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.