Recent content by Karlsen

Oh, sorry. I guess I Didn't think long enough. int(t=1,t=3, 1/t) > A_2 + A_3 > 1, so ln(3) > 1, hence e < 3. bam!

I have to carry out a few steps to show that 2 < e < 3. (e as in 2.71828..) Let f(t) = 1/t for t > 0. (a) Show that the area under y = f(t), above y = 0, and between t = 1 and t = 2 is less than 1 square unit. Deduce that e < 2. This is easy, just integrating and getting ln(2). From...

23*5 - 9 - 8 - 7 - 1 + 6 + 4 That's also one.

Ok, I have a solution for you, but I don't know how to use the math symbols here, so it might look a bit bad. You have x^3 - 6x^2 + 5x - 1 = 0, where a,b,c are real roots. You want a^5 + b^5 + c^5. Now, the sum of all triplets of a,b,c = a*b*c = 1, the sum of all pairs of a,b,c = 5, and the...

Well, yeah, but it's still how sin(x) is defined. (For complex x)

Use product rule first, then you end up differentiating 4^(x^2). A nice formula to know is d/dx ( a^(f(x)) ) = a^f(x) * ln(a) * f'(x), which comes from the chain rule.

It's valid because he wants to solve x^x = x, for x.

The name is Abel. Anyway, a simple proof for e^(pi*i) + 1 = 0 can be obtained with only some basic knowledge of math. Look at the definitions of sin(x) and cos(x): cos(x) = (e^(i*x) + e^-(i*x))/2 sin(x) = (e^(i*x) - e^-(i*x))/(2i) cos(x) + i*sin(x) = (e^(i*x) + e^-(i*x))/2 +...

If you have x = 0, you get 1/0 which is invalid. I also fail to see how you got from the first to the second equation.

You're making it so small that it dissapears, but it's still not equal to zero. Your example, f(x) = 1/x, does not have a limit when x->0. Why? Because if you look at what you said, x = 0.000000001, you get a very big number, and x = 0.0000000000000001 gives you an even bigger number. It can...

Always nice to help a fellow citizen.

In your first try you're finding that 108.7 is 0.956*100 = 95.6% out of 113.7. Look at the increase from 108.7 to 113.7, it's a increase by 5 units. (113.7-108.7). Now, if we want to find how many % 5 is out of 108.7, we do (5/108.7)*100 = 4.59%. So, in general: If a number increases from a...

This is a good explanation, but you should be carefull when you say that the integral works for integers. The gamma-function is not defined for negative integers, it includes division by zero in some way. As you stated, n! = gamma(n+1), which means that (-1)! = gamma(0) which is undefined...

There are oh so many applications of the determinant. I will list a few: * Finding inverse of a matrix * Finding area/volume * Cross product * Eigenvalues / eigenvectors and so on... You might find some of these topics related, but my point is; there's loads of things you can do using...

There's a typo there, it should be sum(n^2)=n(n+1)(2n+1)/6. Although n(n-1)(n-2)/6 will always return a possitve integer for n > 2, it does not give you the desired number.