The Trout Pond Revisited

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Kristopher Bell
5 years ago
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1 The Trout Pond Revisited A. MATERIALS NEEDED Worksheet, calculator, ruler B. OBJECTIVE The student will use the knowledge already gained concerning the calculations of slopes of lines to find average and instantaneous rates of change of the fish population. C. RATIONALE The derivative of a function at a point is defined to be the limit of the slopes of the secant f ( x) f ( a) lines. This idea is frequently expressed with the formula f ( a) = lim, where x a x a the point (a, f (a)) is the fixed point on the curve. In this activity, the students will get practice in determining the average rate of change of a function over small intervals. These average rates of change are approximations of the derivative values of the population function at those points. The students will also calculate average rates of change using a fixed point on the curve. The limiting value of these average velocities as the interval size diminishes is the instantaneous rate of change which is equivalent to the slope of the tangent line at that point and also is the value of the derivative of the function at that point. D. Variations of this activity could include the following: - The students can produce a graph of the data and work with their own graph. - Different numbers may be used in the initial problem to change the character of the function. For instance, change 1000 to 500 to see what this does to the function. Or, change 20% to some other value. - This entire activity does not have to be completed at one time. The problem in this activity is taken from Trout Population Exploration from the Illuminations web site (

2 The Trout Pond Revisited Problem: Each spring, a trout pond is restocked with fish. That is, the population decreases each year due to natural causes, but at the end of each year, more fish are added. Here is the information. - There are currently 3000 trout in the pond. - Due to fishing, natural death, and other causes, the population decreases by 20% each year, regardless of restocking. - At the end of each year, 1000 trout are added to the pond. 1. The population change information given above was used to calculate the number of trout in the pond at the end of each year. Each year s population was rounded to the nearest whole number. Year Trout Population Year Trout Population

3 2. The derivative of a function at a point is defined to be the limit of the slopes of the secant f ( x) f ( a) lines and is expressed with the formula f ( a) = lim. The following two x a x a tables will help to organize the data in order to determine the limit of the slopes of the secant lines. Notice that the ninth year is the fixed point in these calculations, and as a result, the limit will be determined as t approaches 9. Interval [1, 9] [3, 9] [5, 9] [7, 9] [8, 9] Change in Time - t Change in Population - P Average Rate of Change (Slope of Secant Line) a) As the change in time is decreasing, what is the limiting value of the slope of the secant lines based on the information in the above table? Interval Change in Time - t [9, 17] [9, 15] [9, 13] [9, 11] [9, 10] Change in Population - P Average Rate of Change b) As the change in time is decreasing, what is the limiting value of the slope of the secant lines based on the information in the above table? c) Based on the information in the two tables above, make a reasonable conjecture about the value of the derivative of the trout population function when t = 9.

4 3. The derivative of a function at a point can be interpreted as the slope of the line tangent to the graph at that point. A graph of the population data is shown on the next page. Use a straight edge to draw tangent lines to the graph of the population function at the years listed in the table below. After drawing each line, approximate the slope of the line and record the information in the table. Year Slope of Tangent Line a) Compare the slopes of the tangent lines with the average rates of change computed for the same years. Year Average Rate of Change Slope of Tangent Line b) Write a couple of sentences to try to explain the correlation in the values.

5 Trout Population population years

4.2 Rolle's Theorem and Mean Value Theorem

4.2 Rolle's Theorem and Mean Value Theorem Rolle's Theorem: Let f be continuous on the closed interval [a,b] and differentiable on the open interval (a,b). If f (a) = f (b), then there is at least one