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Calculus: An Active Approach with Projects

Stephen Hilbert
Diane Driscoll Schwartz
Stan Seltzer
John Maceli
Eric Robinson
Copyright Date: 2010
Edition: 1
Pages: 332
  • Cite this Item
  • Book Info
    Book Description:

    This volume contains student and instructor material for the delivery of a two-semester calculus sequence at the undergraduate level. It can be used in conjunction with any textbook. It was written with the view that students who are actively involved inside and outside the classroom are more likely to succeed, develop deeper conceptual understanding, and retain knowledge, than students who are passive recipients of information. Calculus: An Active Approach with Projects contains two main student sections. The first contains activities usually done in class, individually or in groups. Many of the activities allow students to participate in the development of central calculus ideas. The second section contains longer projects where students work in groups outside the classroom. These projects may involve material already presented, motivate concepts, or introduce supplementary topics. Instructor materials contained in the volume include comments and notes on each project and activity, guidelines on their implementation, and a sample curriculum which incorporates a collection of activities and projects.

    eISBN: 978-0-88385-972-8
    Subjects: Mathematics

Table of Contents

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  1. Front Matter
    (pp. i-vi)
  2. Preface
    (pp. vii-xviii)
  3. Table of Contents
    (pp. xix-xxiv)
  4. I Activities

    • 1 Graphical Calculus
      (pp. 3-56)

      The first chapter of this book consists of problems and activities designed to introduce you to many of the important ideas of first-year calculus in a way that encourages you to visualize the objects and actions you are studying. We call this approach thecalculus of graphs.

      We see graphs not only in mathematics, but also in the physical sciences, the social sciences, even the daily newspaper. Graphs give us a way of comprehending the world, in part by providing a way to visualize an ongoing process as a whole. That is, a graph can contain the whole past history...

    • 2 Functions, Limits, and Continuity
      (pp. 57-70)

      One of the most useful concepts in mathematics, and in particular in calculus, is thefunction. A function relates each number in a collection of “input” values to a corresponding “output” value. In the early material in this book, we described relationships between variables in a number of ways—verbal descriptions, numerical descriptions, graphical descriptions. Most if not all of these relationships describe functions. In this chapter we will look at functions more formally, and investigate properties that functions might exhibit.

      One tool for describing properties of functions is the concept oflimit. This concept is used in several ways...

    • 3 Derivatives
      (pp. 71-90)

      One of the primary concepts of calculus is the derivative. Derivatives can be used to describe the rate of change of a quantity; for example, the velocity of an object moving away from a fixed point is the rate at which the distance between the object and the fixed point is changing.

      Knowing the derivative enables us to determine a lot about a function, such as the slope of the graph at a point. The derivative also gives information about the shape of the graph of a function: where the function is increasing and decreasing, where it has maxima and...

    • 4 Integration
      (pp. 91-114)

      The concept of integration is one of the two primary notions in calculus. The other is differentiation. While these are two different concepts the fundamental discovery of calculus is that these two concepts are closely related to each other.

      Thedefinite integralplays the lead in this chapter. Integration allows us to use information about the rate at which a quantity is changing to determine the total change of that quantity. For example, the definite integral allows us to determine the total mileage traveled by a car during a specific interval of time if the speed of the car is...

    • 5 Transcendental Functions
      (pp. 115-132)

      Polynomial functions are examples of one type of function calledalgebraic. Functions that are not algebraic are calledtranscendentalfunctions. Some examples of transcendental functions that may already be familiar to you are trigonometric functions, exponential functions and logarithmic functions.

      While polynomial functions are relatively easy to work with, many phenomena are better modeled using transcendental functions.

      Exponential functions model growth (of populations, money, etc.) and decay (radioactive decay, for example).

      Logarithmic functions can be used to solve equations that involve exponential models. Logarithms also are used for graphing data and seeing patterns that may not be observable with normal...

    • 6 Differential Equations
      (pp. 133-150)

      Differential equations, equations that relate the rate of change of a quantity (the derivative) to variables and/or the quantity itself, provide an important and powerful tool to help us model many mathematical situations. Isaac Newton, one of the inventors of calculus, used differential equations to demonstrate that his theory of gravitation implied that the planets in the solar system should have elliptical orbits.

      A remarkable property of differential equations is that a relatively simple equation allows us to investigate short-term and in many cases long-term behavior of the phenomena modeled by the differential equation. In several cases we can characterize...

    • 7 Series
      (pp. 151-178)

      Informally, a series may be thought of as an infinite sum. This chapter will lead you to investigate and better understand series of all sorts.

      In fact, an “infinite sum” is the limit of a sequence of finite sums (called partial sums) as more and more terms in the sum are included. You will look at these partial sums as a way of understanding the behavior of the series (convergence and divergence). You will also see some problems that can be modeled using series.

      More important are power series. Power series can be thought of as “infinite polynomials,” and can...

  5. II Projects

    • 1. Designing a Roller Coaster
      (pp. 181-182)

      You have been hired by Two Flags Over Ithaca to help with the design of their new roller coaster.

      Each individual has been given a path design for astraightstretch (no turns) of a proposed roller coaster. There is a support every 10 feet. A safety rule is that a descent can be no steeper than 80° at any point. In addition each design starts with a 45° incline. (Angles refer to the angle that the path makes with a horizontal line.)

      Each individual will report on her/his design. Your report should include the following data.

      1. Where is...

    • 2. Tidal Flows
      (pp. 183-184)

      You are a team of consulting engineers studying the flow of water from a certain river into a large lake. You have data in the form of a graph detailing the amount of water that has flowed into the lake from the river over a fourteen-day period. The graph is given in Figure 1.

      Your job is to report several results to your client as follows:

      1. First, determine the total volume of water that has flowed from the river in the first five days of the observation period; in the first ten days; in the entire fourteen-day period.


    • 3. More Tidal Flows
      (pp. 185-186)

      Your engineering group has also been retained to aid in a study of the salinity of a bay near the mouth of a tidal river. A tidal river is one that empties into the ocean or an oceanic bay and is so influenced by the tides that its flow is noticeably affected. That is, the change in water level in the body of water into which the river flows affects the flow of water within the river.

      You have collected two kinds of data for a three-day period. The first is a measure of the rate at which the fresh...

    • 4. Designing a Cruise Control
      (pp. 187-187)

      You have been hired as a consultant for a car manufacturer who is designing a cruise control system for a mid-size car. The problem is first broken down into two parts: designing a system to convert real speed into recorded speed; and designing a mechanical apparatus which either slows down or speeds up the car depending upon its recorded speed and a “speed set.” We will concentrate on the former problem.

      This problem of converting real speed into recorded speed is further broken down into two stages. In the first stage, a metal pin is secured to the inside of...

    • 5. Designing a Detector
      (pp. 188-189)

      You are designing a security system for a hospital. The hospital keeps its supply of drugs in a storeroom whose entrance is located in the middle of a 40-foot long hallway. The entrance is a three-foot wide door. The hospital wishes to monitor the entire hallway as well as the storeroom door. You must decide how to program a detector to accomplish this. The detector runs on a track and points a beam of light straight ahead on the opposite wall. The beam reaches from floor to ceiling. Think of the hallway as a coordinate line with the middle of...

    • 6. Gasoline and Advertising
      (pp. 190-191)

      Below is a graph of a hypothetical “demand curve,” which gives the average price per gallon of gasoline as a function of the quantity of gas on the market each day during the third quarter of 2009 in New York State.

      The quantity of gasoline on the market showed several oscillations during the third quarter of 2009. Unfortunately, much of the data has been lost; you do know that the high was 0.75 metric tons on July 2, 2009, and that the low was 0.43 metric tons on September 20, 2009.

      1. Construct a possible graph that shows the quantity...

    • 7. Taxes
      (pp. 192-194)

      Suppose$t\left( x \right)$represents the amount of tax you pay (dollars) if your income is$x$(dollars). There are (at least) three different ways of depicting this tax on a graph:

      graph$y = t\left( x \right)$

      graph$y = $“average tax rate”$\left( {{t_a}\left( x \right)} \right.$=$t\left( x \right)/\left. x \right)$

      graph$y = $“marginal tax rate”$\left( {{t_m}\left( x \right)} \right.$= the tax on the next dollar =$\left( {t\left( {x + 1} \right)} \right. - t\left. {\left( x \right)} \right)/\$ 1$; division by $1.00 converts dollars to a percentage)

      For example, if

      $t(x) = \left\{ \begin{array}{l} 0.1x,{\rm{ 0}} \le x \le 1000 \\ 0.2x - 100,{\rm{ }}x > 1000 \\ \end{array} \right.$


      tax on $500 is 0:1 $500 = $50

      tax on $5000 is 0:2 $5000-$100 = $900

      average tax rate at $500 is $50=$500 = 10%

      average tax rate at $5000 is $900=$5000...

    • 8. Water Evaporation
      (pp. 195-196)

      A water tank was filled with 25,000 gallons of water. Some of the water evaporates each day. The table below gives the amount of water (in thousands of gallons) in the tank$x$months after the tank was filled for one and one half months. So for example if$f\left( 1 \right)$is 21, that would mean that after 1 month there were 21,000 gallons of water in the tank.

      1. Using the table of values for the function, each member will graph the function$f$between 0 and 1.5.

      2. Each member of the group will estimate the slope of the...

    • 9. Glen Canyon Dam
      (pp. 197-200)

      This project is based (loosely) on the Glen Canyon controversy, which was reported on National Public Radio on Saturday, October 20, 1990.

      The amount of electric power required throughout the day by a power grid for distribution is given by the graph below. Early morning when people get up and get ready to go to work, school, etc., and evening when people come home are called “peak power periods.” It would be expensive to build enough power plants to supply peak power demands 24 hours a day, so utilities look for peak power sources. Hydroelectric power is particularly good for...

    • 10. Real Estate Investor
      (pp. 201-201)

      You are an investor, interested in real estate. Part of your interest stems from the fact that you understand that real estate appreciates at about 10% per year. This project will investigate this and related phenomena. What does “10% appreciation per year” mean? Presumably, it means that the value of a property after yearnis 10% more than it was at the end of the year$n - 1$. This may be expressed mathematically by the “difference equation”$v\left( n \right) = 1.10 \cdot v\left( {n - 1} \right)$.

      1. Suppose that the initial value$v\left( 0 \right)$of a house is $100,000. Find the value of the house at the end of...

    • 11. How Much Curve?
      (pp. 202-202)

      What is a reasonable measure of curvature? We would like a criterion that shows the curvature of a large circle to be small relative to a small one, since a large circle does not curve as much as a small one.

      One straightforward approach, which we shall adopt, is to define the curvature of a circle with radiusrto be$k = 1/r$. Then, conceivably, for an arbitrary curve (well, let’s suppose it’s the graph of some function,$y = f(x)$) we could find the curvature at a pointP(with coordinates$({x_0},f({x_0}))$) by finding the circle that best fits a curve at...

    • 12. Mutual Funds
      (pp. 203-204)

      You are investigating mutual funds for the Securities and Exchange Commission. The Tip-Top-Table fund increased its value by $50,000 per month for the first 6 months of 2008, and then its value decreased by $10,000 per month each month after that. The value of the Rags-to-Riches Rule fund changed at the rate of $5,000t$\sin \left( {0.1{t^2} + 1} \right)$per monthtmonths after the start of 2008. The Go-Go Graph fund changed at the rate per month given on the accompanying graph. The Percentage Growth fund increased 4% per month for the first 10 months of 2008; after that it lost 6% per...

    • 13. Rescuing a Satellite
      (pp. 205-206)

      We will investigate whether or not it is possible to rescue an interplanetary probe. The satellite is currently 100,000 miles away from earth. The graph below gives its velocity for the next two years. (After two years the satellite will be useless unless it can be fixed.) The satellite is traveling in a straight line away from earth.

      The rescue ship leaves now and its velocity is$v\left( t \right) = \sqrt {t + 0.1} $thousand miles per hour in$t$years. The rescue ship will also travel in a straight line away from earth, and this is the same as the line on which the satellite...

    • 14. Spread of a Disease
      (pp. 207-208)

      You are working for the Health Department investigating the spread of a sexually transmitted disease in a population. Statistics are compiled monthly on the numbers of individuals who are infected. Since the disease is spread by contact with an infected person, the number of persons who have the disease at the end of any month depends on the number who had the disease the previous month. Your job is to predict the number of persons with the disease in future months.

      Models for the spread of the disease can be thought of in two different ways: (1) as differential equations...

    • 15. Designing an Arch
      (pp. 209-211)

      Your group has been hired as a consultant by a design company. They have a client who is interested in monumental style building so you will investigate the Gateway Arch which is in St. Louis.

      You will need to learn about what are called hyperbolic functions. The hyperbolic sine (denoted by sinh) is defined by

      $\sinh x = \frac{{{e^x} - {e^{ - x}}}}{2}$

      The hyperbolic cosine (denoted cosh) is defined by

      $\cosh x = \frac{{{e^x} + {e^{ - x}}}}{2}$

      The hyperbolic tangent (tanh) is defined by$\tanh x = \sinh x/\cosh x$, and there are corresponding hyperbolic functions for each trig function.

      1. prove that${\cosh ^2}x - {\sinh ^2}x = 1$for allx.

      2. Try to find an identity that relates tanh and...

    • 16. Cleopatra’s Oar
      (pp. 212-213)

      Your consulting firm has been hired by the Museum of National History to check on the authenticity of an oar. A Prof. I. Jones has offered to sell the oar and claims that the oar comes from Cleopatra’s galley. The museum wants your firm to help check that claim by estimating how old the wood in the oar is.

      You should present your conclusions and supporting data in a report to the museum. Your report should make sense to a student in a different section of Calculus 2. Using your report this hypothetical student should be able to explain your...

    • 17. Differential Equations
      (pp. 214-215)

      None of the techniques we have learned have enabled us to solve a differential equation such as

      $\frac{{dy}}{{dt}} = y + t,{\rm{ }}{{\rm{y}}_0} = 1$.

      However, we canapproximatea solution$f$by using the fact that

      $f(t + \Delta t) \approx f(t) + \Delta t \times f'(t)$(for small$\Delta t$).

      Thus, for example,

      $f(0.01) \approx 1 + 0.01 \times 1 = 1.01$.

      (We are using$t = 0,\Delta t = 0.01$.)

      We can now approximate$f(0.02)$:

      $f(0.02) \approx 1.01 + 0.01 \times (1.01 + 0.01) = 1.0202$.

      Continuung, we can approximate$f$from$t = 0$to$t = 1$.

      1. Approximate a solution to${{dy} \mathord{\left/ {\vphantom {{dy} {dt}}} \right. \kern-\nulldelimiterspace} {dt}} = y + t,{\rm{ }}{y_0} = 1$from$t = 0$to$t = 1$using$\Delta t = 0.1$. Sketch the solution.

      2. What could you do to get a better (more accurate) solution to this equation? (Don't do it; explain carefullywhyit would be more accurate.)

      3. Explain...

    • 18. How Many Deer Are There?
      (pp. 216-217)

      The number of deer in a state park was 1,000 six years ago and today it is 2,000. Based upon data from other habitats, the park service estimates that no more than 10,000 deer can inhabit the park. (If there are more than 10,000 deer in the park, overgrazing may change the character of the park and/or large numbers of deer may starve in the winter.) Your group has been hired to assist the park service in predicting both long and short term population trends.

      Your first task is to investigate a family of possible models for the population of...

    • 19. Tax Assessment
      (pp. 218-219)

      A class action suit has been filed against the county board of assessment by a group of landowners unhappy with the assessed value of their properties. Your group will model some assessment schemes and try them out on some lots. Each individual will analyze a single lot. Finally your group will analyze one of the disputed lots.

      Real estate taxes depend on the assessed value of a property. The value of a piece of property is decided by either a single person (the assessor) or a group (the board of assessment). In most cases, the area of a piece of...

    • 20. The Fish Pond
      (pp. 220-221)

      Happy Valley Pond is currently populated by yellow perch. The graph at the right gives an outline of the pond.

      The pond is fed by two springs: spring A contributes 50 gallons of water per hour during the dry season and 80 gallons of water per hour during the rainy season. Spring B contributes 60 gallons of water per hour during the dry season and 75 gallons of water per hour during the rainy season. During the dry season an average of 110 gallons of water per hour evaporates from the pond, and an average of 90 gallons per hour...

    • 21. Drug Dosage
      (pp. 222-223)

      The concentration in the blood resulting from a single dose of a drug normally decreases with time as the drug is eliminated from the body. In order to determine the exact pattern that the decrease follows, experiments are performed in which drug concentrations in the blood are measured at various times after the drug is administered. The data are then checked against a hypothesized function relating drug concentration to time.

      Suppose a single dose of a certain drug is administered to a patient at timet= 0, and that the drug concentration in the patient’s blood is measured immediately...

    • 22. Investigating Series
      (pp. 224-225)

      In this project, you will experiment with some infinite sequences and their limits.

      Starting with a given sequence of numbers$\{ {b_1},{b_2}, \ldots \} $, you will construct a new sequence$\{ {a_1},{a_2}, \ldots \} $as follows:

      ${a_1} = {b_1}$

      ${a_2} = {b_2} - {b_1}$

      ${a_3} = {b_3} - {b_2}$



      ${a_n} = {b_n} - {b_{n - 1}}$




      Starting with each of the following sequences as${b_n}$:

      (I)$\frac{2}{1},\frac{8}{3},\frac{{26}}{9},\frac{{80}}{{27}},\frac{{242}}{{81}},\frac{{728}}{{243}},\frac{{2186}}{{729}}, \ldots $and

      (II)$\frac{3}{4},\frac{6}{6},\frac{{9}}{8},\frac{{12}}{{10}},\frac{{15}}{{12}},\frac{{18}}{{14}},\frac{{21}}{{16}}, \ldots $

      1. Compute the first six elements of the sequence${a_n}$.

      2. Graph${a_n}$versusnand${b_n}$versusnon the same set of coordinate axes. Plot at least the first six values for each sequence. Visually determine the limit of each sequence, if it exists, and place...

    • 23. Topographical Maps
      (pp. 226-228)

      Attached is a section of a topographical map of Tompkins County (the county in which Ithaca College is situated). The light lines on the map are lines of constant altitude: that is, if you traveled along one of these lines, your altitude would remain constant. In mathematics, we call such lineslevel curves. With some thought and practice, one can picture a three-dimensional surface (in this case, the surface of the earth) by looking at a two-dimensional map on which the level curves have been indicated.

      1. One point on your map is labeled “O” (for origin). From that origin,...

  6. III Instructor Notes for Activities

    • 1 Graphical Calculus
      (pp. 231-246)

      Topic: Graphical modeling

      Summary: Students observe a physical motion and record the position function as a graph.

      Time required: 20 minutes

      Threads: graphical calculus, modeling

      Question 1: You should toss the chalk in the air and catch it. You may need to repeat the action for students who want to take a closer look as they construct their graphs.

      Question 2: Discuss student responses before going on. Some will not understand how to represent the passage of time on the graph, though most will see that displacement is a vertical distance (because of the particular choice of observation). The questions...

    • 2 Functions, Limits, and Continuity
      (pp. 247-250)

      Topics: Function, domain, range

      Summary: Introduces the mathematical concept of function. This activity can be used by students while the included materials are being presented or discussed in class.

      Time required: 30 minutes

      Thread: multiple representation of functions

      Question 2: Many students will already be familiar with the vertical line test. The others can “discover” it here.

      Question 5: Mention that the dependent variable is a “dummy variable.” Ask the students to find$f\left( z \right)$,$f\left( {{u^2}} \right)$, and$f\left( {x + 1} \right)$.

      Topics: Continuity

      Background assumed: Piecewise-defined functions

      Summary: Postage is used to illustrate the concept of continuity. This leads naturally to the concept of...

    • 3 Derivatives
      (pp. 251-254)

      Topics: Derivative, linear approximation

      Summary: This activity should help the students see that the marginal cost is the derivative of the cost function.

      Background assumed: None required; if the class has already seen the idea of secant lines this activity can easily be related to the concept.

      Time required: 10–15 minutes

      Resources needed: None; a calculator is useful for the calculations.

      Threads: approximation and estimation, multiple representation of functions

      Question 2: The curve seems to be increasing and concave down near (80, 1894) since the averages—the slopes of secant lines—are decreasing as the intervals over which the average is computed...

    • 4 Integration
      (pp. 255-262)

      Topic: Integration

      Summary: Introduces Riemann sums as estimates for the integral. Since the function is monotone, upper and lower estimates for the distance can be obtained.

      Time required: 20–25 minutes (more if you do the problems)

      Threads: distance and velocity, approximation and estimation, multiple representation of functions

      Question 7: As long as the speed does not decrease the analysis is still correct. In this case, instead of the car speeding up for the entire ten seconds, all that is needed is that the car never slows down during the ten seconds.

      Question 8: Explain that in this case the average...

    • 5 Transcendental Functions
      (pp. 263-268)

      Topics: Trigonometric functions and their derivatives

      Background assumed: Some ideas about the graphical relationship between velocity and position.

      Time required: 50 minutes

      Threads: distance and velocity, graphical calculus, modeling

      1. This is a good time to review radian measure and mention that we will always use radian measure in calculus. If students ask why you can mention the fact that this unit for angles will make calculation much simpler. Be sure to pick up this thread when you get to the derivative of sine and cosine.

      2. Remind the students that one of the main ways we investigate hard problems...

    • 6 Differential Equations
      (pp. 269-272)

      Topic: The geometric relationship between a differential equation and its solution.

      Summary: Students develop the geometric method of using direction fields or slope fields to approximate the solutions to a differential equation. The notion of isoclines is used in this development.

      Background assumed: Introduction to differential equations—students should realize that a first-order differential equation specifies the slope of a solution at each point.

      Time required: 40 minutes

      Threads: graphical calculus, approximation and estimation

      This activity is self-contained.

      Topics: Differential equations, direction fields

      Summary: Students draw a direction field and use it to draw and analyze solutions to the logistic...

    • 7 Series
      (pp. 273-280)

      Topic: Series of constants

      Summary: Introduces the idea of partial sums and convergence of sequences of partial sums

      Background assumed: Students should be familiar with convergence of numerical sequences.

      Time required: 20 minutes

      Resources needed: If you are inclined to use technology, you need something that can perform rational arithmetic.

      Thread: approximation and estimation

      Note that the series in this problem is not geometric.

      Topic: Limit of a series

      Summary: This is one way to introduce convergent series.

      Background assumed: Limit; series ideasnot required

      Time required: 10–15 minutes

      Thread: approximation and estimation

      Question 1. The class will usually...

  7. IV Instructor Notes for Projects

    • 1. Designing a Roller Coaster
      (pp. 283-284)

      Summary: This project uses the concepts of increasing, decreasing, and concavity to analyze the path of a roller coaster. Students construct graphs of the first and second derivatives of the path of the roller coaster and can use these graphs to find the steepest part of the path. They also relate the slope of the path to the angle the path makes with a horizontal line. Other mathematical concepts include constructing a curve that satisfies given constraints and then optimizing curves with respect to certain conditions.

      Background assumed: Increasing, decreasing and concavity for graphs; constructing slope graphs from a given...

    • 2. Tidal Flows
      (pp. 284-285)

      Topics: Rate of change

      Summary: This project is intended to introduce rate of change as slope, in a context other than distance-velocity.

      Background assumed: Students should be comfortable with the idea of representing a function just as a graph, and with the idea of estimation from a graph.

      Time required: About 2 weeks, as a group project.

      Threads: graphical calculus, approximation and estimation, modeling


      1. In question 1, they need only read data from the graph. In question 2 they will estimate the slope of the graph by calculating rate of change of the volume over shorter and shorter...

    • 3. More Tidal Flows
      (pp. 285-285)

      Topics: Preview of the Fundamental Theorem of Calculus

      Summary: Introduces the idea of using Riemann sums to estimate aggregates from a rate graph.

      Background assumed: Students should be comfortable with the idea of representing a function just as a graph, and with the idea of estimation from a graph.

      Time required: About 2 weeks, as a group project

      Threads: graphical calculus, approximation and estimation, modeling


      1. This can be used as an extension of the previous project,Tidal Flows, although the two are independent. If used early in the first semester it gives a gentle preview of the Fundamental...

    • 4. Designing a Cruise Control
      (pp. 285-286)

      Summary: This project is designed for early use in Calculus 1. It serves as an application of the concepts of composition of functions, piecewise-defined functions, and graphical analysis of functions. The project also serves to motivate the concepts of limits and continuity. Students are forced to see that a function is a mathematical “device” that determines a unique output for a given input.

      Background assumed: Functions as represented by graphs and formulas; composition of functions.

      Time required: Two weeks

      Threads: distance and velocity, graphical calculus, multiple representation of functions, approximation and estimation, modeling


      1. In part 1, students can...

    • 5. Designing a Detector
      (pp. 286-286)

      Topic: Periodic functions

      Summary: This is an example of an open-ended project. Students model the position of a detector both graphically and analytically. This project illustrates the mathematical modeling process very well. The initial “solutions” will go through a number of refinements as students modify their assumptions. Students also encounter a mathematics problem where there is no “correct” solution.

      Background assumed: Derivatives, relationship between position and velocity

      Time required: About two weeks

      Threads: modeling, multiple representation of functions, graphical calculus, distance and velocity


      1. Part A. Many groups will makexa piecewise-linear function oft. This lets them...

    • 6. Gasoline and Advertising
      (pp. 286-287)

      Topics: Composition of functions, Iteration of functions

      Summary: In Part A, the students investigate the composition of functions that are given graphically. They look for relations between the extrema of the composition and the extrema of the functions that are composed. In Part B, they investigate iterations and the stability of the equilibria of the iterations. In Part A, the function in question 1 is open ended in the sense that several functions can satisfy the given conditions. In Part B, the students may work with a table of values as well as graphically.

      Background assumed: Composition of functions


    • 7. Taxes
      (pp. 287-287)

      Topics: Graphical differentiation and integration

      Summary: The most important aspect of this project is the relationship between the tax function and the marginal tax function. Another important theme is multiple representation of functions; functions arise in algebraic and graphical contexts throughout. Students have to use and construct some piecewise-defined functions. There are several open ended aspects.

      Background assumed: Students must have seen the relationship between a function and its “rate graph” and “area graph.”

      Time required: About two weeks

      Threads: multiple representation of functions, graphical calculus


      1. Part A is essentially a warm up intended to get the students...

    • 8. Water Evaporation
      (pp. 288-288)

      Topics: Derivatives, differences, functional equations

      Summary: This project requires students to work with a function given in tabular form. It also involves relating the tabular form to graphical and algebraic properties.

      Background assumed: Increasing and decreasing functions, concavity

      Resources needed: At least a calculator. A spreadsheet can be used to compute the table of differences.

      Thread: multiple representation of functions


      The given function is$25{e^{ - .2x}}$. We have given each group different data by simply using$25{e^{kx}}$with values of k such as –.1,–.2, . . .

      In part A, question 5, be sure that the class knows that...

    • 9. Glen Canyon Dam
      (pp. 288-288)

      Topics: Difference equations, differential equations, rates of return

      Summary: The project involves setting up and solving difference equations. Students also relate the difference equations to differential equations. The idea of rate of return is introduced and students calculate rates of return and investigate the long term behavior of rates of return.

      Background assumed: Solving difference equations and simple differential equations

      Time required: about 2 weeks

      Threads: approximation and estimation, modeling

      Question 4.The rate of return can change from year to year. If you wish, you can have individualswork out the rate of return for a particular value of the down...

    • 10. Real Estate Investor
      (pp. 288-288)

      Topics: Difference equations, differential equations, rates of return

      Summary: The project involves setting up and solving difference equations. Students also relate the difference equations to differential equations. The idea of rate of return is introduced and students calculate rates of return and investigate the long term behavior of rates of return.

      Background assumed: Solving difference equations and simple differential equations

      Time required: about 2 weeks

      Threads: approximation and estimation, modeling

      Question 4.The rate of return can change from year to year. If you wish, you can have individuals work out the rate of return for a particular value of the...

    • 11. How Much Curve?
      (pp. 289-289)

      Topics: Curvature

      Summary: Students determine the curvature of the graph of a function in two different ways.

      Background assumed: Differentiation, point-slope form of a line, arc length, derivative of arctan , chain rule

      Time required: Two weeks

      Threads: graphical calculus

      Comments: As noted, the notation in the general case — even for$y=x^2$— can be overwhelming.

      One group of students, realizing that the derivative and second derivative were the only important quanties, assumed that (or, equivalently, translated the function so that) the best-fitting circle was centered at the origin. This led to a very elegant algebraic solution to the general case. (In this...

    • 12. Mutual Funds
      (pp. 289-289)

      Topic: Fundamental theorem of calculus

      Summary: This project focuses on the representation of functions via tables, algebraic expressions, verbal descriptions, and graphs. Students are asked to write several functions in several ways. The relationship between a rate of change and total value is explored. While this hints at the fundamental theorem of calculus, this project may be done near the beginning of Calculus 1 before this theorem is mentioned or along with an initial discussion of the theorem.Questions 6, 7, 8, and 9 have the students comparing rates of increase and decrease, considering how these rates are affected under basic...

    • 13. Rescuing a Satellite
      (pp. 289-290)

      Topics: Integration, graphical integration

      Summary: Students use Riemann sums to estimate integrals and decide whether a rocket ship will catch a satellite within two years.

      Background assumed: Riemann sums, integrals as area under a curve

      Threads: multiple representation of functions, distance and velocity, approximation and estimation


      1. This is a nice application of the intermediate value property.

      2. Some groups may have trouble with the units. Both velocities are given in miles per hour aftertyears. The graphical approach (distance traveled is area under the velocity curve) illustrates the difficulty. One block would have an area of$0.25.\frac{1}{{12}}$...

    • 14. Spread of a Disease
      (pp. 290-290)

      Topics: Differential equations, iteration

      Summary: Students compare itertion with the use of a differential equation to model spread of a disease. They examine the role of the constants in each model, and look for long run equilibrium.

      Background assumed: Solution of differential equations by separation of variables.

      Resources needed: A computer or calculator could be programmed to provide a way for students to explore further the effects of changing the constants in each model.

      Time required: About two weeks, for all parts

      Threads: modeling, approximation and estimation


      The objectives of this project are:

      1. To have the students solve...

    • 15. Designing an Arch
      (pp. 290-291)

      Topics: Hyperbolic functions, differential equations, approximation by polynomials

      Summary: Students work with hyperbolic functions to design an arch and investigate properties of the arch.

      Background assumed: Differential equations, arc length, (hyperbolic functions are not assumed)

      Time required: Three weeks

      Threads: approximation and estimation, modeling


      Part 1. Students should be able to derive these properties of hyperbolic functions from the definitions

      Part 4. The St. Louis Arch will not fit the model from part 2c) but several groups were able to approximate the arch by estimating and varying...

    • 16. Cleopatra’s Oar
      (pp. 291-291)

      Topics: Differential equations

      Summary: Students work with the concept of half-life and differential equations to estimate the age of an oar.

      Background assumed: Separable differential equations

      Time required: Two weeks

      Threads: modeling


      Part 1. You can easily add another substance if you have groups of four. If you wish you can have groups submit Part 1 before the final report is due, to make sure they understand the model....

    • 17. Differential Equations
      (pp. 291-291)

      Topic: Numerical and exact solutions to linear differential equations

      Summary: Students develop Euler’s method for approximating the solution to a differential equation. They also are led through variation of constants and substitution as methods for finding an exact solution to a differential equation of the form$dy/dt = L(t,y)$where$L(t,y)$is linear in t and y.

      Background assumed: Solving differential equations by separation of variables

      Time required: Two weeks

      Threads: graphical calculus, approximation and estimation


      1. The easiest explanation for (Part 1) 3. is to appeal to the slope field.

      2. Any$k{\rm{ < }} - 1$will work for (Part 1) 4.


    • 18. How Many Deer Are There?
      (pp. 292-292)

      Topics: Differential equations

      Summary: Students solve several differential equation models and use the models to analyze the population of deer in a park. In Part 4 they need to use an approximate method.

      Background assumed: Separable differential equations, Euler’s method

      Time required: Two - three weeks

      Threads: approximation and estimation, modeling


      Part 1. This part ensures that students can solve standard separable DE and analyze a family of models. We usually assign a due date of 1 week for this part and return it before the final report is due.

      Part 2. This can be done in many different...

    • 19. Tax Assessment
      (pp. 292-292)

      Topics: Graphical and numerical integration

      Background assumed: Integration

      Summary: This project requires students to use integrals to calculate the value of various lots and to find how to equitably divide these lots.

      Threads: approximation and estimation, modeling


      1. You can use different worth functions for different groups if you wish. If you do use different functions for each group you can illustrate the advantage of solving problems with parameters when you discuss solutions. For example, if$w(x) = a + Bx$, what values of A and B make sense and what will happen to the fair division point as A and B change?...

    • 20. The Fish Pond
      (pp. 293-293)

      Topics: Differential equations, numerical integration

      Background assumed: Integration (including numerical integration), separable differential


      Summary: Students estimate the volume of water in a pond from a table of depth values. They use the volume to decide how much salt is in the pond by solving differential equations.

      Threads: modeling, approximation and estimation, multiple representation of functions


      1. One of our report guidelines is to assume the reader is a student in a different calculus class. To decide whether or not another student could understand your report try this criterion: Given a different set of depth measurements, could the student...

    • 21. Drug Dosage
      (pp. 293-293)

      Topics: Exponential functions, partial sums and convergence, geometric series.

      Summary: This project deals with models of drug concentration levels. It introduces numerical series and the concept of convergence.

      Background assumed: Familiarity with the exponential function, and ability to solve the differential equation$y' = - ky$.

      Time required: About two weeks for all parts

      Threads: modeling, approximation and estimation


      1. This project introduces numerical series in a concrete context. The students use graphs to learn about the ideas of convergence and divergence by examining the concentration levels in the blood of a drug that has been administered repeatedly.

      2. The case of...

    • 22. Investigating Series
      (pp. 294-294)

      Topics: Sequences, series, and Taylor polynomials

      Summary: This project emphasizes pattern recognition and exploration with sequences and series. It also has students work with multiple representations of sequences, namely, graphical and numerical representations. The project works well as an introduction to series with students who have been introduced to sequences and are familiar with the concept of convergence. The idea in Part 1 is to represent the sequence${b_n}$as a partial sum of the sequence$\left\{ {{a_1},{a_2}, \ldots } \right\}$. By viewing a sequence of partial sums in “closed form” as the sequence$\left\{ {{b_1},{b_2}, \cdots } \right\}$, students should begin to believe that a sequence of...

    • 23. Topographical Maps
      (pp. 294-294)

      Topics: Level curves (intuitively), partial derivatives, directional derivatives (intuitively).

      Summary: This project is intended to introduce visualization of surfaces, partial derivatives and directional derivatives, as a preview of vector calculus.

      Background assumed: Derivative as a rate of change

      Time required: About one week

      Thread: graphical calculus


      1. In attempting to sketch what they can see from their vantage point, students need to realize that they cannot see objects in the landscape that are, for example, behind hills.

      2. In estimating the rate of change in their altitude, they need to distinguish betweeny, the distance north,z, the altitude,...

  8. V Appendices

    • A Sample Curriculum
      (pp. 297-302)
    • B “Sample Guidelines for Projects”
      (pp. 303-304)
    • C Guide to the Threads
      (pp. 305-306)
  9. About the Authors
    (pp. 307-307)