Proposals for calculus modules/labs. Version 3. Under construction.
These "labs" are intended (for now at least) as introductory and experimental to students. They should provide hands-on simulations but preferably be accessible before the topic has been studied and be used by the students to develop their intuition and understanding about the basic principles involved. In addition, the industrial/naval aspects should give them some appreciation of "what it's good for".
I. Torque wrench. Partially implemented at http://www.usna.navy.mil/MathDept/mdm/wrench9b/wrench9b.html .
Mathematical/pedagogical goal: to improve student understanding of the crossproduct
Place in syllabus: beginning of vectors (was start of calculus 3, now is end of calculus 2 and is reviewed at start of calculus 3).
Real life application: to torque wrench, used in tightening bolts, perhaps related to gyroscopes
To do: add descriptive links about torque, check out http://torquerepair.com/ , . . .
II. Ocean Waves
Mathematical/pedagogical goal: to improve students understanding of sine & cosine, periodicity, and how functions change by shifting and stretching, and the function concept in general.
Place in syllabus: first chapter, calculus 1.
Real life application - ships in ocean. (Also internal waves for subs?) Sound waves (sonar)? Electromagnetic waves (radar)? Other periodic functions - tides, sunrise, engines, …
Types of questions/experiments Given y=a+b*sin(c*(t-d)), graphed, they can change parameters a,b,c,d, say initially a=1,b=2,c=2,d=0 (y=1+2sin(2t)). See http://www.physics.upenn.edu/courses/gladney/mathphys/java/sect1/subsubsection1_1_4_2.html middle of page.
Our version so far 09 June '99
To do - research descriptive links, make java applet to sketch curve allowing input of a,b,c,d , embed in web page.
III. Velocity/Speed - graph of ship's position (traveling in straight line) versus time. Choose second point to get secant line. Up to user to estimate slope of secant, and from that, slope of tangent to guess instantaneous speed. Also do average speed.
See http://www.math.gatech.edu/~carlen/applets/archived/ClassFiles/Secant.html
Surfing copy.Mathematical/pedagogical goal: understand concept of derivative as limit of difference quotients (mostly geometrically but also algebraically), understand derivative as instantaneous vice average rate of change.
Place in syllabus: Section 2.1 Velocity introduction, section 2.6 tangent lines and velocity.
Real life applications: ship speed, plane speed, bomb speed, parachutist speed, bullet (see Schattschneider p.447, Ebersole p. 295, Johnston #2, p.199), diving tower (Larson p. 159).
IV. Integration - approximating area
Mathematical/pedagogical goal: understand the interpretation of integral as area, concept of limit of approximating rectangles/trapezoids, estimation of same
Place in syllabus: Start of calculus II or end of calculus I - Section 5.1 Areas, Section 5.2 Definite Integral, Section 5.8 Approximate Integration
Real life applications: Finding areas of: forested region for marines to search, deck of ship, …
Type of activities: Draw left/right/midpoint/trap approx? Estimate area. (see Schattschneider p. 447)
V. Volume - using slicing and integration.
Mathematical/pedagogical goal: understand the interpretation/application of integral as volume. Understand the slicing method and perhaps disk method.
Place in syllabus: Calculus II, Section 6.2 Volumes
Real life applications: Finding volume of fuel tank, displacement of submarine, …
Something like http://www.ies.co.jp/math/java/renshi/renshi.html perhaps. Show shape, slices, resulting area graph?
VI. Related rates.
Mathematical/pedagogical goal: understand how rates can be related.
Place in syllabus: Calculus I, Section 4.1 Related Rates.
Real life applications: Ships on perpendicular courses, what about rate of change of distance between them?
Perhaps animate situation, ask easy questions like sign of answer, formula for distance, etc.? Or aircraft. See Freilich p. 173.
Other possibilities:
A. Exponential function - similar to waves, except exp vice sin. Decay (nuclear) or temperature (so asymptotic limit is non-zero) or parachutist - terminal velocity, growth - economics/interest/compounding (somewhat like#2).
B. Max/min problem like VI. See Freilich p. 160 and #38 p. 352, Goldstein & Lay pp. 147-9,153,167,168.
C. Integration - inertial navigation, Freilich p. 238, broken odometer Johnston p. 200.
D. Graphing f, f' - Economics, cost, average cost, marginal cost, see Cannone-Williams pp. 265-9.
E. Parametric equations: bullet/baseball trajectories with and without air resistance, Ostebee & Zorn p. 367.
URL: http://www.nadn.navy.mil/MathDept/cdp/Proposal2.html