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This 5 page Class Notes was uploaded by Corina Johnson on Monday February 22, 2016. The Class Notes belongs to MATH 1311 at University of Houston taught by Yifan Wang in Spring 2016. Since its upload, it has received 22 views. For similar materials see Elementary Mathematical Modeling in Mathematics (M) at University of Houston.
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Date Created: 02/22/16
Math 1311 Section 1.2 Functions Given by Tables Sometimes we work with functions for which we do not know a formula. In this case, many times we use a table of values to show the value of the function for specific values of the independent variables. Why tables? Advantages Tables are how we can organize experimental data for which the relationship is not known in formula form. Tables provide an easy to see relationship between quantities, since evaluating a formula is not always easy. It is often easier to spot trends in tabular data. Disadvantages There are only finitely many values of the function known; we do not necessarily know what happens in the gaps. To use the table to guess what will happen in the future, you must assume that there is a pattern and that it will continue. Average Rate of Change is a ratio of the change in function values to the change in values of the independent variable. Average rate of change of a function ???? over the interva1 ???? 2????is defined to be ???? ????)2 − ????(???? 1 ???????????????????????????? ???????????????? ???????? ???????????????????????? = ???? − ???? 2 1 And AROC has units: the units of ???? divided by the units of the ???? variable. This is a VERY IMPORTANT concept throughout this course. Limiting Values Information about physical situations can sometimes show that limiting values are to be expected for functions that model those physical situations. The limiting value may be estimated from a trend established by the data. There are 5 basic skills we need to learn from this section. They are 1. Reading a Table of Values 2. Averaging between two data points 3. Finding the Average Rate of Change for an Interval 4. Using the Average Rate of Change to Estimate the Values between data points 5. Recognizing a Limiting Value if it exists We are going to practice these skills by looking at the tables for several physical situations Example 1: Women Employed Outside the Home Year 1943 1946 1970 1980 1990 2000 ????, # in 18.7 16.8 31.5 45.3 56.8 66.3 millions a. Explain what ????(1946) means and give its value. b. Express the number of women employed outside the home in 1985 in function notation and find its value. c. Find the average rate of change for the interval 1980 to 1990. d. Use the average rate of change from part c. to estimate the number of women working outside the home in 1983. e. Find the average rate of change for this function from 1943 to 1946. f. Use the average rate of change from part e. to estimate the number of women working outside the home in 1945. g. The actual value ????(1945) is 19.0. Why is our average rate of change estimate so far off? h. Does the function appear to be increasing, decreasing or tending toward single value as the time goes on? Example 2: The following table shows the number of deer in a wildlife preserve, ????, as a function of the number of years since the herd was introduced. Years, ???? ???? 0 12 5 154 10 379 15 410 20 412 25 412 a. What does ????(15) mean and find its value? b. Find the average rate of change of the deer population in the first five years on the reserve. c. Use the AROC in part b. to estimate the deer population 2 years after the herd was introduced. d. Make a chart that shows the average rate of change for each time interval in the table above. Time interval AROC, deer per year 0 to 5 5 to 10 10 to 15 15 to 20 20 to 25 e. Is there a limiting value for ????(????)? What is it? f. What do you notice about the average rate of change as the value of ???? gets close to the limiting value? Example 3: Gross Domestic Product Year = t 1996 2000 2002 G = GDP, 7.81 9.82 10.45 in millions of dollars a. ????(1996) b. Estimate the gross domestic product in 1998 and express your answer in function notation. c. Find the average rate of change of ???? from 2000 to 2002. d. Use the AROC from part c. to estimate the gross domestic product in 2010.
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