Recent content by TadeusPrastowo

That makes sense. Ah, I see the point now. Thank you very much.

Taking an analogy to instantaneous rate-of-change of car's position (i.e., car's speed), the rate-of-change of position is very beneficial because speed relates to the magnitude of the car's kinetic energy where the greater the energy is, the more fatal a car crash becomes. So, I think the...

My https://www.amazon.com/dp/0073532320/?tag=pfamazon01-20 (p. 176 Example 7.1) pointed out that an investment ##p(t) = 100\,2^t## (##t## in year) that doubles the capital every year starting with an initial capital of $100, has an (instantaneous) rate-of-change ##\frac{\text{d}}{\text{d}t} p(t)...

Now I understand. Thank you very much for your patience. I took a look again at the formal definition of limit as ##x## grows unbounded and saw that indeed the requirement for ##\lim_{x\to \infty} f(x)## to exist is to have a horizontal asymptote, not a slant asymptote (i.e., a slant asymptote...

Specifically, I want to know the sound method to algebraically manipulate a rational function definition in finding the equation of the asymptote as ##x## grows without bound. For example: 1. Always start with a long division because of this reason. 2. Then, for the resulting terms, divide by...

I don't agree. In my opinion, analytically, as ##x## gets very large, the algebraic manipulation of the function definition must reveal the nature of the asymptote: whether it is horizontal or slanted. As the numerical calculation shows that my alternative obtained by extracting the greatest...

My https://www.amazon.com/dp/0073532320/?tag=pfamazon01-20 gives a rule of thumb to divide by the highest power in the denominator for the following problem to demonstrate a slant (oblique) asymptote: \lim_{x\to\infty} \frac{4x^3+5}{-6x^2-7x} = \lim_{x\to\infty}...

Many many many thanks for pointing out the concept of being connected vs. being disconnected to me. Yes, that's exactly what I want to say by my word "gapless". Once again, thank you very much. I really appreciate your enlightenment.

In my own words, by "gapless" I mean a domain such as the set ##\mathbb{R}## of real numbers. ##\mathbb{R} = \mathbb{Q} \bigcup \overline{\mathbb{Q}}##. That is, for any gap between the closest pair of rational numbers, there are infinitely many irrational numbers filling in the gap making the...

Based on the following problem from http://math.uchicago.edu/~vipul/teaching-0910/151/applyingformaldefinitionoflimit.pdf: f(x) = \begin{cases} x^2 &, \text{ if }x\text{ is rational} \\ x &, \text{ if } x\text{ is irrational} \end{cases} is shown to have the following limit: \lim_{x\to 1}f(x)...

I have tried to continue the derivation after correcting the dr/dt as you pointed out. I did (\boldsymbol\omega \times \mathbf{r}_p)_k by calculating \boldsymbol\omega \times \mathbf{r}_p first as matrix multiplication [\omega]_\times [r_p] and then taking the k-element. Then, I continued...

No, it is not \boldsymbol\omega \times \boldsymbol\omega but \boldsymbol\omega \times [I] \boldsymbol\omega where [I] is a second-order tensor, not a scalar. To be exact, we are dealing with the product of matrices [\omega]_\times \left([I] [\omega]\right). See Wikipedia article on cross-product...

Homework Statement The Wikipedia article on spatial rigid body dynamics derives the equation of motion \boldsymbol\tau = [I]\boldsymbol\alpha + \boldsymbol\omega\times[I]\boldsymbol\omega from \sum_{i=1}^n \boldsymbol\Delta\mathbf{r}_i\times (m_i\mathbf{a}_i). But, there is another way to...

Homework Statement Proof the following: \frac{\text{d}\boldsymbol\{\mathbf{I}\boldsymbol\}}{\text{d}t} \, \boldsymbol\omega = \boldsymbol\omega \times (\boldsymbol\{\mathbf{I}\boldsymbol\}\,\boldsymbol\omega) where \boldsymbol\{\mathbf{I}\boldsymbol\} is a tensor...

Ah, yes, you are right about it, haruspex! Thank you very much!