Projects: Calculus: The Language of Change / Edition 2

Projects: Calculus: The Language of Change / Edition 2

by Keith D. Stroyan
     
 

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ISBN-10: 0126730318

ISBN-13: 9780126730319

Pub. Date: 12/28/1998

Publisher: Elsevier Science & Technology Books

Projects for Calculus is designed to add depth and meaning to any calculus course. The fifty-two projects

presented in this text offer the opportunity to expand the use and understanding of mathematics. The wide range of topics will

appeal to both instructors and students. Shorter, less demanding projects can be managed by the independent learner, while

Overview

Projects for Calculus is designed to add depth and meaning to any calculus course. The fifty-two projects

presented in this text offer the opportunity to expand the use and understanding of mathematics. The wide range of topics will

appeal to both instructors and students. Shorter, less demanding projects can be managed by the independent learner, while

more involved, in-depth projects may be used for group learning. Each task draws on special mathematical topics and

applications from subjects including medicine, engineering, economics, ecology, physics, and biology.

Product Details

ISBN-13:
9780126730319
Publisher:
Elsevier Science & Technology Books
Publication date:
12/28/1998
Edition description:
2ND
Pages:
326
Product dimensions:
7.41(w) x 9.23(h) x 0.65(d)

Table of Contents

1 INTRODUCTION TO THE SCIENTIFIC PROJECTS
2(8)
0.1 What Good Is It?
2(1)
0.2 How Much Work Are They?
2(1)
0.3 How to Write a Project
2(1)
0.4 Help on the World Wide Web
3(2)
PROJECT 1. Linear Approximation of CO(2) Data
5(5)
2 EPIDEMIOLOGICAL APPLICATIONS
10(23)
1.1 Review of the S-I-R Model
10(1)
1.2 Basic Assumptions
11(1)
1.3 Derivation of the Equations of Change
11(2)
PROJECT 2. The 1968-69 New York Hong Kong Flu Epidemic
13(4)
PROJECT 3. Vaccination for Herd Immunity
17(1)
3.1 Herd Immunity
18(1)
3.2 The Contact Number Data
18(1)
3.3 Project Issues
19(1)
3.4 Vaccine Failures
19(2)
PROJECT 4. S-I-S Diseases and the Endemic Limit
21(1)
4.1 Basic Assumptions
21(1)
4.2 The Continuous S-I-S Variables
22(1)
4.3 Parameters for the SIS Model
22(1)
4.4 The Importance of the Contact Ratio
23(1)
4.5 Conjectures
24(1)
4.6 Conclusions
25(2)
PROJECT 5. Max-Min in S-I-R Epidemics
27(6)
3 THE ROLE OF RULES FOR DERIVATIVES
33(10)
PROJECT 6. The Expanding Economy
33(2)
PROJECT 7. The Expanding House
35(3)
7.1 Volume Expansion Explained by Calculus
38(5)
4 APPLICATIONS OF THE INCREMENT APPROXIMATION f[t + Delta t] - f[t] = f' [t] Delta t + Epsilon . Delta t
43(14)
PROJECT 8. A Derivation of Hubble's law
43(4)
PROJECT 9. Functional Linearity
47(4)
PROJECT 10. Functional Identities
51(3)
10.1 Additive Functions
54(3)
5 DIFFERENTIAL EQUATIONS FROM INCREMENT GEOMETRY
57(16)
PROJECT 11. The Tractrix
57(2)
PROJECT 12. The Isochrone
59(2)
12.1 Conservation of Energy
61(2)
PROJECT 13. The Catenary
63(1)
13.1 The Catenary Hypotheses
63(1)
13.2 Parameters
63(1)
13.3 Variables
64(1)
13.4 The Equation for Tension
64(4)
13.5 Optimizing Length and Strength
68(5)
6 LOG AND EXPONENTIAL FUNCTIONS
73(24)
PROJECT 14. The Canary Resurrected
73(2)
PROJECT 15. Drug Concentration and "Biexponential" Functions
75(1)
15.1 Primary Variables of the Model
76(1)
15.2 Parameters of the Model
76(1)
15.3 The Formulas for Concentration
77(3)
15.4 Comparison with Mythical Data
80(1)
15.5 Comparison with Real Data
80(1)
PROJECT 16. Measurement of Kidney Function by Drug Concentration
81(1)
16.1 Variables and Parameters
81(1)
16.2 Overview of the Project
81(1)
16.3 Drug Data
82(5)
PROJECT 17. Numerical Derivatives of Exponentials
87(4)
PROJECT 18. Repeated Exponents
91(6)
PROJECT 19. Solve dx = r[t]x[t] dt + f[t]
7 THEORY OF DERIVATIVES
97(26)
PROJECT 20. The Mean Value Math Police
97(1)
20.1 The Mean Value Theorem for Regular Derivatives
97(2)
20.2 The Theorem of Bolzano
99(1)
20.3 The Mean Value Theorem for Pointwise Derivatives
100(1)
20.4 Overall Speed IS an Average
101(2)
PROJECT 21. Inverse Functions and Their Derivatives
103(1)
21.1 Graphical Representation of the Inverse
104(1)
21.2 The Derivative of the Inverse
105(6)
21.3 Nonelementary Inversion
111(2)
PROJECT 22. Taylor's formula
113(1)
22.1 The Increment Equation and Increasing
114(1)
22.2 Taylor's formula and Bending
115(1)
22.3 Symmetric Differences and Taylor's formula
116(2)
22.4 Direct Computation of Second Derivatives
118(1)
22.5 Direct Interpretation of Higher Order Derivatives
118(5)
8 APPLICATIONS TO PHYSICS
123(34)
PROJECT 23. The Falling Ladder (or Dad's Disaster)
123(2)
23.1 Air Resistance on Dad (Optional)
125(2)
PROJECT 24. Falling with Air Resistance: Data and a Linear Model
127(2)
24.1 Terminal Velocity
129(1)
24.2 Comparison with the Symbolic Solution
130(1)
PROJECT 25. Bungee Diving
131(1)
25.1 Forces Acting on the Jumper before the Cord Is Stretched
131(2)
25.2 Forces Acting on the Jumper after He Falls L Feet
133(2)
25.3 Modeling the Jump
135(2)
PROJECT 26. Planck's Radiation Law
137(1)
26.1 The Derivation of Planck's law of Radiation
138(1)
26.2 Wavelength Form and First Plots
138(3)
26.3 Maximum Intensity in Terms of a Parameter
141(4)
PROJECT 27. Fermat's principle Implies Snell's law
145(2)
27.1 Reflection of a Curved Mirror
147(2)
27.2 Computation of Reflection Angles
149(8)
9 APPLICATIONS IN ECONOMICS
157(18)
Project 28. Monopoly Pricing
157(1)
28.1 Going into Business
157(3)
28.2 Going into Politics
160(1)
PROJECT 29. Discrete Dynamics of Price Adjustment
161(1)
29.1 The Story
161(3)
29.2 The Basic Linear Model
164(1)
29.3 Taxation in the Linear Economy (Optional)
164(1)
29.4 A Nonlinear Economy (Optional)
165(1)
29.5 Taxation in the Nonlinear Economy
166(3)
PROJECT 30. Continuous Production and Exchange
169(1)
30.1 Why Trade?
169(6)
10 ADVANCED MAX-MIN PROBLEMS
175(18)
PROJECT 31. Geometric Optimization Projects
175(1)
31.1 Distance between Lines
175(2)
31.2 Distance between Curves
177(1)
31.3 An Implicit-Parametric Approach
178(1)
31.4 Distance from a Curve to a Surface
179(2)
PROJECT 32. Least Squares Fit and Max-Min
181(1)
32.1 Introductory Example
181(1)
32.2 The Critical Point
182(1)
32.3 The General Critical Equations
183(2)
PROJECT 33. Local Max-Min and Stability of Equilibria
185(1)
33.1 Steepest Ascent
185(1)
33.2 The Second Derivative Test in Two Variables
186(7)
11 APPLICATIONS OF LINEAR DIFFERENTIAL EQUATIONS
193(24)
PROJECT 34. Lanchester's Combat Models
193(2)
34.1 The Principle of Concentration
195(1)
34.2 The Square Law
195(1)
34.3 Guerrilla Combat
196(1)
34.4 Operational Losses (Optional)
197(4)
PROJECT 35. Drug Dynamics and Pharmacokinetics
201(2)
35.1 Derivation of the Equations of Change
203(7)
35.2 Where Do We Go from Here?
210(1)
35.3 Periodic Intravenous Injections (Optional Project Conclusion)
210(3)
35.4 Steady Intravenous Flow
213(1)
35.5 Intramuscular Injection -- a Third Compartment
214(3)
12 FORCED LINEAR EQUATIONS
217(20)
PROJECT 36. Forced Vibration -- Nonautonomous Equations
217(1)
36.1 Solution of the Autonomous Linear Equation
218(1)
36.2 Transients -- Limiting Behavior
218(1)
36.3 Superposition for the Spring System
218(2)
36.4 Equations Forced by Gravity
220(1)
36.5 Equations Forced by Sinusoids
221(2)
36.6 Nonhomogeneous IVPs
223(2)
PROJECT 37. Resonance -- Maximal Response to Forcing
225(1)
37.1 Some Useful Trig
225(1)
37.2 Resonance in Forced Linear Oscillators
226(1)
37.3 An Electrical Circuit Experiment
227(1)
37.4 Nonlinear Damping
227(2)
PROJECT 38. A Notch Filter -- Minimal Response to Forcing
229(1)
38.1 The Laws of Kirchhoff, Ohm, and Coulomb
229(2)
38.2 Steady-State Solution
231(2)
38.3 A Check on a[t] 2
233(1)
38.4 Where's the Min?
233(4)
13 APPLICATIONS IN ECOLOGY
237(24)
PROJECT 39. Logistic Growth with Hunting
237(1)
39.1 Basic Fertility
237(1)
39.2 Logistic Growth
238(1)
39.3 Voodoo Discovers the Mice
239(2)
PROJECT 40. Predator-Prey Interactions
241(1)
40.1 Bunny Island
241(4)
40.2 Rabbit Island
245(2)
PROJECT 41. Competition and Cooperation between Species
247(1)
41.1 Biological Niches
247(4)
41.2 Cooperation between Species
251(2)
PROJECT 42. Sustained Harvest of Sei Whales
253(1)
42.1 Carrying Capacity, Environmental and Mathematical
253(2)
42.2 The Actual Carrying Capacity
255(6)
14 DERIVATIONS WITH VECTORS
261(36)
PROJECT 43. Wheels Rolling on Wheels
261(1)
43.1 Epicycloids
262(2)
43.2 Cycloids
264(1)
43.3 Hypocycloids
265(2)
PROJECT 44. The Perfecto Skier
267(1)
44.1 The Mountain's Contribution
268(2)
44.2 Gravity and the Mountain
270(1)
44.3 The Pendulum as Constrained Motion
271(1)
44.4 The Explicit Surface Case
271(2)
PROJECT 45. Low-Level Bombing
273(4)
45.1 Significance of Vector Air Resistance
277(2)
PROJECT 46. The Pendulum
279(1)
46.1 Derivation of the Pendulum Equation
280(2)
46.2 Numerical Solutions of the Pendulum Equation
282(2)
46.3 Linear Approximation to the Pendulum Equation
284(3)
46.4 Friction in the Pendulum (Optional)
287(1)
46.5 The Spring Pendulum (Optional)
287(4)
PROJECT 47. Using Jupiter as a Slingshot
291(1)
47.1 Setting up the Problem: Scaling and Units
292(1)
47.2 Newton's Law of Gravity
293(1)
47.3 Newton's F = ma Law
294(1)
47.4 Numerical Flights out of the Solar System
294(3)
15 CHEMICAL REACTIONS
297(16)
PROJECT 48. Stability of a Tank Reaction
297(1)
48.1 Mass Balance
297(1)
48.2 Arrhenius' Law
298(1)
48.3 Heat Balance
298(2)
48.4 Stability of Equilibria
300(2)
48.5 Forced Cooling (Optional)
302(1)
PROJECT 49. Beer, Coke, & Vitamin C
303(1)
49.1 Enzyme-mediated Reactions
303(1)
49.2 Molar Concentration and Reaction Rates
303(2)
49.3 The Briggs-Haldane Dynamics Approximation
305(1)
49.4 The Michaelis-Menten Dynamics Approximation
306(1)
49.5 Blood Ethanol
307(1)
49.6 Blood CO(2)
308(5)
16 MORE MATHEMATICAL PROJECTS
313(13)
PROJECT 50. Rearrangement of Conditionally Convergent Series
313(4)
PROJECT 51. Computation of Fourier Series
317(6)
PROJECT 52. The Big Bite of the Subtraction Bug
323(3)
17 ADDITIONAL PROJECT REFERENCES
326
1001.1 Lowering the Water Table
326(1)
1001.2 The Light Speed Lighthouse
326(1)
1001.3 Horizontal, Vertical, and Slant Asymptotes
326

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