Derivative Word Problem - US Population

In summary, a Derivative Word Problem involves using the derivative to solve real-world problems, such as analyzing the population growth of the United States. To solve such a problem, data on the current population and growth rate over a specific time period is needed. The derivative is used to find the rate of change of the population and can be applied to predict future growth and inform policy decisions. However, there are limitations to this approach, such as assuming a constant rate of growth and not accounting for other factors like immigration and emigration.
  • #1
bobraymund
27
0

Homework Statement



(a) Use a graphing calculator or computer to fit an exponential function to the data. Graph the data points and the exponential model. How good is the fit?

(b) Estimate the rates of population growth in 1800 and 1850 by averaging slopes of secant lines.

(c) Use the exponential model in part (a) to estimate the rates of growth in 1800 and 1850. Compare these estimates with the ones in part (b).

(d) Use the exponential model to predict the population in 1870. Compare with the actual population of 38,558,000. Can you explain the discrepancy?


Homework Equations



Year Population
1790 3,929,000
1800 5,308,000
1810 7,240,000
1820 9,639,000
1830 12,861,000
1840 17,063,000
1850 23,192,000
1860 31,443,000


The Attempt at a Solution



I understand part B, you would just do the population in 1800 - population in 1790 all divded by 1800 - 1790... etc. etc..

However, part A and consequently parts C and D are confusing me...

Any help would be GREATLY appreciated.

Thanks,
Bob
 
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  • #2
You are supposed to use a computer or calculator to do a best fit exponential to the data. You could do this by hand, but it would be tedious and they don't want that. Once you have that fit then you have a model of uniform growth rate over that time. The actual growth rate over any single decade may differ from that average. In particular for d) you are going to find the population growth rate is considerably smaller than you might expect for the period 1860-1870. Did something happen in the US in the decade that might explain it? It's all just about comparing an average growth model with individual data points and figuring out why some of them might not fit the average behavior.
 
  • #3
Ok, but how exactly would I fix an exponential to the data?

Our teacher gave us the solution as this:

solution.jpg


Perhaps you could explain exactly what they did?

How did they get those numbers in part a?
 
  • #4
They typed the population numbers into a calculator or a computer program. They expect you to be able to do the same. Don't you have one or the other?
 
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  • #5
Yeah, ha ha, but excuse my "noobness". This is my first calculus course and am still trying to figure out my TI-89.

How would I graph the population numbers on my calculator? Would I use the table feature?

Thank you for your patience,
Bob
 
  • #6
bobraymund said:
Yeah, ha ha, but excuse my "noobness". This is my first calculus course and am still trying to figure out my TI-89.

How would I graph the population numbers on my calculator? Would I use the table feature?

Thank you for your patience,
Bob

Sorry, I can't help you there. I don't own a calculator that does much more than multiply or divide. I did google and find this. http://www.prenhall.com/esm/app/cal...cuments/TI-89/desc_pages/ti89techskills2.html Does that help? If not I'm sure you can find lots of other stuff on the web.
 
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  • #7
Dick said:
Sorry, I can't help you there. I don't own a calculator that does much more than multiply or divide. I did google and find this. http://www.prenhall.com/esm/app/cal...cuments/TI-89/desc_pages/ti89techskills2.html Does that help? If not I'm sure you can find lots of other stuff on the web.

I tried following the instructions on site to plot the points on my TI89 but to no avail.

Any further help is kindly welcome.

-Bob
 
  • #8
I'll post a call for help from other homework helpers who might know this stuff.
 
  • #9
What instructions did you follow? Those for a linear regression? You are supposed to do an exponential fit. The section of the web page Dick found on nonlinear regression talks about exactly this kind of problem.
 
  • #10
D H said:
What instructions did you follow? Those for a linear regression? You are supposed to do an exponential fit. The section of the web page Dick found on nonlinear regression talks about exactly this kind of problem.

Thank you! I've finally figured it out thanks to the instructions that Dick linked to on that site. Thank you so much for pointing that out.

Ha, I am super happy right now you all have no idea. I was trying to figure this out forever, thank you!

-Bob
 
  • #11
For future reference, the manual for the TI-89 is online:

http://www.education.ti.com/downloads/guidebooks/graphing/89ti/TI89TitaniumGuidebook_Part2_EN.pdf [Broken]

It would be worthwhile for you to download the manual to your computer.

The ExpReg function (for fitting an exponential to data) is found on page 819.
 
Last edited by a moderator:

1. What is a Derivative Word Problem?

A Derivative Word Problem involves using the derivative, which is the rate of change of a function, to solve a real-world problem. In this case, we are using it to analyze the population growth of the United States.

2. What data is needed to solve a Derivative Word Problem on US Population?

To solve a Derivative Word Problem on US Population, we would need data on the current population of the US, as well as data on the population growth rate over a specific time period.

3. How is the derivative used to analyze the US population growth?

The derivative is used to find the rate of change of the US population over a specific time period. By taking the derivative of a function that represents the population growth, we can determine how fast the population is increasing or decreasing at any given point in time.

4. What are some potential applications of using a Derivative Word Problem on US Population?

A Derivative Word Problem on US Population can be used to predict future population growth and inform policy decisions related to infrastructure, healthcare, and education. It can also be used to determine the optimal time to invest in certain industries or regions based on population trends.

5. Are there any limitations to using a Derivative Word Problem on US Population?

As with any mathematical model, there are limitations to using a Derivative Word Problem on US Population. It assumes a constant rate of growth, which may not always be accurate. It also does not take into account factors such as immigration and emigration, which can significantly impact population growth.

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