Wikipedia:Reference desk/Archives/Mathematics/2010 July 12

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

July 12

order of growth

To show that log x grows more slowly then x^c (c > O) is it sufficient to show that log x/x^c goes to 0? How does that mathematically imply that log x is less then x^c for sufficiently large x. Thanks-Shahab (talk) 06:38, 12 July 2010 (UTC)

I can answer the second part. If $\lim _{x\rightarrow \infty }{\frac {\log x}{x^{c}}}=0$, then for every $\epsilon >0$ there exists $N\in \mathbb {R}$ such that for all $x>N$, $\left|{\frac {\log x}{x^{c}}}\right|<\epsilon$. Choosing $\epsilon =1$ gives us that for large enough $x$ (larger than a finite positive real number $N$), ${\frac {\log x}{x^{c}}}<1$, which implies that $\log x<x^{c}$ (since $x^{c}>0$ whenever $x$ is positive). RJaguar3 | u | t 06:44, 12 July 2010 (UTC)

Thanks for answering both parts.-Shahab (talk) 06:50, 12 July 2010 (UTC)

Calculus/Derivatives

Find the equations of both lines that are tangent to the curve y=1+x^2 and are parallel to the line 12x-y=1

—Preceding unsigned comment added by KRmiwaD93 (talk • contribs) 07:38, 12 July 2010 (UTC) I need some help finding the equation. Could I have an explanation please? Thanks.

What have you tried so far? 71.141.88.179 (talk) 08:08, 12 July 2010 (UTC)

The question didn't actually ask for both lines did it? Are you sure it didn't say the line that is both tangent to.. and parallel to ...? Dmcq (talk)

That's the way I read it. —Anonymous Dissident^Talk 08:45, 12 July 2010 (UTC)

If the line has to be parallel to $12x-y=1$ then it must have the same gradient. You then need to find out at which value of x the curve $y=1+x^{2}$ has that gradient. Readro (talk) 09:17, 12 July 2010 (UTC)

Well one can't have 'both lines' as no two points on the quadratic have the same tangent direction whereas a straight line only has the one tangent direction. Dmcq (talk) 11:32, 12 July 2010 (UTC)

In general, there is only one tangent to a given parabola that is parallel to a given line. The exception is that there are no tangents parallel to the axis of the parabola.--RDBury (talk) 13:36, 12 July 2010 (UTC)

Rewrite the linear part as $y=12x-1$, and then you mentioned derivatives in the title, what does differentiating with respect to x give and what use is that? Dmcq (talk) 14:33, 12 July 2010 (UTC)

Trig question

Resolved

I'm having a bit of trouble trying to solve the following identity verification:

[ (sec - tan)^2 + 1 ] divided by [(sec)(csc) - (tan)(csc)] entire fraction set equal to 2tan

I have gotten as far as converting the denominator in terms of sine and cosine, but I'm a bit stuck from there. Any help is appreciated! Thanks, WordyGirl90 12:07, 12 July 2010 (UTC)

If you express the numerator in terms of cos and sin too it works out pretty simply - don't forget a famous identity involving the squares of cos and sin. AndrewWTaylor (talk) 12:57, 12 July 2010 (UTC)

If you divide said famous identity by the square of cos x then you get another identity in terms of tan and sec, which you can use if you fancy cancelling some terms first before converting to sin and cos. Readro (talk) 13:06, 12 July 2010 (UTC)

Thanks guys. A classmate solved it basically through factoring. I ended up using his method. Thanks anyway! WordyGirl90 19:58, 12 July 2010 (UTC)

integers solutions

Resolved

Hello. I have the following equation in three variables: 3x + 4y = 7z. I want to find out solutions to it which satisfy the following criteria: one they are all positive integers, secondly, they are all distinct, thirdly x,y,z are all less then 30. The only ways I know are to set it up as an integer programming problem, and use a software which I have, or to write a computer program which uses brute force. I want to then find out solutions for 2x + 5y = 7z, and in general I want to find out a way to solve ax + by = (a+b)z. What would be the appropriate way? Thanks-Shahab (talk) 14:25, 12 July 2010 (UTC)

$x\equiv y{\pmod {7}}$. For the general case, $x\equiv y{\pmod {a+b}}$. Gandalf61 (talk) 14:40, 12 July 2010 (UTC)

...assuming (without loss of generality) that a and b are coprime.—Emil J. 14:47, 12 July 2010 (UTC)

To write it more explicitly, integer solutions of the equation are exactly triples of the form (x, x + k(a + b), x + kb), where x and k are integers.—Emil J. 14:51, 12 July 2010 (UTC)

Gandalf61 how do I solve $x\equiv y{\pmod {7}}$ and from where did you get it. Thanks to you both for responding so fast.--Shahab (talk) 15:18, 12 July 2010 (UTC)

$x\equiv y{\pmod {7}}$ just means that x and y leave the same remainder when you divide them by 7 - see modular arithmetic. For a parametric solution, just pick two integers x and k and use Emil J.'s solution (x, x + 7k, x + 4k). Geometrically, these solutions all line on the plane spanned by the vectors (1,1,1) and (0,7,4). Gandalf61 (talk) 15:46, 12 July 2010 (UTC)

No, no...I mean how did you deduce that x,y such that $x\equiv y{\pmod {7}}$ give the solutions.-Shahab (talk) 16:37, 12 July 2010 (UTC)

On the one hand, if 3x + 4y = 7z, then 7 | 3x + 4y, hence also 7 | 4(y − x). As 7 and 4 are coprime, this implies 7 | y − x, i.e., $x\equiv y{\pmod {7}}$. On the other hand, if this conguence holds, then z = (3x + 4y)/7 = x + 4(y − x)/7 is an integer, and solves 3x + 4y = 7z.—Emil J. 16:43, 12 July 2010 (UTC)

Thanks, its all clear now-Shahab (talk) 16:51, 12 July 2010 (UTC)