Recent content by AMenendez

Everyone likes donuts, so volume of a torus is a pretty good application of calculus. Let's say you want to know how many calories are in the donut you're about to eat, but you only know the number of calories per unit volume. To find the total volume, you can rotate the cross section of the...

We can think of a series of small rotations starting at 1 and continuing by infinitely small shifts until having traveled the entire length of the curve connecting 1 and -1 in the complex plane. Each shift is expressed as: ##1 + i \delta##, where ##\delta## is a small angle. The total number...

Only mathematicians really care about the correctness of notation. In the case you presented, ##x(t) = \frac{1}{2}at^2 + v_0 t + x_0## ##x_0## is just the initial value for the function, just like you might write a linear equation like: ##y(x) = ax + y_0## Though I don't know how many people...

There really isn't a ``rule" for which ##x##-value to pick for the intial ``guess" in Newton's Method. Just look at the function and use some analytic techniques (like FactChecker mentioned) to try to reason where the roots might be, then pick the nearest integer around there.

Stewart isn't a terribly rigorous introduction to calculus; if you want something more mathematically pure and rigorous, Michael Spivak's ``Calculus" is a favorite in the math community for its emphasis on theory and proofs. Learning ##\delta - \epsilon## proofs isn't crucial to understanding...

Nah, just because there is a solution doesn't always mean it's unique. Take something like: ##\frac{dy}{dx} = 3y^{2/3}##, where y(0) = 0. ##f(x,y)## is continuous about (0 , 0), but ##\frac{\partial f}{\partial y}## is not continuous about (0 , 0); this violates the conditions for...

Girl, are you sin(x)? Well, I'm cos(x); why don't you get on top of me so that we can make tan(x)? ;)

So, the original problem is ``flawed"? So, that's it, there's no solution? Or, should we keep trying?

*EDIT* I misspoke above; I listed two eigenvalues, whereas a ##3 \times 3## matrix (with a cubic characteristic polynomial) would have three eigenvalues. Of course, in the case I listed above, two of the eigenvalues, ##\lambda_2## and ##\lambda_3## would be the same (λ2 = λ3 = -9). I was...

Well, you could have something like: ## \lambda^3 + 18\lambda^2 + 81\lambda## Where you have eigenvalues ##\lambda_1 = 0## and ##\lambda_2 = -9##. So, I certainly think you can have an eigenvalue of zero and other nonzero eigenvalues. And also, I'm not sure what you mean by the determinant being...

So, the standard way of adding / subtracting numbers is to first identify a ``least common denominator". So, say I had: ##\frac{1}{2} + \frac{3}{7}## Well, what is the smallest number that is a multiple of both 2 and 7? It's 14. So now, how do we write the original fractions with denominators of...

The idea behind it is that, when we differentiate a function, we're sort of 'secretly' using the chain rule. Sometimes, we have equations that are explicitly defined as a function of ##x##, like: ##y = 5x - 2## (1) Of course, we could rearrange it so that it is 'implicitly' defined as a...

Fair enough. Thanks for pointing that out.

As Mark44 pointed out... Find all of the vertical asymptotes (set the denominator equal to zero and solve for ##x##--this will tell you the ##x##-values through which the vertical asymptotes go). Once you do this, you have a rough idea of where the graph of the function is "split" into pieces.

First, Euler's Identity has nothing to do with the Reimann-Zeta function. Euler's Identity is just ##e^{i \pi} + 1 = 0## Euler's Formula is probably what you're thinking of (which is where the above identity comes from): ##e^{ix} = \cos x + i \sin x## But there's no exponential function (it...