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

- ProblemPlus 3. max-min gutter cross-section.
- ProblemPlus 8. maximum of a function

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.