He gave me a half hour lecture illustrating the basics of integration and differentiation. It was the most useful half hour of my life, and it got me through the semester. Prof. Gomez pretty much saved my @ss.
No, my public high school had not offered calculus in any way, shape or form. My fellow freshmen couldn't believe I had come to Caltech utterly ignorant of calculus.
It took until my sophomore year to get my act together. Fortunately, freshman year at Caltech was pass/fail, and I barely scraped by that year.
1. attend all lectures
2. do all of the homework on time
3. make sure I understood how to do 100% of the homework problems
4. get help with anything I didn't understand
5. write down everything that the prof wrote on the blackboard
6. clean up my handwriting so it was legible
7. lectures/homework/studying always came first
Ever since, when working with someone who has difficult-to-decipher handwriting I advise them that their handwriting is just fine for themselves, so when you need others to read it correctly, in those situations approach it like calligraphy instead.
Naturally I typed my homework and did "calligraphy" on exams.
Also the homework contributed so little credit toward your final grade that a single wrong answer on a midterm or final counted against you mathematically more so than turning in no homework at all, so some good students treated it like they could make up for many hours shortage on homework in just five minutes of the time spent acing the exams. But it turned out to be impossible to get an A for the course if all of the homework was not completed and all correct as well. This was by design and for students to figure out for themselves. They didn't give very many A's in Chemistry.
The point of the homework was to teach you the material. Skip it at your peril. Mastering it meant you were reasonably prepared for the exams.
As far as I recall, I was the only freshman who did not know calculus. I can only infer from that that I did not get the message, or ignored it. I certainly received no useful guidance from high school.
Similarly to you, I was educated in North America where ironically integrals were not part of the curriculum, but when I arrived in Europe I had to take 3 entrance exams: English (the exam is very awkward for the native speakers, you get 2 questions: what is your name and why are you hear after they notice that you do not have an accent), math (their curriculum with the missing content I did not have) and German.
As I understand, if a local does not have a matura in the required fields, they have to take similar examinations which should prepare them.
I think you had a very nice professor though.