the high school math project bottles and divers of change.pdfanalytical geometry by descartes and...

The High School Math Project —Focus on Algebra

http://www.pbs.org/mathline

Bottles and Divers(Rates of Change)

Objective

Students calculate the rate of change between two points on a curve by determiningthe slope of the line joining those two points. They then extend this method toestimate the instantaneous rate of change at a given point by taking two points veryclose 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 ofalgebra. This lesson has an introductory activity that uses bottles of various shapesto help students understand the concept of rate of change. The lesson then developsthe concept of the average rate of change between two points on a curve. Finally, itexamines instantaneous rate of change: rate of change function is used to estimatethe rate of change at a particular point. This function is used along with thecalculator to make computing the rate of change fast and easy for students. Thedevelopment of this function helps students understand what is meant byinstantaneous rate of change and lays a firm foundation or the study of thederivative 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

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HSMP —Bottles and Divers Lesson Guide • http://www.pbs.org/mathline

• 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 experiencedealing with the abstract concept of rate of change. Each group of students isgiven one container. They use a graduated cylinder to measure amounts ofwater, then add the water to the container and measure the height of thewater. After this data is collected, students make a scatterplot of their data.

If possible, each group should conceal their container from the other groupsas they are collecting data and creating the scatter plot. To help them concealthe containers as the students do their measuring, you may wish to makescreens of poster board folded in half and taped to a desk. A large box with twosides cut away also works well.

The containers should represent a variety of shapes. Your science departmentmay have a selection of beakers, flasks, long stem funnels, and graduatedcylinders that could be used, or students could bring in a variety of bottles orcontainers. In assigning the containers, keep in mind that containers' shapesaffect the difficulty level of the activity: beakers, graduated cylinders, and anyother type of prism are easier; an Erlenmeyer flask is moderately difficult; andthe Florence flask and the long stem funnel are generally more difficult, or atleast more time consuming.

As the groups complete their measurements and scatterplots, have them tapethese to the blackboard or a wall. When all groups are finished, lead the entireclass through a discussion of the graphs that are on display. For each graphyou might ask questions similar to the following. Have each group discussthem before you ask for responses directed to the entire class.

ä What do you think the shape of the container is like?ä Is the graph similar to any other graph that is displayed?ä How do you think the shape differs from the shape of the container

whose graph is similar to it?ä Could you write an algebraic rule for the graph?

After the groups have discussed these questions, have a representative fromone of the groups go to the board or the overhead and sketch the shape his orher group believes the graph represents. Let other groups either confirm the

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sketch or challenge it by giving their alternative and justifications for adifferent sketch. As the class goes through this discussion, remind the groupwhose shape is being discussed that they are not to give any hints. After thesketch has been made and discussed, the group who did the plot should showtheir container and either affirm the sketch or explain why the sketch is notcorrect.

2. The Bungee Jumper: Class discussion of the "Think About This Situation"section of Introduction: The Bungee Jumper serves as the introduction to thelesson on rates of change for functions with algebraic rules. This activity isdesigned to engage the students and to have them talk about some ways inwhich they could use an algebraic rule to help determine a rate of change.Students could mention determining an average rate of change or using thealgebraic 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, thegraph, and the table of values as they explore this problem. After all of thegroups have had time to work and discuss the problem, have them presenttheir findings using the blackboard or newsprint. Discuss each part of theproblem, moving from small group discussions to whole class discussions.Problem number 4 should be thoroughly discussed because it allows studentsto generalize the procedure that the previous problems had been leadingthem to discover. Students might need more guidance with thisgeneralization.

As students work with the rate-of-change estimate function with theircalculator, it will be helpful for them to understand that they can enter that

equation in Y1 as Y1 =Y2(x + 0.1) − Y2(x − 0.1)

0.2, where Y2 is the function for

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 ofthe most important means of assessing students is by listening to their comments asthey work with their classmates. In this lesson, students are grouped in threes tokeep the students from pairing off as discussions take place. This aides the teacher inlistening to a group work, but it also means that there will be more groups tocirculate among. (Many instructional decisions, like this one, involve trade-offs.)

The teacher mentions using a variety of means to assess the students, includingclass discussions, projects, tests, and quizzes. An important part of the written testsand quizzes involves a more open-ended approach in which the teacher asks thestudents to not only give an answer, but also to explain their reasoning. This

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emphasis on problem-solving and reasoning as well as using multiple means ofassessing students are in line with the NCTM standards, and they are an importantpart 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 datacollection and plotting before others.

Imagine filling each of the containers shown below by adding water in 10equal-size amounts. Sketch the predicted (v o l u m e , height) graphs for eachpair 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 additionalactivities that allow the students to use the difference equation in theircalculators to quickly determine rates of change in addition to exploring somedifferent 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 standardsrecommend 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

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examined from both a graphical and a numerical perspective. This lesson is aninformal exploration of rates of change and the derivative.

The derivative and related developments represent one of the most importantadvances in the history of mathematics. The seventeenth century discoveries ofanalytical geometry by Descartes and infinitesimal calculus by Newton and Leibnitzare the beginning of modern mathematics. The concepts of limits and the derivativeare key ideas in calculus.

Some students do not have the opportunity to explore the concepts involved withthe derivative in an informal way before they begin the formal study of calculus. Asa result, these students may rely on memorizing formulas and rules rather thanclearly understanding the concepts involved. Explorations such as those in thislesson allow students to investigate and explore the important mathematicswithout the formality that they will encounter later. These explorations and theapplications on which the investigations are based help give students a solidmathematical foundation.

In many calculus texts, the derivative of the function y = f(x) is defined as lim∆ x →0

∆y

∆x

which can also be written as lim∆ x →0

f (x1 +∆ x) − f (x1)

∆x. How does this differ from

lim∆ x →0

f (x1) − f (x1 − ∆x)

∆x or

lim∆ x →0

f (x1 +∆ x) − f (x1 −∆ x)

2∆x?

In the video lesson, the students calculate an approximation for the derivativebecause they are actually using a particular value for ∆x. Help your students realizethat there are several ways that an approximation may be calculated. They candetermine 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 pointand a point slightly below the given point. Also help students to understand thatthey can get a closer approximation to the instantaneous rate of change by usingsmaller values for ∆x.

As students as they calculate an approximation for the derivative, help them realizethat the more they zoom in to a particular area of a graph, the more that particularpart of the graph will approximate a straight line. Have the students use the zoomfeature of their calculator to examine the graphical implications of calculating therate of change over a very small period as an approximation of the derivative or theinstantaneous rate of change at a particular point.

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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 ofLearning is an approach that has taken the best of what we know about therelationships between how we think and how we learn and created a framework tohelp teachers plan and deliver instruction. Dimension 3 relates to thinking whichextends and refines knowledge, and offers a list of different types of questions whichpromote different kinds of thinking and learning.

The following chart is presented as a resource. Many teachers create a cue card orchart for themselves as a handy reference in planning or teaching. You might createa goal for yourself of ensuring that you use at least four types of questioning in eachlesson.

Comparison Identifying and articulating similarities and differences betweenthings.How are these things alike? different?

Classification Grouping things into definable categories on the basis oftheir attributes.Into what groups could you organize these things? What are therules for membership?

Induction Inferring unknown generalizations or principles from observationor analysis.Based on the observations we have listed, what conclusions mightyou draw?

Deduction Inferring unstated consequences and conditions from givenprinciples and generalizations.If we accept these generalizations as true, what conclusionsnecessarily follow?

Error Analysis Identifying and articulating errors in your own thinking or inthat of others.How is the reasoning in this argument misleading?

Constructing Building a system of support or proof for an assertion.Support What facts would support this claim?

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Abstracting Identifying and articulating the underlying theme or generalpattern of information.What pattern do you see?

Analyzing Identifying and articulating your personal perspectives on issues asPerspective well as others' perspectives.

How might someone else see this, and what reasoning might theyuse to support their position?

Resources

Coxford, Art, James Fey, Christian Hirsch, and Harold Schoen, ContemporaryMathematics in Context, Course 4. Chicago, IL: Everyday LearningCorporation, 1998.

Everyday Learning Corporation, Two Prudential Plaza, Suite 1175, Chicago, IL 60601,1-800-382-7670.

Internet location: http://www.npac.syr.edu:80/REU/reu94/williams/calc-index.htmlCyberCalc, an interactive learning environment for calculus, contains threechapters—Review of Topics Needed for Calculus, Limits and Continuity, andThe Derivative.

Internet location: http://forum.swarthmore.edu/mathed/calculus.reform.htmlThis location is a source of information on calculus reform—issues, projects,conferences, workshops, and a bibliography.

Internet location: http://www.maths.tcd.ie/pub/HistMath/People/RBallHist.htmlThis site contains bibliographies of many mathematicians of the seventeenthand eighteenth centuries, including Newton and Leibnitz. It also has anaccount of the controversy between Newton and Leibnitz over the inventionof calculus.

Internet location: http://www.everydaylearning.comEveryday Learning Corporation publishes Contemporary Mathematics inContext. You can request catalogs or send email to the company from this site.

Internet location: http://www.wmich.edu/math-stat/cpmp/Information about the Core Plus Mathematics Project is available at thissite. Sample lessons are available as well as an overview of the project.

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Ideas for Online Discussion

(Some ideas may apply to more than one standard of the NCTM Professional Standards for TeachingMathematics.)

Standard 1: Worthwhile Mathematical Tasks

1. The NCTM Standards advocate to providing high school students with theunderpinnings of calculus. How does the content of this lesson address thisstandard? How do the activities of the lesson help students understand themathematical concepts?

Standard 2: The Teacher’s Role in Discourse

2. Discourse refers to ways of representing, thinking, talking, and agreeing anddisagreeing. Teachers send messages about types of knowledge and thinkingthat are valued by the ways that they handle discourse. What are some of theways 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 amathematics program that has encouraged group work and activeparticipation. What evidence do you see that demonstrates their ability towork in groups and discuss meaningful mathematics? What do you look foras 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 forpromoting discussion and reasoning about mathematics. The use of toolssuch as calculators and computers is encouraged in addition to the use ofmore conventional mathematical symbols and other means such asdrawings, diagrams, invented symbols, and analogies. What tools helped topromote mathematical thinking and the discussion of mathematical conceptsin the video lesson?

Standard 5: Learning Environment

5. One of the key components of the learning environment is the idea that itsupports serious mathematical thinking. Teachers and students have agenuine respect for ideas and reasoning, and students have time to puzzleand think. A positive learning environment helps students be successfulmathematical thinkers. What are some of the important ways of creatingsuch a learning environment?

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Standard 6: Analysis of Teaching and Learning

6. Part of the reason for analyzing teaching and learning is to ensure that everystudent is mastering important mathematics and developing a positiveattitude toward mathematics. How do you assess the attitudes of students?What suggestions could you give to help other teachers assess the attitudestheir students have about mathematics?

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Graphically Speaking

On a typical day, you may fill and empty several different kinds of bottles, jars, andcans. In the process, you might have to look at the height of the contents in thosecontainers and estimate their volumes. With typical cylindrical jars or cans, thatestimation 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 roughlythe 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 eachgroup knows only its own shaped container's shape.

• Within your group, estimate the rate at which the water will rise in yourcontainer as it is added in equal amounts. Sketch a graph of (v o l u m e ,height) data that matches your ideas.

• Using a measuring cup or graduated cylinder, fill your container by addingequal amounts of water in 10 to 15 steps. Measure the height of the waterat each step. Record your (v o l u m e , height) data in a table.

• Make a scatterplot of the (v o l u m e , height) data for your container, anddisplay the resulting graph for the other groups to study.

• Now study the graphs of (v o l u m e , height) data from the other groups.Without looking at their containers, sketch the bottle shapes that youbelieve their graphs represent. Compare your sketches with the actualcontainers.

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 LearningCorporation, Two Prudential Plaza, Suite 1175, Chicago, IL 60601, 800-382-7670.

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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.

2 4 6 8 Elapsed Time in Jump in Seconds

100

60

40

20

Jumper's Heightin Feet

80

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?

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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 positionwith fairly simple function rules. Those rules can also be used to estimate thevelocity 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) = 30 – 4.9t2 (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 afunction of time in flight is what it tells about how fast the diver will be travelingwhen he or she hits the water. In problem 1 you found the average speed fromtakeoff to hitting the water (approximately 2.47 seconds) was about 12 meterspersecond. But common sense and the study of tables and graphs for the heightfunction 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?

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Try your ideas. Then compare your strategies and results to the approach outlinedin problem 2.

2. Estimate the average speed of the diver for these time intervals in his flight:a. t = 1 to t = 2.47b. t = 2 to t = 2.47c. t = 2.4 to t = 2.47d. t = 2.46 to t = 2.47

What do the answers to parts (a–d) suggest about the diver's speed when hehits the water?

3. Now consider the question of estimating the diver’s speed at several otherpoints in his dive. Try several different ways of producing what you believeare 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 itmore convenient for you to calculate the rate of change and slope of the graphof f(x) at any point by a simple substitution. One way to estimate values for arate 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 accurateestimates? To estimate rate of change for other functions?

c. Test this rate of change estimation rule by completing the followingtable for h(x) = 30 – 4.9x2

x h (x) = 30 – 4.9x2 D(x) = h(x + 0.1) − h(x − 0.1)

0.2

0

0.5

1

2

2.4

2.467

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The Bungee Jumper

Think about the bungee jumpergraph shown below. Onefunction rule that has a graph likethat it is

h(t) = 32 + 80cos(2 t − 0.7)

(t +1)

This rule gives height in feet as afunction of time in seconds. Itlooks like a complicated functionrule, but once you’ve entered it inyour graphing calculator you canexplore many interestingquestions about the jumper’sfalling and bouncing.

2 4 6 8 Elapsed Time in Jump in Seconds

100

60

40

20

Jumper's Heightin Feet

80

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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, 82.951)

b. (1.0, 42.7)

c. (1.5, 10.679)

d. (2.0, 5.6679)

5. Explain how the differences among results in questions (3) and (4) areillustrated by the shape of the graph of h (t).

6. Find what you believe to be the point at which the bungee jumper isfalling at the greatest velocity, and explain how you can locate that pointby inspection of a table and a graph of the height function h (t).

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The Fish Population

Suppose that the given graph represents the pattern of growth in the fishpopulation of a lake that was stocked in 1985. Use your calculator to create a table ofvalues 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 wasincreasing most rapidly as time passed?

2. This graph is modeled well by the function rule P(t) = 100

1 + 20(0.5t ). How

can you use this function rule to locate precisely the time of maximumpopulation 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 isthe 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 oftime? Why or why not?

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Graphically SpeakingSelected Answers

The following are sample data and scatterplots for variouscontainers. The scale was kept the same for all of the scatterplots. If you wish to have your students use the same scale,you will need to determine an appropriate one ahead of timeand give it to the students. Also, students might need to bereminded that volume is to be placed along the horizontalaxis and height along the vertical axis.

Container # 1

Container # 2

Container #3

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Container #4

Container #5

Container #6

The following answers are for the "Extensions and Adaptations Exercises."

(a) (b) (c)

(d) (e)

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Introduction: The Bungee Jumper

Selected Answers

Think About This Situation

a. If you want to estimate the speed at any particular time, you couldcompute the average rate of change of the function h (t) over asmall interval containing that time.

b. You could use the rule to create the graph from which you couldestimate the times when the graph is at a peak (top of the bounce)or a valley, (bottom of the bounce). Alternatively, you couldestimate the times when the jumper reached the bottom or top ofa bounce by looking for times when the speed estimate is zero.

c. The times when the jumper was traveling at his or her maximumspeed could be estimated from the graph by looking for the placeswhere the graph is steepest—that is, where the magnitude (orabsolute value) of the slope is greatest.

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The Diver ProblemSelected Answers

1. a. The diver hits the surface when the height h(t) = 30 – 4.9t2 equals 0.That is, when t is approximately 2.47 seconds.

b.h(0) − h(2.47)

0 − 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 orshe hits the water. This increase in speed is shown by the continualincrease in the steepness (or slope) of the graph.

2. a.h(1) − h(2.47)

1− 2.47≈

25.1− .10559

1− 2.47≈− 17 meters per second. Thus, the average

speed is 17 m/s. (–17 meters per second is measure of velocity, whichtakes into account direction). Students might arrange the terms in thenumerator and denominator so that both are positive, but at this pointit is reasonable for them to be more consistent in how they calculatethe differences. Students can then begin to think about when the rateof change is positive, and when it is negative, and what that means.

b.h(2) − h(2.47)

2 − 2.47≈

25.1− .10559

2 − 2.47≈ – 21.9 meters per second. Average speed

is approximately 21.9 meters/sec.

c.h(2.4) − h(2.47)

2.4 − 2.47≈

1.776 − .10559

2.4 − 2.47≈− 23.9 meters per second. Average

speed is approximately 23.9 meters/sec.

d.h(2.46) − h(2.47)

2.46 − 2.47≈

.34761 − .10559

2.46 − 2.47≈ − 24.2 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 of24.2 meters per second.

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3. a. Here are 3 methods students might use:h(1) − h(0.99)

1− 0.99≈ − 9.75 m/s

h(1.01) − h(1)

1.01− 1≈− 9.85 m/s

h(1.01) − h(0.99)

1.01− 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)

2 − 1.99≈ −19.55 m/s

h(2.01) − h(2)

2.01− 2≈− 19.65 m/s

h(2.01) − h(1.99)

2.01− 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)

0.01− 0≈− 0.049 m/s

h(0.001) − h(0)

0.001− 0≈− 0.0049 m/s

It appears that the instant he takes off from his dive, his speed is 0meters per second.

4. a. D(x) = f (x + 0.1)f(x 0.1)

0.2gives a good estimate of the instantaneous rate

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 0.001 as the numberadded to and subtracted from x in the numerator and 0.002 as the denominator. To estimate derivatives for other functions simply substitute a different function for h (x).

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c. Test this rate of change estimation rule by completing the following table

for h (x) = 30 – 4.9x2

x h (x) = 30 – 4.9x2D(x) =

h(x + 0.1) − h(x − 0.1)

0.2

0 30 0

0.5 28.775 –4.9

1 25.1 –9.8

2 10.4 –14.7

2.4 1.776 –23.52

2.467 0.17816 –24.18

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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

a. From t = 0 to t = 0.5 is h(0.5) − h(0)

0.5 − 0≈ −20.5 feet per second.

b. From t = 0.5 to t = 1.0 is h(1.0) − h(0.5)

1.0 − 0.5≈ −80.5 feet per second.

c. From t = 1.0 to t = 1.5 is h(1.5) − h(1.0)

1.5 − 1.0≈ −64 feet per second.

d. From t = 1.5 to t = 2 is h(2) − h(1.5)

2 − 1.5≈ −10 feet per second.

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4. Responses may vary slightly, depending on the method used. Otherapproaches will produce similar results. The jumper’s estimated velocity at

a. (0.5, 81) is h(0.5) − h(0.49)


b. (1.0, 41) is h(1.0) − h(0.99)


c. (1.5, 8.7) is h(1.5) − h(1.49)


d. (2.0, 3.7) is h(2) − h(1.99)

2 − 1.99≈ 17 feet per second.

5. In question 3, students should note that although the time intervals are thesame, the average velocity is different for each interval because the change inthe jumper’s height is not the same over each interval (that is, the slope of thegraph is not constant). From 0–0.5 seconds, the graph is not as steep as it isfrom 0.5–1.0 seconds. Thus, since the average velocity is the slope of the lineconnecting the points, the average velocity for the first time interval is less (inabsolute 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 velocityis negative because the graph is decreasing, but that at 2 seconds, the velocity ispositive because the graph is increasing. Also, the steeper the graph is, thelarger the velocity is (in absolute value). So, for example, since the graph issteeper at 1 second than at 0.5, 1.5, or 2 seconds, the velocity is greater (inabsolute value).

6. The point at which the bungee jumper is falling at the greatest velocity isapproximately (0.8, 59.6). This point can be located by looking for the part of thegraph where the magnitude of the slope is greatest. From a table, you couldlocate this point by looking for the time interval where there is the greatestchange in height.

PBS MATHLINE®


The Fish PopulationSelected Answers

1. at about 4.3 years

2. Enter P(t) = 100

1 + 20(0.5t) as Y1 and the equation for the approximation of

the derivative in the diver problem for Y2. Use the table function todetermine when Y2 has a maximum value, or use the CALC feature of thegraphing calculator to determine when Y2 has a maximum value close to4.3. The population at that time will be the value in Y1.

3.

Year Rate ofChange

Appearance of the Graph

2 9.6283 The slope is rising, but not rapidly.

4 17.108 The slope at this point is fairly largecompared to the rest of the graph.

6 12.573 The slope is still positive, but it is beginningto slow down.

8 4.6611 The slope is still positive, but is has slowedway down and seems to be approaching alimit.

4. The growth rate continues to slow down during this time period as thepopulation approaches the maximum capacity for the lake.

5. This model would probably not be accurate for a twenty year period of timebecause there are many other variables involved, such as development,weather, and pollution .

the high school math project bottles and divers of change.pdfanalytical geometry by descartes and...

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