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ii) Make f(t) = e^t. Then simplify out, once integral is simplified, sub in x=1.If you don't mind me expanding on this
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I'm guessing you're not looking for a proof by induction?
Yep.I'm guessing you're not looking for a proof by induction?
If you simplify the LHS, it becomes:
Quite clever, I'm unsure about how to prove that:If you simplify the LHS, it becomes:
My first thought is to construct an infinite product that actually equals to. The most simple form would be:
Multiplying all the 2's together will add the powers to get:
Now for this to equal, we want:
The easiest way to do this is to let:
such that:
So we get the infinite product:
Which in expanded terms is:
Now lets consider what we can do with this infinite product:
1) Since all terms are less than 1, then by multiplying by an extra term decreases the actual product each time. Hence the product will be a minimum when there are infinite terms.
2) Since the product is a minimum when there are infinite terms, then all we have to do is show that the infinite product of the result that needs to be proven is larger than the infinite product we constructed.
Now trying to figure out how to finish this off...
(added in mark count as an estimated length of soln)
Similar to what I had in mind:
LHS - RHS, make it a common denominator, the result simplifies into proving:
(don't feel the obligation to answer the questions I'm posting in order, answer whichever ones you want to)
(and gives you the equality condition of all numbers being equal).LHS - RHS, make it a common denominator, the result simplifies into proving:
This can be done by expanding the LHS, you will get, an n number of 1's
Andpairs of reciprocals, i.e. x1/x2 + x2/x1, an nC2 number of these pairs
And we know that:
That means in the above expression:
proof is then complete.
Oops yeah hehe.(and gives you the equality condition of all numbers being equal).
I think this counts under Polynomials:soz to kill the party but does anyone have any harder Q's on topics other than Harder 3 unit? i mean it is the HSC 4U Marathon after all!
As no-one else seems to be answering it:I think this counts under Polynomials:
First lets model the particle travelling in circular motion with parametric equations:Here is a mechanics one:
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a) (I should of specified that a > 0, b > 0, c > 0)Just some inequalities, they are not in order of difficulty, nor will the answer to one necessarily help with the others:
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