Math 3F03 Kritik Assignment # 1
北美数学代写 Solve the following problems, clearly outlining and explaining your thought process, in such a way that you and your peers can easily ···
Creation Phase. 北美数学代写
Solve the following problems, clearly outlining and explaining your thought process, in such a way that you and your peers can easily understand your solutions. You can either write very neatly on paper, and scan your solutions with your phone (use the Dropbox app rather than just a photo,) or (even better) type your solutions (using for exemple Overleaf, which will be described in Pip’s tutorial this Friday,) and upload to your Kritik account. This step is due by 11:00pm on Wednesday Sept 15.
Consider the ODE:
x˙ = f (x),
where f ( ) is a C1 function on R with the property that f (c) = 0 fr(c) = 0 (namely f does not vanish on intervals). In this assignment you will show that, contrarily to the case of second order ODE’s, solutions to first order ODE’s on the line cannot oscillate. You will proceed via a contradiction argument.
So, suppose that for some T > 0, there is a solution x(t) with x(t) = x(t + s), 0 < s < T and
x(t) = x(t + T ).
北美数学代写
A.Showthat ∫ t+T f (x(τ )) dx dτ = 0, by using an appropriate
B.Obtaina contradiction to the hypothesis on x(t) by using (A) and that ∫ b (f (x(τ )))2 dτ ≥ 0,
∀ a < b, with equality only when f (x(τ )) ≡ 0 for τ ∈ (a, b).
C.Concludethat there cannot be any non-trivial periodic solutions to x˙ = f (x) with f as
above, and that non-trivial solutions to first order ODE’s on the line cannot have oscillations (more
precisely that it is not possible to find t1 /= t2 with x(t1) = x(t2) and x(τ ) x(t1), ∀τ ∈ (t1, t2).)
Evaluation Phase. 北美数学代写
After the due date you will receive 4 submissions of solutions to the above problems, which you are to evaluate using the criteria in the rubric: Results, Explanation, Interpre- tation and Presentation. The written form of the solution and a clear explanation of the approach used is as important as the mathematical content.
The idea is to see how your classmates solve and present the results, and to give feedback to your peers. You will be required to make a written criticism of each submission, indicating places where the student has done well and where improve- ment is needed. (You will be evaluated on how useful your remarks are!)
The ideal is what you would expect in a textbook: a complete and perfectly written solution which explains everything in a concise and clear way. You will receive a solution which you may use to help you in the evaluation phase. Remember that there may be more than one correct and clear method for proving any statement in mathematics.
The Evaluate step is due by 11:00pm on Monday September 20.
Feedback stage. 北美数学代写
Finally the Feedback step is due by 11:00pm on Wednesday September 22.
Important note: Keep in mind that there are more than one correct. And clear method for proving any statement in mathematics. Whenever a novel approach or a clear explanation impresses you, don’t hesitate to express it. I expect that you will find this experience interesting and rewarding. And that you will evaluate your peers in the same way that. You would appreciate receiving feedback about your results. So be respectful of the feelings of others when making criticism of their work. And write your comments as you would like to receive them: useful and constructive.
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