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2016 HSC
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Advanced Mathematics (Hons)/Computer Science @ UNSW (2017– )
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.
Whilst it is fairly obvious that r^n/n! converges to zero if r is positive (in fact regardless of sign but we can wlog take it positive), I don't think it is a fact that should be glossed over by first year students.
In any case, the methods used to bound such a term can be useful in analysing far more subtle "remainder" terms.
Also unanswered. I will get people started:
For k=2 it is not possible.
Proof:
Assume f is a cts function taking every value exactly twice.
There must exist two zeros a < b of the function f.
Between a and b f must have fixed sign (otherwise the intermediate value theorem would provide the existence of a third zero).
Without loss of generality, we assume f is positive in (a,b).
Then f must attain a positive maximum m at some p in (a,b) (Extreme value theorem applied to [a,b]).
Now this means that f(x)=2m must have its solutions OUTSIDE the interval [a,b], without loss of generality we may assume that one such solution q is greater than b. Then by the intermediate value theorem, the equation f(x)=m/2 must have:
-at least one solution in (a,p)
-at least one solution in (p,b)
-at least one solution in (b,q).
This contradiction completes the proof.
As for k=3, it IS actually possible.
Picture the square [0,1]x[0,1] and draw line segments between O, (1/3,1), (2/3,0), (1,1). (Looks like a zigzag.)
Now replicate this curve in each square [k,k+1]x[k,k+1].
The resulting curve is the graph of a continuous function with the sought property.
What about for higher k?
One can also consider the same problem dropping the condition of surjectivity.
(For which positive integers k is it possible to find a continuous function f defined on R that takes every value in its range exactly k times?)
Use the squeeze theorem to find the derivative of e^{x} at x=0
Note you must differentiate from first principles.
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.
Here is a question related to a basic tool in the analysis I do.
Last edited by seanieg89; 13 May 2016 at 11:24 AM.
Related:
I didn't even get up to the last part of my quadrature question but I feel this last part is a good marathon question so I'll drop it here... I could not get it out though.
Good exercise question from a past paper.
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