Math Is Fun Forum

Hi!

First of all, I think your equations do not define a ring, but a sphere... At least you write nothing about being a hole centered at k and of any radius.

Anyway, the following website has the solution and, apparently, you got close to the solution!
http://plus.maths.org/issue1/puzzle/index.html

Hope it helps
Jose

Hi Greaterpathmagician

The function means that you asign the value 4 to the power of x to y = f(x).

So, for 1.a (I see some light lines, so you may have done, but just in case)

(i) 4^2 = 16
(ii) 4^1.5 = 8
(iii) 4^0.5 = 2

And for 1.b, you have to search the values the other way arround. Call the vertical axis y, and then:

(i) y = 1 -> x = 0
(ii) y = 4 -> x = 1
(iii) y = -2 -> you cannot find it. Indeed, 4^x is never < 0
(iv) y = 10 -> x around 1.7

Hope it helps!

Hi!

According to what you write, you're defining an element wise derivative, after all!

I would say:

Jose

Hi!

I have no books with me to check the demo, but I believe you can show it following a couple of links from Wikipedia:

1. Jordan normal form, http://en.wikipedia.org/wiki/Jordan_form

First of all, since A is a general matrix, it may not be diagonalizable (in this case it would happen when it doesn't have n independent eigenvectors). But then we can use its normal Jordan form:

2. Now lets work with the Jordan form J of A, and from http://en.wikipedia.org/wiki/Dunford_decomposition we know:

where D is diagonal and N is nilpotent (i.e. N^m = 0 for some integer m).

3. Then, from the above definition and the "Generalization" in http://en.wikipedia.org/wiki/Matrix_exponential:

If D commutes with N (DN = ND), we can write:

4. In this case, still following Wikipedia:

5. Now, using the previous partial results:

Anybody, please correct me.

Hope it helps, anyway

Jose

I agree again bobbym

Corrected

Thanks bobbym!

However, mathkeep you're right, jaja. I think I mixed things up a little, and I was lucky, since I wrote the first equation down using:

But, the expression that you give is all the time with the power of n. The solutions are correct, though, but just skip the substitution using z...

I am going to think about the problem in general again!

Indeed, it would be something like:

But this is not so easy any more!

Jose

This is the only way I see for 19, but it is a little weird!

First consider:

where the substitution used has been alpha x^2 = u^2

Then,

Jose

I would like to propose another integral, which took me really long to solve! (I would grade it at least with **)

If anyone feels this should be another post, since both integration by parts and substitution need to be used, just let me know

Otherwise, have fun with the problem. Here we go:

where

Enjoy!

I found 16 is indeed much easier this way:

where for the first step I have used x = exp(ln(x)), and the substitution is then 4x^2 ln(3) = t^2. Finally, the last step is done evaluating the erf function.

Jose

Hi again!

The exact solutions are really long to type, but you can easily find them using:
http://www.quickmath.com/webMathematica3/quickmath/page.jsp?s1=equations&s2=solve&s3=advanced

Just type
1/4 + 4*z + 17*z^2 + 8*z^3 + z^4 = 3*z^2 *(1-z^2)
in the equations box

and z in the variables box. Then click solve!

If you want the result for x, use the original equation directly.

By the way, bobbym is right extending the real solutions + 2 pi n. What about the complex solutions?

Jose

Hi

I would do the following:

Using sin(x) = z

If I did the calculations correctly, reordering and solving first for z and then for x, you should obtain the answer directly given by the quickmath website:

Jose

(Edited, since I didn't write the last equation correctly!)

Hi!

I think the easiest way is to solve it backwards:

I hope I did the calculations OK!

Regarding the last question, you already know one answer, since (x - 1) is a factor, the solution is x = 1.

Use this website to expand f(x) and solve for other x:
http://www.quickmath.com

Jose