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I am in the process of redesigning the calculus course that I have taught five or six times. What I would like to know is if anyone has some really good examples or exercises that I could either do in class or give as a project. In particular, I've found that I don't have many good examples/exercises that illustrate the awesomeness of the main theorems (Intermediate Value Theorem, Mean Value Theorem, etc.). All levels of difficulty are certainly appreciated. The intent is to have material that I can present or assign here and there throughout the course that goes beyond basic calculus and will challenge even those to whom math comes naturally.

An example of what I'm looking for is something like showing a continuous function on $S^1$ has to map two antipodal points to the same value.

I am teaching Calc I, but Calc II examples are welcome. I believe that Calc III is easy to make interesting.

EDIT: In response to Qiaochu Yuan, Calc I and II together form all of single variable calculus. For Calc I: limits, differentiation, Riemann integration (improper as well). For Calc II: sequences, series, polar coordinates, parametric coordinates.

EDIT: In response to Pete L. Clark, you may assume that these are freshman math majors. We go through $\epsilon$-$\delta$ proofs and such. We may skip the more involved proofs like L'Hopital's Rule. I run my The old book for this course without a text, but was Stewart's "Calculus: Early Transcendentals" is a close fitTranscendentals", but I don't follow any book when I teach.

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I am in the process of redesigning the calculus course that I have taught five or six times. This is a course intended for math majors. What I would like to know is if anyone has some really good examples or exercises that I could either do in class or give as a project. In particular, I've found that I don't have many good examples/exercises that illustrate the awesomeness of the main theorems (Intermediate Value Theorem, Mean Value Theorem, etc.). All levels of difficulty are certainly appreciated.

An example of what I'm looking for is something like showing a continuous function on $S^1$ has to map two antipodal points to the same value.

I am teaching Calc I, but Calc II examples are welcome. I believe that Calc III is easy to make interesting.

EDIT: In response to Qiaochu Yuan, Calc I and II together form all of single variable calculus. For Calc I: limits, differentiation, Riemann integration (improper as well). For Calc II: sequences, series, polar coordinates, parametric coordinates.

EDIT: In response to Pete L. Clark, you may assume that these are freshman math majors. We go through $\epsilon$-$\delta$ proofs and such. We may skip the more involved proofs like L'Hopital's Rule. I run my course without a text, but Stewart's "Calculus: Early Transcendentals" is a close fit.

An example of what I'm looking for is something like showing a continuous function on $S^1$ has to map two antipodal points to the same value.
EDIT: In response to Pete L. Clark, you may assume that these are freshman math majors. We go through $\epsilon$-$\delta$ proofs and such. We may skip the more involved proofs like L'Hopital's Rule. I run my course without a text, but Stewart's "Calculus: Early Transcendentals" is a close fit.