PROJECTS FOR CALCULUS III

The following take home problems test the student's mathematical knowledge as well creativity. I recommend the students be given at least a week to tackle these problems. Working in teams may also be beneficial to the students. Suggestions from people at Ithaca College who have used team projects may be helpful. Students should submit their work, whenever appropriate, as Maple worksheets. Please send comments to rakesh@udel.edu .

IMPORTANT : A log of the projects from the list below, used in Math 243, is kept so that instructors can determine if a praticular project was assigned recently. So any Math 243 instructor who uses one of the projects below is requested to send a message to rakesh@udel.edu so that it may be entered in the log.

• VECTORS
• Chicago-Bombay Determine the length of the shortest path from Chicago to Bombay - and related questions. ( latex version )
• Friction Determine the minimum force required to stop a weight from sliding down an inclined plane. ( latex version )
• Equilibrium Determine the equilibrium position of an object attached by ropes going over pulleys to 3 other objects. ( latex version )
• Suspended object If an object is suspended by three ropes determine the tension in the ropes. ( latex version )
• Parabolic Mirrors Show that rays parallel to the axis of a paraboloid mirror are focussed at a point and determine the focus. ( latex version )

 
• PROJECTILE MOTION
• Range Find the range of a gun assuming the shell experiences gravity and air resistance.

( latex version )
• Target Practice Find the elevation of a gun to reach a target behind a hill. ( latex version )

 
• SURFACE
• Parameterisations (ps file) Determine the parametric equation for a cycloid, a torus, a mobius strip. ( latex version needs eps sketch )
• Drawing Solids Using Maple to plot a solid bounded by several surfaces. ( latex version )

 
• DIFFERENTIATION
• Implicit Differentiation Using implicit differentiation detremine the rate at which the angles of a triangle change as the length of the sides is changed. ( latex version )
• Calculus of Variation Find the minimum value of a functional by approximating the unknown function by (a) a polynomial (b) a piecewise linear function and using optimization methods learnt in M243. ( latex version )
• Catenary A chain made up of three rods linked together is suspended. Determine the equilibrium position of the rods - the equilibrium position has the least potential energy. For a chain made up of more than three rods the equations get too complicated and require special numerical techniques. ( latex version )
• (Plywood Box) Given a sheet of plywood of size a feet by b feet. What is the box of largest volume that can be constructed from this sheet, assuming that each side of the box is made up of only one piece of plywood? Remark : This is perhaps a difficult question to analyze.

 
• INTEGRATION
• Integration and Summation Approximate the mass of an inhomogeneous plate by subdiving the plate into small squares etc. ( latex version )
• Cartography Given the vertices of the polygonal approximation ofthe State of Delaware determine the area, the perimeter, and the "center" of the state. Green's theorem leads to an interesting and quick way for computing these quantities. Remarks : I have not yet been able to find any source for the polygonal approximation to the state. ( latex version )

 

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