I can't word properly yet.
Calculus & Analysis Marathon & Questions
This is a marathon thread for single-variable calculus & analysis (mainly real and maybe complex analysis). Please aim to pitch your questions for first-year/second-year university level maths. Excelling & gifted/talented secondary school students are also invited to contribute.
(mod edit 7/6/17 by dan964)
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Thought it'd be a good idea to start a marathon for users to post and answer first year uni calculus problems.
First question:
Last edited by dan964; 7 Jun 2017 at 4:40 PM.
Bachelor of Science (Advanced Mathematics) @ USYD
I can't word properly yet.
Last edited by leehuan; 24 Apr 2016 at 5:05 PM.
Bachelor of Science (Advanced Mathematics) @ USYD
lel these look like tute or lecture qns
(I actually remember the above question haha)
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Last edited by leehuan; 24 Apr 2016 at 5:40 PM. Reason: I may not have put in the add on question myself after Trebla...even though he meant USyd
Is this thread's main intention as a Q&A, or as a challenge thread?
(A bit tricky)
For which positive integers k is it possible to find a continuous function f:R->R that such that
f(x)=y
has exactly k solutions x for every real number y?
Indeed that is the essence of the question, I don't want to spoil it too quickly. It is very good exercise to reach a conclusion yourself and try to rigorously justify it. (The main tools at your disposal being properties of continuous functions like the intermediate and extreme value theorem).
Bachelor of Science (Advanced Mathematics) @ USYD
I hope...
For completeness sake
For this question, because infinity is not a measurable quantity (and I'm not sure if you can assume that it is an integer) we can't say for sure what the behaviour of f is if f(x)=y yields an infinite number of solutions.
At least, my simple brain can't visualise f anymore
Not even sure what I was thinking at the time now
Here's my attempt:
Since x is positive, we can invoke the following inequalities:
It is easy to show that the left hand side is bounded from above by 3/2, and the right hand side is bounded from below by the same value.
It is then sufficient to show the left bound is monotonically increasing, and the right bound is monotonically decreasing, for sufficiently large x.
If I am a conic section, then my e = ∞
Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.
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