Calculator/Math Thread

**ben1996123** · 07-04-2012

Can't

have any value, not just 0 or 1?

$0^{0}=0^{n}\times 0^{-n}=\frac{0^{n}}{0^{n}}=\frac{0}{0}$ , which is indeterminate and can have any value, eg.

$\lim_{x\rightarrow 1} \frac{x}{x-1}-\frac{1}{ln(x)}=\lim_{x\rightarrow 1} \frac{xln(x)-x+1}{xln(x)-ln(x)}\rightarrow \frac{0}{0}$

Using L'hopital's rule, this limit must be the same as $\lim_{x\rightarrow 1} \frac{\frac{d}{dx}(xln(x)-x+1)}{\frac{d}{dx}(xln(x)-ln(x))}$

$\frac{d}{dx}(xln(x)-x+1)=\frac{d}{dx}(x)ln(x)+\frac{d}{dx}(ln(x))x-1=ln(x)+\frac{x}{x}-1=ln(x)$

$\frac{d}{dx}(xln(x)-ln(x))=\frac{d}{dx}(x)ln(x)+\frac{d}{dx}(ln(x))x-\frac{d}{dx}(ln(x))$ $=ln(x)+\frac{x}{x}-\frac{1}{x}=ln(x)+1-\frac{1}{x}$

$\lim_{x\rightarrow 1} \frac{\frac{d}{dx}(xln(x)-x+1)}{\frac{d}{dx}(xln(x)-ln(x))}=\lim_{x\rightarrow 1} \frac{ln(x)}{ln(x)+1-\frac{1}{x}}\rightarrow \frac{0}{0}$

Using L'hopital's rule again:

$\lim_{x\rightarrow 1} \frac{ln(x)}{ln(x)+1-\frac{1}{x}}=\lim_{x\rightarrow 1} \frac{\frac{d}{dx}(ln(x))}{\frac{d}{dx}ln(x)+1-\frac{1}{x}}$

$\frac{d}{dx}(ln(x))=\frac{1}{x}$

$\frac{d}{dx}(ln(x)+1-\frac{1}{x})=\frac{1}{x}+\frac{1}{x^{2}}$

$\lim_{x\rightarrow 1} \frac{\frac{d}{dx}(ln(x))}{\frac{d}{dx}ln(x)+1-\frac{1}{x}}=\frac{\frac{1}{x}}{\frac{1}{x}+\frac{1}{x^{2}}}=\frac{1}{2}$

So in this case, $\frac{0}{0}=\frac{1}{2}$ , which also means that (in this case) $0^{0}=\frac{1}{2}$ . Maybe I'm not noticing something, but I think this is correct.

**cmowla** · 07-04-2012

Originally Posted by ben1996123

Can't

have any value, not just 0 or 1?

Oops! I edited my post just as you quoted me! Well, the reason I edited my post is because I realized that I made the mistake of dividing by zero (so 0^0 doesn't equal 0/0 because you would have to accept that 0/0 is defined when it is not). Besides, the limit as x->a of a function (or in the case of L'hopital's rule) doesn't actually equal the function evaluated at a unless the function is defined at x=a. So I don't think we can say that $\underset{x\to 1}{\mathop{\lim }}\,\left( \frac{x}{x-1}-\frac{1}{\ln \left( x \right)} \right)$ approaches 0/0 because that doesn't make sense. We say that "..." 0/0 in order to say that we are currently not looking (analyzing) the given function in the correct manner.

So sorry for the confusion, but I do think that my argument is correct for 0^0.

**CarlBrannen** · 07-04-2012

I voted for Basic Algebra. The toughest class I ever had used a text titled "Basic Algebra I" and "Basic Algebra II". It's also known as "Jacobson".

It has something to do with Rubik's cubes. Section 1.7 is about orbits and cosets:

[IMG]

[/IMG]

**DYGH.Tjen** · 07-07-2012

Went to a small competition yesterday, will probably be cake to some of you so here are the questions

hope I can get steps/workings plus solutions within the hour :P Thanks!

1. Find the area of a convex quadrilateral which has perpendicular diagonals of lengths 15 and 18.

2. A sequence a0, a1, a2 ... is defined by a0 = 2012 and ak+1 = ak^2 +1 for all k ≥ 0. Find the last digit of a2012. whaaat I can't type subscripts. Hope you guys know what I mean. On a side note, what program/online tool do you use to type math symbols and the like?

3. Given three distinct positive integers a,b,c such that a!b!c! = 10!, find a+b+c.

4. An obtuse triangle has side lengths d,30,40 where d is an integer. How many possible values of d are there?

5. The product of two four-digit positive integers is 4^8 + 6^8 + 9^8. What is the smaller integer?

6. A positive integer ending with 8888 has 16 odd positive factors. How many even positive factors does this number have?

Part B

1. Given a triangle ABC and a point M on the side BC. Let [1 and [2 be the circumcircles of triangles ABM and ACM respectively. The perpendicular bisector of BC intersects AM at P. Line BP intersects [1 at D (different from B), and line CP intersects [2 at E (different from C). Prove that ED is parallel to BC.

Note: the circumcircle of a triangle XYZ is the unique circle that passes through X, Y and Z.

2. Let Tn = 1 + 1/n - 1/(n^2) - 1/(n^3)

Find the smallest integer k such that T2T3T4...Tk > 2012

3. There are 15 keys on a keyboard, arranged in a straight line. Two keys are one semitone apart if they are adjacent, and are one tone apart if they are separated by one key. A general scale is a sequence of keys from left to right such that the first key is the leftmost key, and two consecutive keys are either one tone or semitone apart.

Find the number of general scales with 8 keys.

**Stefan** · 07-07-2012

3. Given three distinct positive integers a,b,c such that a!b!c! = 10!, find a+b+c.
I will. As soon as they're actually given.

6. A positive integer ending with 8888 has 16 odd positive factors. How many even positive factors does this number have?
48

**DYGH.Tjen** · 07-07-2012

Reasons/workings/steps please Stefan? :P Especially for Q6?

**Stefan** · 07-07-2012

You can divide the number by 2 three times, so its prime factorization includes 2^3. For every odd factor x, you get even factors x*2, x*4, x*8. Thus 16*3=48 even factors.

On a side note, what program/online tool do you use to type math symbols and the like?
http://www.speedsolving.com/forum/sh...Forum-and-Wiki

**mr. giggums** · 07-07-2012

Originally Posted by Stefan

3. Given three distinct positive integers a,b,c such that a!b!c! = 10!, find a+b+c.
I will. As soon as they're actually given.

3, 5, 7

So the answer is 15.

**Stefan** · 07-07-2012

Bah, you should have only given me the three numbers.

**DYGH.Tjen** · 07-09-2012

Originally Posted by DYGH.Tjen

Went to a small competition yesterday, will probably be cake to some of you so here are the questions

hope I can get steps/workings plus solutions within the hour :P Thanks!

1. Find the area of a convex quadrilateral which has perpendicular diagonals of lengths 15 and 18.

2. A sequence a0, a1, a2 ... is defined by a0 = 2012 and ak+1 = ak^2 +1 for all k ≥ 0. Find the last digit of a2012. whaaat I can't type subscripts. Hope you guys know what I mean. On a side note, what program/online tool do you use to type math symbols and the like?

3. Given three distinct positive integers a,b,c such that a!b!c! = 10!, find a+b+c.

4. An obtuse triangle has side lengths d,30,40 where d is an integer. How many possible values of d are there?

5. The product of two four-digit positive integers is 4^8 + 6^8 + 9^8. What is the smaller integer?

6. A positive integer ending with 8888 has 16 odd positive factors. How many even positive factors does this number have?

Part B

1. Given a triangle ABC and a point M on the side BC. Let [1 and [2 be the circumcircles of triangles ABM and ACM respectively. The perpendicular bisector of BC intersects AM at P. Line BP intersects [1 at D (different from B), and line CP intersects [2 at E (different from C). Prove that ED is parallel to BC.

Note: the circumcircle of a triangle XYZ is the unique circle that passes through X, Y and Z.

2. Let Tn = 1 + 1/n - 1/(n^2) - 1/(n^3)

Find the smallest integer k such that T2T3T4...Tk > 2012

3. There are 15 keys on a keyboard, arranged in a straight line. Two keys are one semitone apart if they are adjacent, and are one tone apart if they are separated by one key. A general scale is a sequence of keys from left to right such that the first key is the leftmost key, and two consecutive keys are either one tone or semitone apart.

Find the number of general scales with 8 keys.

=O bump, can anyone help to complete those questions? :/ I thought they would be easy considering we have people like qq and Ben here :P

@qqwref @ben1996123

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