jcarlson

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Posts posted by jcarlson

Inverse of a continuous function.

$f(x) = tan^{-1}(x)$

The inverse of this function is:

$x = tan^{-1}(f(x))$

$f(x) = tan(x); -\frac{\pi}{2} < x < \frac{\pi}{2}$

The domain is restricted on the inverse because $f(x) = tan^{-1}(x)$ is restricted to the range $-\frac{\pi}{2} < f(x) < \frac{\pi}{2}$. Thus the function $f(x) = tan^{-1}(x)$ is continuous on the domain of all real numbers, but its inverse is not.

*edit*

This is the case for every function whose domain is all the real numbers, but whose range is restricted due to a horizontal asymptote or an absolute minimum. The inverse will not be continuous on the domain of all real numbers. Perhaps an even better example would be the function $f(x) = x^{2}$, whose inverse, $f(x) = \sqrt{x}$ is not only not defined for x < 0, but isnt a function at all, by the vertical line test.

There isn't any reason mathematically why the differential of time couldnt be negative just like the differential of any other variable in a function. There is no definition in mathematics that I know of that states that the variable "time" is any different from any other variable. However when you apply mathematics like this to physics you also have to take into account the properties of what you are applying the math to, not just the math itself. In the case of time, it is unidirectional, and therefore obtaining a differential of time that was negative in a physics problem would be nonsensical.

Ending the 0.999~ = 1 debates

$.\overline{9}$ is most certainly a number. Not only that, its a real number, and also a rational number. It is merely a different representation of the same real number that 1 represents.

Another force besides the attraction between the two particles would be needed to accelarate one particle in a direction tangent to an ellipse around the other particle. Then theoretically, if the velocity of the orbiting object is the correct magnitude and direction in relation to the distance between the two objects at a given point, they should start orbiting another.

Java IDEs

On a serious note,I have only ever used Borlands JBuilder. I have heard that the new version of JBuilder by Borland supports J2SE 1.5, and I have also heard good things about NetBeans, although I've never used it.

Help me help a kid with math (or, justifying negative exponents)

also:

3^3 = 27

3^2 = 9 or 27/3

3^1 = 3 or 9/3

so it follows that if the pattern continues:

3^0 = should equal 3/3, or 1

3^-1 = 1/3

Ending the 0.999~ = 1 debates

Yes, but in base 11, assuming a is the 11th digit, .aaa... does equal 1.

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