Calculus Homework Generic Outline
Steve Nuchia, University of Tulsa
You are writing to convince a fellow student of the correctness of your answer. Your (imaginary) audience knows all the math you know but has not studied the problem you are solving. There is an art to deciding when detail is needed to avoid the "then a miracle occurs" step and when detail just gets in the way of a clear presentation. Learning that art is an important objective in this course.
A good write-up begins after you have solved the problem. Nobody wants to read the history of how you discovered the solution. It may have been a lot of work but that is not an excuse for making your reader work hard! Once you really understand the problem, ask yourself what steps are really key and what mathematical and real-world ideas are behind those steps. Only then should you begin to write.
The following outline should serve as a starting point and perhaps a kind of checklist. Not every problem will fit exactly – you should adapt the outline to the problem and not the other way around!
Often the problem assigned will seem to be ambiguous or incomplete. Sometimes it actually is ambiguous! Your restatement of the problem plays a critical role here. If you have misinterpreted the problem but you correctly solve the problem as you understand it you will probably receive full credit. I have given students full credit for turning in good solutions to the wrong problem entirely.
Imagine that you have been asked by your boss or a client to solve this problem. If you give them the solution in the form "here is the answer to the question you asked" and you have in fact misunderstood the question, you will have given an incorrect answer. On the other hand if you say, "here is how I understand your question, and here is my solution" then your boss will have a fighting chance to notice that your understanding of the problem does not match her own.
Good writing does not happen in a vacuum. It results from practice informed by reading and constructive criticism. As you read mathematical arguments, not just in the Calculus textbook but in all your technical courses, pay attention to the way they are written. Some will be good, some bad; many will be hard for you to understand. You can learn a lot by thinking about why a written mathematical argument is hard for you to follow.