Synopsis:
You are with your friends in the Outback trying to figure out the population of Australia. You and your team have collected data over the past couple of years which show a steady increase of people, because they cannot resist what Australia has to offer including the adorable koalas and kangaroos. The information below shows the population measured in Australia, beginning in 2007 and ending in 2013. The table gives the population, y, at time t, in years.
Table: Population of Humans in Australia Over 7 Years
| t (years) |
1
|
2
|
3
|
4
|
5
|
6
|
7
|
| y=a(t) (population in millions) |
21.02
|
21.38
|
21.78
|
22.07
|
22.32
|
22.68
|
23.13
|
What is the instantaneous rate of change for the population of Australia in 2010 (Year 4)?
Average Rate of Change
(4, 22.07) and (3, 21.78)—> (21.78-22.07)/ (3-4)=0.29
(4, 22.07) and (5, 22.32)—> (22.32-22.07)/ (5-4)=0.25
(4, 22.07) and (6, 22.68) —> (22.68-22.07)/ (6-4)=0.305
To find the average rate of change, I have calculated the slopes between two points. Beginning at the point (4, 22.07), I have found the average rate of change for three secant lines. Since I found these slopes, I have also found the average rate of change for each line. After calculating the ARC of these three secant lines, I found that the results were getting closer and closer to the IRC. If you had information for smaller intervals, you could calculate a secant slope that was closer to the slope of the tangent.
Instantaneous Rate of Change
(4,22.07) and (2, 21.28)—> (21.28-22.07)/ (2-4)=0.395
The line that is tangent meets the ordered pair of (4,22.07). I found another set of numbers on the tangent line to use in my calculations to find the instantaneous rate of change in 2010. The instantaneous rate of change is 0.395. IRC, slope of a point, and slope of the tangent line are all different ways to talk about derivatives. This means that in 2010 (year 4), the population of Australia had an increase of 0.395 million people. Since the values of calculations from part d are getting smaller and smaller and closer to this answer, this shows that the slope of the secant lines are getting closer to the tangent line. 0.395 is the instantaneous rate of change because that is the answer you get when finding the slope of the tangent line.
Hey Christine! I really liked the topic you picked to explore, it was very unique! Nice job on the calculations and your explanation of how the results related to your specific example. The graphing on the computer really made things more clear. Nice Job!
ReplyDeleteYou choose a very different topic and I liked it. You organized your work well and was able to explain it in a clear way. Your graph was also very good and helped me see the example in a visual way. Good job
ReplyDeletenice job, christine! i like the back story you created to get your reader interested in your project! your graphics are great as well as your explanations. additionally, your calculations are all accurate. your tangent line is so precise!
ReplyDeleteall i would add in your explanation is to note that in 2010, the population is increasing at a rate of .395 million people per year.
prof little