Determining Change: Derivatives

Project 4.1: Fertility Rates



The Statistical Abstract of the United States (1995 Edition) reports the following fertility rates for whites in the United States.

TABLE 4.51


Fertility rate




























  1. Find a table of fertility rate data in a current edition of the Statistical Abstract, and summarize in your own words the meaning of "fertility rate." Assign units to the fertility rate column in Table 4.51.
  2. Find a piecewise model that fits the given fertility rate data. Write the derivative function for this model. Construct a table of fertility rates and the rate of change of fertility rates for all years from 1970 through 1994. Discuss the points at which the derivative of your model does not exist (and why it does not exist), and explain how you estimated the rates of change at these points.
  3. Using the data in a current edition of the Statistical Abstract, complete the same analysis as in Task 2 for the fertility rates for blacks in the United States.
  4. Add 1996 and any other recent data for the fertility rate of whites. (Note: Recent editions of the Statistical Abstract occasionally update older data points. Therefore, you should check the data for pre-1996 as well and change any updated values so that your data agree with the most recent Statistical Abstract.) Find a piecewise model for the updated data. Use this new model to calculate the rates of change that occurred in the years since 1995.


Write a report discussing your findings and their demographical impact. Include your mathematical computations as an appendix.


Project 4.2: Superhighway



The European Communities have decided to build a new superhighway that will run from Berlin through Paris and Madrid and end in Lisbon. This superhighway, like some others in Europe, will have no posted maximum speed so that motorists may drive as fast as they wish. There will be three toll stations installed, one at each border. Because these stations will be so far apart and motorists may not anticipate their need to stop, there has been widespread concern about the possibility of high speed collisions at these stations. There have been editorials protesting the installation of the toll stations, and a private-interest group has lobbied to delay the building of the new superhighway. In response to the concern for safety, the Committee on Transportation has determined that flashing warning lights should be installed at an appropriate distance before each toll station. Your firm has been contracted to study known stopping distances and to develop a model for predicting where the warning lights should be installed.


  1. Getting Started Find data that give stopping distances as a function of speed and cite the source for the data. (Hint: Drivers' handbooks and Department of Transportation documents are possible sources of data on stopping distance.) Present the data in a table and as a graph. Find a model to fit the data. Justify your choice. Before using your model to extrapolate, consult someone who could be considered an authority to determine whether the model holds outside of the data range. Consult a reliable source to determine probable speeds driven on such a highway. On the basis of your model, make a recommendation about where the warning lights should be placed. Justify your recommendation. Keep in mind that the Committee on Transportation does not wish to post any speed-limit signs. Suggest what other precautions could be taken to avoid accidents at the toll stations.
  2. Proceeding Find rates of change of your model for at least three speeds, one of which should be the speed that you believe to be most likely. Interpret these rates of change in this context. Would underestimating the most likely speed have a serious adverse affect? Support your answer.


(Bear in mind that you are reporting to Europeans who wish to see all results in the metric system. However, because you work for an American-based company, you must also have the English equivalent.)

  1. Prepare a written report of your study for the Committee on Transportation.
  2. Prepare a press release for the Committee on Transportation to use when it announces the implementation of your safety precautions. The press release should be succinct and should answer the questions Who, What, When, Where, and Why.
  3. (Optional) Prepare a brief (15-minute) presentation of your results. You will be presenting it to members of the Committee on Transportation.

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