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.

a.      Calculation of Slope of the Line:

                                               i.     Change in Y / Change in X -> 3.6-2.1/1970-1950 -> 1,591,172,616/20 -> 79,558,630

b.     What the calculations means mathematically in terms of the experiment/application:  The calculations are not exactly the same results that were calculated with the rate of change earlier, but that is okay. The calculations are somewhat similar and are reasonably close when considering the fact that the points were estimated and the tangent may not have been drawn perfectly. Overall, the calculations done earlier, which resulted in our limit being around 81 million, should be similar to the calculations here as instantaneous rate of change is the same as finding the slope at a point (limit of the difference quotient of the derivative). The end results are not exactly the same, but close enough. From this, we can see that the limit is towards 81 million. We are changing (in our case, increasing) the population roughly 81 million people a year.

2.     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). 

a.      The value from part g is the IRC due to several of reasons. First, as seen in part d, we noted that

                                               i.     4.4-3.6/1980-1970 = 757,876,182/10= 75,787,618.2

                                              ii.     5.3-3.6/1990-1970 = 1,629,644,051/20= 81,482,202.55

                                            iii.     6.1-3.6/2000-1970= 2,436,527,812/30= 81,217,593.73

From these calculations, we noted that the limit was close to 81 million because the calculations were not increasing that much in the 81 millions, thus we can conclude that is the limit.

Now in part g, we noted that the IRC was 79,558,630 we can see that the IRC is getting closer to the limit of 81,000,000. This shows that secant is getting closer and closer to the tangent line. We can conclude that our population is increasing by 81 million people a year.  Also, in the end it doesn’t matter what calculation we did as we can see that the numbers are coming closer to 81 million. Thus, we can determine that 81 million is our limit, especially since IRC, derivative, and slopes are all synonyms of one another.

End Notes:
3.     The data was collected from: http://www. geohive.com/earth/his_history3.aspx
4.     In the blog post during the rate of change calculations, I began with a "3.6" and other numbers in decimals, this was only to abbreviate the long population number. The actual population number was always used during calculations.