﻿ MAT2500 Challenge Problems

# MAT2500 Challenge Problems

For those who actually enjoy mathematics, the Stewart Calculus textbook has Problems Plus at the end of each chapter which require just a bit more thinking compared to the homework problems. bob added some others to the list below.

Here are a few selected problems with additional instructions whose successful solution without collaboration, if written up as a clear self-contained mathematical report (or using a clearly documented Maple worksheet), can substitute at a 10/10 for one of your lowest quiz grades. Just getting the answer does not cut it. [I have solved each of these problems myself in order to assess their reasonableness for students.]

chapter 12. vector algebra

• ProblemPlus 8. maximum volume solid region with given projections onto the 3 coordinate planes: square, triangle, circle.

chapter 13. space curves

• ProblemPlus 3. projectile motion.
• Figure out how to parametrize this star shaped parametrized curve [gif] by observing the relative frequencies of the rotating circles and the usual parametrizations of those circles, and find the exact angle needed to rotate this star so it is "upright".
• Figure out how to parametrize the motion of a corner of the square wheel which rides horizontally on a cycloid roadbed.

chapter 14. surfaces

chapter 15. multivariable scalar integration

• ProblemPlus 7. maximal area circumscribed ellipse.
• ProblemPlus 13. volume between an ellipsoid and a tilted plane passing thru 3 of its axis intercepts.
This is a nice problem, and helpful for really understanding triple integration, but the direct approach only leads to an unsuccessful attempt, which forces you to a numerical integration in a concrete example with fixed scale parameters, and hence to no general formula in terms of the arbitrary scale parameters.
[Hint: this is only do-able if you make an obvious change of variables to factor out the parameters!]
• Use approximate Riemann sum discrete double integration to determine the volume of the 3d heart and what percentage it fills of the minimal box which contains it. This is a nontrivial challenge and you have to do a bit of Maple programming with numerical procedures to get this to work.

chapter 16. vector field integration

• ProblemPlus 1. alert astro majors! explore solid angle for understanding areas on the celestial sphere, with the divergence theorem.
• ProblemPlus 2. maximizing a closed loop line integral. also find its maximum value.
• ProblemPlus 5. a typical vector analysis identity, proven by brute force expansion.
• ProblemPlus 6. looks very interesting, but I have not myself looked at how to work it yet.