The High School Math Project Focus on Algebra. Bottles and Divers. (Rates of Change)

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1 The High School Math Project Focus on Algebra Bottles and Divers (Rates of Change) Objective Students calculate the rate of change between two points on a curve by determining the slope of the line joining those two points. They then extend this method to estimate the instantaneous rate of change at a given point by taking two points very close to and on either side of the given point. Overview of the Lesson The concept of slope as the rate of change is an important one in the study of algebra. This lesson has an introductory activity that uses bottles of various shapes to help students understand the concept of rate of change. The lesson then develops the concept of the average rate of change between two points on a curve. Finally, it examines instantaneous rate of change: rate of change function is used to estimate the rate of change at a particular point. This function is used along with the calculator to make computing the rate of change fast and easy for students. The development of this function helps students understand what is meant by instantaneous rate of change and lays a firm foundation or the study of the derivative in calculus. Materials graphing calculator overhead unit overhead projector class data chart on newsprint or blackboard a collection of various bottles, beakers, and flasks For each group of four: markers newsprint one container ruler Page 1

2 measuring device such as a small graduated cylinder screening device such as poster board or a box Graphically Speaking acitivty sheet Introduction: The Bungee Jumper discussion questions The Diver Problem activity sheets The Bungee Jumper activity sheets The Fish Population activity sheet Procedure 1. Graphically Speaking: The activity allows students to get hands-on experience dealing with the abstract concept of rate of change. Each group of students is given one container. They use a graduated cylinder to measure amounts of water, then add the water to the container and measure the height of the water. After this data is collected, students make a scatterplot of their data. If possible, each group should conceal their container from the other groups as they are collecting data and creating the scatter plot. To help them conceal the containers as the students do their measuring, you may wish to make screens of poster board folded in half and taped to a desk. A large box with two sides cut away also works well. The containers should represent a variety of shapes. Your science department may have a selection of beakers, flasks, long stem funnels, and graduated cylinders that could be used, or students could bring in a variety of bottles or containers. In assigning the containers, keep in mind that containers' shapes affect the difficulty level of the activity: beakers, graduated cylinders, and any other type of prism are easier; an Erlenmeyer flask is moderately difficult; and the Florence flask and the long stem funnel are generally more difficult, or at least more time consuming. As the groups complete their measurements and scatterplots, have them tape these to the blackboard or a wall. When all groups are finished, lead the entire class through a discussion of the graphs that are on display. For each graph you might ask questions similar to the following. Have each group discuss them before you ask for responses directed to the entire class. What do you think the shape of the container is like? Is the graph similar to any other graph that is displayed? How do you think the shape differs from the shape of the container whose graph is similar to it? Could you write an algebraic rule for the graph? After the groups have discussed these questions, have a representative from one of the groups go to the board or the overhead and sketch the shape his or her group believes the graph represents. Let other groups either confirm the HSMP Bottles and Divers Lesson Guide Page 2

3 sketch or challenge it by giving their alternative and justifications for a different sketch. As the class goes through this discussion, remind the group whose shape is being discussed that they are not to give any hints. After the sketch has been made and discussed, the group who did the plot should show their container and either affirm the sketch or explain why the sketch is not correct. 2. The Bungee Jumper: Class discussion of the "Think About This Situation" section of Introduction: The Bungee Jumper serves as the introduction to the lesson on rates of change for functions with algebraic rules. This activity is designed to engage the students and to have them talk about some ways in which they could use an algebraic rule to help determine a rate of change. Students could mention determining an average rate of change or using the algebraic rule to create a table of values or a graph. 3. The Diver Problem: Have students work in groups to complete the problem. They might need to be reminded to consider the symbolic representation, the graph, and the table of values as they explore this problem. After all of the groups have had time to work and discuss the problem, have them present their findings using the blackboard or newsprint. Discuss each part of the problem, moving from small group discussions to whole class discussions. Problem number 4 should be thoroughly discussed because it allows students to generalize the procedure that the previous problems had been leading them to discover. Students might need more guidance with this generalization. As students work with the rate-of-change estimate function with their calculator, it will be helpful for them to understand that they can enter that Y2(x + 0.1) Y2(x 0.1) equation in Y1 as Y1 =, where Y2 is the function for 0.2 which they are exploring rates of change. As they explore different functions, they simply have to enter the new function in Y2. Assessment As in many of the other video lessons in which the students work in groups, one of the most important means of assessing students is by listening to their comments as they work with their classmates. In this lesson, students are grouped in threes to keep the students from pairing off as discussions take place. This aides the teacher in listening to a group work, but it also means that there will be more groups to circulate among. (Many instructional decisions, like this one, involve trade-offs.) The teacher mentions using a variety of means to assess the students, including class discussions, projects, tests, and quizzes. An important part of the written tests and quizzes involves a more open-ended approach in which the teacher asks the students to not only give an answer, but also to explain their reasoning. This HSMP Bottles and Divers Lesson Guide Page 3

4 emphasis on problem-solving and reasoning as well as using multiple means of assessing students are in line with the NCTM standards, and they are an important part of the many changes that are occurring in the algebra classroom. Extensions & Connections The following is a nice activity for groups that finish Graphically Speaking data collection and plotting before others. Imagine filling each of the containers shown below by adding water in 10 equal-size amounts. Sketch the predicted (volume, height) graphs for each pair of containers. Plot each pair of graphs on a separate set of coordinate axes. (a) (b) (c) (d) (e) The Bungee Jumper and The Fish Population are included as additional activities that allow the students to use the difference equation in their calculators to quickly determine rates of change in addition to exploring some different function models. Mathematically Speaking The NCTM Standards give direction on the content of high school mathematics, indicating that the underpinnings of calculus be included. The standards recommend that maximum and minimum points of a graph, the limiting process, the area under a curve, the rate of change, and the slope of a tangent line be HSMP Bottles and Divers Lesson Guide Page 4

5 examined from both a graphical and a numerical perspective. This lesson is an informal exploration of rates of change and the derivative. The derivative and related developments represent one of the most important advances in the history of mathematics. The seventeenth century discoveries of analytical geometry by Descartes and infinitesimal calculus by Newton and Leibnitz are the beginning of modern mathematics. The concepts of limits and the derivative are key ideas in calculus. Some students do not have the opportunity to explore the concepts involved with the derivative in an informal way before they begin the formal study of calculus. As a result, these students may rely on memorizing formulas and rules rather than clearly understanding the concepts involved. Explorations such as those in this lesson allow students to investigate and explore the important mathematics without the formality that they will encounter later. These explorations and the applications on which the investigations are based help give students a solid mathematical foundation. y In many calculus texts, the derivative of the function y = f(x) is defined as lim x 0 x f (x which can also be written as lim 1 + x) f(x 1 ). How does this differ from x 0 x lim x 0 lim x 0 f (x 1 ) f (x 1 x) or x f (x 1 + x) f(x 1 x)? 2 x In the video lesson, the students calculate an approximation for the derivative because they are actually using a particular value for x. Help your students realize that there are several ways that an approximation may be calculated. They can determine the slope of the line through points on either side of the given point, through the point and a point slightly above the given point, or through the point and a point slightly below the given point. Also help students to understand that they can get a closer approximation to the instantaneous rate of change by using smaller values for x. As students as they calculate an approximation for the derivative, help them realize that the more they zoom in to a particular area of a graph, the more that particular part of the graph will approximate a straight line. Have the students use the zoom feature of their calculator to examine the graphical implications of calculating the rate of change over a very small period as an approximation of the derivative or the instantaneous rate of change at a particular point. HSMP Bottles and Divers Lesson Guide Page 5

6 Tips From Ellen Cues for Asking Good Questions A key ingredient in reform mathematics is enhancing discourse in the classroom, and effective questioning can be a key to effective discussion. Dimensions of Learning is an approach that has taken the best of what we know about the relationships between how we think and how we learn and created a framework to help teachers plan and deliver instruction. Dimension 3 relates to thinking which extends and refines knowledge, and offers a list of different types of questions which promote different kinds of thinking and learning. The following chart is presented as a resource. Many teachers create a cue card or chart for themselves as a handy reference in planning or teaching. You might create a goal for yourself of ensuring that you use at least four types of questioning in each lesson. Comparison Classification Induction Deduction Identifying and articulating similarities and differences between things. How are these things alike? different? Grouping things into definable categories on the basis of their attributes. Into what groups could you organize these things? What are the rules for membership? Inferring unknown generalizations or principles from observation or analysis. Based on the observations we have listed, what conclusions might you draw? Inferring unstated consequences and conditions from given principles and generalizations. If we accept these generalizations as true, what conclusions necessarily follow? Error Analysis Identifying and articulating errors in your own thinking or in that of others. How is the reasoning in this argument misleading? Constructing Support Building a system of support or proof for an assertion. What facts would support this claim? HSMP Bottles and Divers Lesson Guide Page 6

7 Abstracting Analyzing Perspective Identifying and articulating the underlying theme or general pattern of information. What pattern do you see? PBS MATHLINE Identifying and articulating your personal perspectives on issues as well as others' perspectives. How might someone else see this, and what reasoning might they use to support their position? Resources Coxford, Art, James Fey, Christian Hirsch, and Harold Schoen, Contemporary Mathematics in Context, Course 4. Chicago, IL: Everyday Learning Corporation, Everyday Learning Corporation, Two Prudential Plaza, Suite 1175, Chicago, IL 60601, Internet location: CyberCalc, an interactive learning environment for calculus, contains three chapters Review of Topics Needed for Calculus, Limits and Continuity, and The Derivative. Internet location: This location is a source of information on calculus reform issues, projects, conferences, workshops, and a bibliography. Internet location: This site contains bibliographies of many mathematicians of the seventeenth and eighteenth centuries, including Newton and Leibnitz. It also has an account of the controversy between Newton and Leibnitz over the invention of calculus. Internet location: Everyday Learning Corporation publishes Contemporary Mathematics in Context. You can request catalogs or send to the company from this site. Internet location: Information about the Core Plus Mathematics Project is available at this site. Sample lessons are available as well as an overview of the project. HSMP Bottles and Divers Lesson Guide Page 7

8 Ideas for Online Discussion (Some ideas may apply to more than one standard of the NCTM Professional Standards for Teaching Mathematics.) Standard 1: Worthwhile Mathematical Tasks 1. The NCTM Standards advocate to providing high school students with the underpinnings of calculus. How does the content of this lesson address this standard? How do the activities of the lesson help students understand the mathematical concepts? Standard 2: The Teacher s Role in Discourse 2. Discourse refers to ways of representing, thinking, talking, and agreeing and disagreeing. Teachers send messages about types of knowledge and thinking that are valued by the ways that they handle discourse. What are some of the ways that discourse is handled by the teacher is this lesson? Standard 3: Students Role in Discourse 3. The students in this lesson are seniors in high school. They have been in a mathematics program that has encouraged group work and active participation. What evidence do you see that demonstrates their ability to work in groups and discuss meaningful mathematics? What do you look for as you watch and assess the group work of your own students? Standard 4: Tools for Enhancing Discourse 4. The NCTM Standards encourage teachers to use a variety of means for promoting discussion and reasoning about mathematics. The use of tools such as calculators and computers is encouraged in addition to the use of more conventional mathematical symbols and other means such as drawings, diagrams, invented symbols, and analogies. What tools helped to promote mathematical thinking and the discussion of mathematical concepts in the video lesson? Standard 5: Learning Environment 5. One of the key components of the learning environment is the idea that it supports serious mathematical thinking. Teachers and students have a genuine respect for ideas and reasoning, and students have time to puzzle and think. A positive learning environment helps students be successful mathematical thinkers. What are some of the important ways of creating such a learning environment? HSMP Bottles and Divers Lesson Guide Page 8

9 Standard 6: Analysis of Teaching and Learning PBS MATHLINE 6. Part of the reason for analyzing teaching and learning is to ensure that every student is mastering important mathematics and developing a positive attitude toward mathematics. How do you assess the attitudes of students? What suggestions could you give to help other teachers assess the attitudes their students have about mathematics? HSMP Bottles and Divers Lesson Guide Page 9

10 Graphically Speaking On a typical day, you may fill and empty several different kinds of bottles, jars, and cans. In the process, you might have to look at the height of the contents in those containers and estimate their volumes. With typical cylindrical jars or cans, that estimation is relatively easy. But what about containers like those shown below? To test your estimation ability, try the following experiment with your classmates. Collect an assortment of bottles and jars of different shapes, but roughly the same volume. Get enough containers so there is one for each group, with a few left over. Have your teacher pick one container for each student group so that each group knows only its own shaped container's shape. Within your group, estimate the rate at which the water will rise in your container as it is added in equal amounts. Sketch a graph of (volume, height) data that matches your ideas. Using a measuring cup or graduated cylinder, fill your container by adding equal amounts of water in 10 to 15 steps. Measure the height of the water at each step. Record your (volume, height) data in a table. Make a scatterplot of the (volume, height) data for your container, and display the resulting graph for the other groups to study. Now study the graphs of (volume, height) data from the other groups. Without looking at their containers, sketch the bottle shapes that you believe their graphs represent. Compare your sketches with the actual containers. This material is from the pre-publication version of Year 4 of Contemporary Mathematics in Context. The published version of the material will be available in August, 1999 from Everyday Learning Corporation, Two Prudential Plaza, Suite 1175, Chicago, IL 60601, HSMP Bottles and Divers Lesson Guide Page 10

11 Introduction: The Bungee Jumper In many problems about rates of change it s possible to express the relations among variables by symbolic function rules. For example, principles of physics can be used to model the flight of a bungee jumper with a rule giving jumper height as a function of elapsed time in the flight Jumper's Height in Feet Elapsed Time in Jump in Seconds Think About This Situation If you were given the rule for a function h(t) that predicts a bungee jumper s height at any time t in his or her jump, how would you use that rule to: a. Estimate the jumper s speed of fall or rise at any time? b. Estimate the times when the jumper reached the bottom or top of a bounce? c. Estimate the times when the jumper was traveling at his or her maximum speed? HSMP Bottles and Divers Lesson Guide Page 11

12 The Diver Problem As you know from experience, very few objects in motion travel at constant speeds. However, in many of those situations it is possible to predict the object's position with fairly simple function rules. Those rules can also be used to estimate the velocity and acceleration of the moving objects. 1. The Mexican cliff divers are among the most spectacular high divers; they leap off of rocky outcroppings into ocean bays. If one of these divers jumps from a spot that is 30 meters above the water, his or her height can be modeled well by a function with the rule h (t) = t 2 (height in meters and time in seconds). Use the function rule to answer these questions as accurately as possible: a. How long will it take the diver to reach the surface of the water? b. What will be the average speed of the diver, from takeoff to hitting the water? c. How will the diver s speed change during his flight, and how is that change shown in the shape of the (t, h (t)) graph? One of the most interesting features of the equation that expresses diver height as a function of time in flight is what it tells about how fast the diver will be traveling when he or she hits the water. In problem 1 you found the average speed from takeoff to hitting the water (approximately 2.47 seconds) was about 12 metersper second. But common sense and the study of tables and graphs for the height function tells you that the diver s speed increases throughout his flight. How would you go about estimating the diver s speed when he hits the water? HSMP Bottles and Divers Lesson Guide Page 12

13 Try your ideas. Then compare your strategies and results to the approach outlined in problem Estimate the average speed of the diver for these time intervals in his flight: a. t = 1 to t = 2.47 b. t = 2 to t = 2.47 c. t = 2.4 to t = 2.47 d. t = 2.46 to t = 2.47 What do the answers to parts (a d) suggest about the diver's speed when he hits the water? 3. Now consider the question of estimating the diver s speed at several other points in his dive. Try several different ways of producing what you believe are good estimates: a. What is the diver's speed exactly 1 second into the dive? b. What is the diver's speed exactly 2 seconds into the dive? c. What is the diver's speed just as he or she takes off from the dive? 4. Is there a way to make this calculation easier? A simple rule would make it more convenient for you to calculate the rate of change and slope of the graph of f(x) at any point by a simple substitution. One way to estimate values for a rate of change function is to use the rule D(x) = f (x + 0.1) f(x 0.1). 0.2 a. Why will D(x) produce the desired estimates? b. How can you modify the rule for D(x) to make more accurate estimates? To estimate rate of change for other functions? c. Test this rate of change estimation rule by completing the following table for h(x) = x 2 x h (x) = x 2 D(x) = h(x + 0.1) h(x 0.1) 0.2 HSMP Bottles and Divers Lesson Guide Page 13

14 The Bungee Jumper Think about the bungee jumper graph shown below. One function rule that has a graph like that it is h(t) = cos(2 t 0.7) (t +1) This rule gives height in feet as a function of time in seconds. It looks like a complicated function rule, but once you ve entered it in your graphing calculator you can explore many interesting questions about the jumper s falling and bouncing Jumper's Height in Feet Elapsed Time in Jump in Seconds HSMP Bottles and Divers Lesson Guide Page 14

15 1. Estimate, as accurately as possible, the coordinates of these points on the graph: a. the starting point b. the bottom of the first fall c. the top of the first bounce up d. the bottom of the second fall e. the top of the second bounce up 2. Use your estimates from question 1 to calculate the jumper s average velocity (feet per second) in each of these segments of his or her trip: a. from start to bottom of the first fall b. from bottom of first fall to top of first bounce up c. from top of first bounce to bottom of second fall d. from bottom of second fall to top of second bounce up 3. Estimate the average velocity of the jumper in each of these time intervals: a. from t = 0 to t = 0.5 b. from t = 0.5 to t = 1.0 c. from t = 1.0 to t = 1.5 d. from t = 1.5 to t = 2 4. Estimate, as accurately as possible, the jumper s velocity at each of these points: a. (0.5, ) b. (1.0, 42.7) c. (1.5, ) d. (2.0, ) 5. Explain how the differences among results in questions (3) and (4) are illustrated by the shape of the graph of h (t). 6. Find what you believe to be the point at which the bungee jumper is falling at the greatest velocity, and explain how you can locate that point by inspection of a table and a graph of the height function h (t). HSMP Bottles and Divers Lesson Guide Page 15

16 The Fish Population Suppose that the given graph represents the pattern of growth in the fish population of a lake that was stocked in Use your calculator to create a table of values giving the rate of change for each of the first 20 years. 1. At what point in time does it appear that the fish population was increasing most rapidly as time passed? This graph is modeled well by the function rule P(t) = (0.5 t ). How can you use this function rule to locate precisely the time of maximum population growth rate and the population at that time? 3. Give the rate of change for the fish population at 2, 4, 6, and 8 years. How is the rate of change at those particular times reflected in the graph? 4. Describe what happens to the fish population between 10 and 20 years. 5. Do you think this model would be accurate for a twenty year period of time? Why or why not? HSMP Bottles and Divers Lesson Guide Page 16

17 Graphically Speaking Selected Answers The following are sample data and scatterplots for various containers. The scale was kept the same for all of the scatter plots. If you wish to have your students use the same scale, you will need to determine an appropriate one ahead of time and give it to the students. Also, students might need to be reminded that volume is to be placed along the horizontal axis and height along the vertical axis. Container # 1 Container # 2 Container #3 HSMP Bottles and Divers Lesson Guide Page 17

18 Container #4 Container #5 Container #6 The following answers are for the "Extensions and Adaptations Exercises." (a) (b) (c) (d) (e) HSMP Bottles and Divers Lesson Guide Page 18

19 Introduction: The Bungee Jumper Selected Answers Think About This Situation a. If you want to estimate the speed at any particular time, you could compute the average rate of change of the function h (t) over a small interval containing that time. b. You could use the rule to create the graph from which you could estimate the times when the graph is at a peak (top of the bounce) or a valley, (bottom of the bounce). Alternatively, you could estimate the times when the jumper reached the bottom or top of a bounce by looking for times when the speed estimate is zero. c. The times when the jumper was traveling at his or her maximum speed could be estimated from the graph by looking for the places where the graph is steepest that is, where the magnitude (or absolute value) of the slope is greatest. HSMP Bottles and Divers Lesson Guide Page 19

20 The Diver Problem Selected Answers 1. a. The diver hits the surface when the height h(t) = t 2 equals 0. That is, when t is approximately 2.47 seconds. b. h(0) h(2.47) 12 Thus the average speed of the diver from takeoff to hitting the water is approximately 12 meters per second. c. As time increases, the diver's speed will gradually increase until he or she hits the water. This increase in speed is shown by the continual increase in the steepness (or slope) of the graph. 2. a. b. c. h(1) h(2.47) meters per second. Thus, the average speed is 17 m/s. ( 17 meters per second is measure of velocity, which takes into account direction). Students might arrange the terms in the numerator and denominator so that both are positive, but at this point it is reasonable for them to be more consistent in how they calculate the differences. Students can then begin to think about when the rate of change is positive, and when it is negative, and what that means. h(2) h(2.47) meters per second. Average speed is approximately 21.9 meters/sec. h(2.4) h(2.47) meters per second. Average speed is approximately 23.9 meters/sec. d. h(2.46) h(2.47) meters per second. Average speed is approximately 24.2 meters/sec. Parts (a d) suggest that the diver hits the water with an approximate speed of 24.2 meters per second. HSMP Bottles and Divers Lesson Guide Page 20

21 3. a. Here are 3 methods students might use: h(1) h(0.99) 9.75 m/s h(1.01) h(1) 9.85 m/s h(1.01) h(0.99) 9.8 m/s A good estimate of the diver s speed is 9.8 meters per second. b. As in part (a), students may try any number of methods, including: h(2) h(1.99) m/s h(2.01) h(2) m/s h(2.01) h(1.99) 19.6 m/s A good estimate of the diver s speed is 19.6 meters per second. c. Here are two methods that students might try: h(0.01) h(0) m/s h(0.001) h(0) m/s It appears that the instant he takes off from his dive, his speed is 0 meters per second. 4. a. f (x + 0.1)f(x0.1) D(x) = gives a good estimate of the instantaneous rate 0.2 of change at (x, f(x)) because it determines the rate of change between a point just above and just below that point. b. D(x) can be modified to make more accurate estimates by changing the numbers that are used. For example, you could use as the number added to and subtracted from x in the numerator and as the denominator. To estimate derivatives for other functions simply substitute a different function for h (x). HSMP Bottles and Divers Lesson Guide Page 21

22 c. Test this rate of change estimation rule by completing the following table for h (x) = x 2 x h (x) = x 2 h(x + 0.1) h(x 0.1) D(x) = HSMP Bottles and Divers Lesson Guide Page 22

23 The Bungee Jumper Selected Answers 1. Responses may vary, but should be fairly close to the following: a. The coordinates of the starting point are approximately (0, 93). b. The coordinates of the bottom of the first fall are approximately (1.8, 4). c. The coordinates of the top of the first bounce up are approximately (3.4, 50). d. The coordinates of the bottom of the second fall are approximately (5, 19). e. The coordinates of the top of the second bounce up are approximately (6.6, 42). 2. Based on the values above, the jumper s average velocity from a. the start to bottom of the first fall is 49.4 feet per second. b. the bottom of first fall to top of first bounce up is 28.8 feet per second. c. the first bounce to bottom of second fall is 19.4 feet per second. d. the bottom of second fall to top of second bounce up is 14.4 feet per second. 3. The average velocity of the jumper in the time interval h(0.5) h(0) a. From t = 0 to t = 0.5 is 20.5 feet per second b. From t = 0.5 to t = 1.0 is c. From t = 1.0 to t = 1.5 is h(1.0) h(0.5) h(1.5) h(1.0) feet per second. 64 feet per second. d. From t = 1.5 to t = 2 is h(2) h(1.5) feet per second. HSMP Bottles and Divers Lesson Guide Page 23

24 4. Responses may vary slightly, depending on the method used. Other approaches will produce similar results. The jumper s estimated velocity at h(0.5) h(0.49) a. (0.5, 81) is 65 feet per second b. (1.0, 41) is c. (1.5, 8.7) is h(1.0) h(0.99) h(1.5) h(1.49) feet per second. 40 feet per second. d. (2.0, 3.7) is h(2) h(1.99) feet per second. 5. In question 3, students should note that although the time intervals are the same, the average velocity is different for each interval because the change in the jumper s height is not the same over each interval (that is, the slope of the graph is not constant). From seconds, the graph is not as steep as it is from seconds. Thus, since the average velocity is the slope of the line connecting the points, the average velocity for the first time interval is less (in absolute value) than the average velocity for the second time interval. In question 4, students should note that at 0.5, 1.0, and 1.5 seconds the velocity is negative because the graph is decreasing, but that at 2 seconds, the velocity is positive because the graph is increasing. Also, the steeper the graph is, the larger the velocity is (in absolute value). So, for example, since the graph is steeper at 1 second than at 0.5, 1.5, or 2 seconds, the velocity is greater (in absolute value). 6. The point at which the bungee jumper is falling at the greatest velocity is approximately (0.8, 59.6). This point can be located by looking for the part of the graph where the magnitude of the slope is greatest. From a table, you could locate this point by looking for the time interval where there is the greatest change in height. HSMP Bottles and Divers Lesson Guide Page 24

25 The Fish Population Selected Answers 1. at about 4.3 years Enter P(t) = (0.5 t as Y1 and the equation for the approximation of ) the derivative in the diver problem for Y2. Use the table function to determine when Y2 has a maximum value, or use the CALC feature of the graphing calculator to determine when Y2 has a maximum value close to 4.3. The population at that time will be the value in Y1. 3. Year Rate of Change Appearance of the Graph The slope is rising, but not rapidly The slope at this point is fairly large compared to the rest of the graph The slope is still positive, but it is beginning to slow down The slope is still positive, but is has slowed way down and seems to be approaching a limit. 4. The growth rate continues to slow down during this time period as the population approaches the maximum capacity for the lake. 5. This model would probably not be accurate for a twenty year period of time because there are many other variables involved, such as development, weather, and pollution. HSMP Bottles and Divers Lesson Guide Page 25