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  1. rolpsy

    Mathematical Curiosities.

    probably another reason why some people prefer Leibniz's notation. it reads very nicely i just thought of a few questions that i think are interesting. hopefully others find them interesting/surprising(?) too. i'll wait a few days then answer them if no one else has (but hopefully this won't...
  2. rolpsy

    Mathematical Curiosities.

    still sounds quite hand-wavy. perhaps it can be formalised in terms of nonstandard analysis..? ah this is more like it. though admittedly I've always found that first formula utterly indecipherable (this incarnation looks especially nasty) a few more qns to hopefully get people...
  3. rolpsy

    Mathematical Curiosities.

    interesting thread! something that used to annoy me: If dy/dx isn't a fraction, then why is it that \frac{dy}{dx} \times \frac{dx}{dy} = 1 \text{ and } \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} (inverse function theorem and chain rule) What are precise statements and how...
  4. rolpsy

    Mathematical Curiosities.

    this isn't true as stated… 2 + √1 = 1 + √4 so 2 = 1 and 1 = 4, right? you need a, b, c, d to be rational and one of b and d to not be a perfect square edit: take a look at the post above
  5. rolpsy

    HSC 2012-14 MX2 Integration Marathon (archive)

    Re: MX2 Integration Marathon \\\text{Let } I(m,n) = \int^{1}_{0} x^m(1-x)^n \, \mathrm{d}x \text{ where } n \text{ and } m \text{ are positive integers.}\\\begin{align*}\text{Integrating by parts with } u &= (1-x)^n \hspace{7mm} \text{ and } \hspace{6.5mm} \mathrm{d}v = x^m \...
  6. rolpsy

    HSC 2012-14 MX2 Integration Marathon (archive)

    Re: MX2 Integration Marathon \int^{\pi}_{0} \frac{\mathrm{d}x}{1 + \cos^2(x)}
  7. rolpsy

    HSC 2013 MX2 Marathon (archive)

    Re: HSC 2013 4U Marathon Hint: Choose \alpha as the largest root(s) and consider the discriminant of \frac{x^3 + ax^2 + bx + c}{x - \alpha}
  8. rolpsy

    HSC 2013 MX2 Marathon (archive)

    Re: HSC 2013 4U Marathon oops haha yep
  9. rolpsy

    HSC 2013 MX2 Marathon (archive)

    Re: HSC 2013 4U Marathon \\\text{Let } a, b, c \text{ be real numbers such that the roots of the cubic equation }\\[2pt]x^3 + ax^2 + bx + c = 0 \text{ are real}\\[5pt]\text{Prove these roots are bounded above by } \frac{2\sqrt{a^2 - 3b} - a}{3}
  10. rolpsy

    HSC 2013 MX2 Marathon (archive)

    Re: HSC 2013 4U Marathon n(n+1)n! lol
  11. rolpsy

    Help with parametric question!

    There should really be absolute value signs around that: \left | p - q \right | = \left |1 + pq \right |  (unless some other condition is placed on p and q), because the angle between two lines takes the absolute value: \tan\left ( \theta \right ) = \left | \frac{m_{1} - m_{2}}{1 +...
  12. rolpsy

    Centrifugal Force?

    Centrifugal force does exist. (however centrifugal force does not exist in an inertial frame of reference) There's a decent explanation near the end of The Student's Guide to HSC Physics under 'extra content' p.s. i think this comic is relevant
  13. rolpsy

    Polynomials Help!

    Did you get –80/27?
  14. rolpsy

    Polynomials Help!

    \begin{align*}\alpha^3 + \beta^3 &= (\alpha + \beta)(\alpha^2 - \alpha\beta + \beta^2)\\&=(\alpha + \beta)\left (( \alpha + \beta)^2 - 3\alpha\beta \right )\\&=\cdots\end{align*}
  15. rolpsy

    Limiting sum question?

    \begin{align*}\text{Height} &= 3 + 2\frac{2}{5} + 1\frac{23}{25} + \cdots\\&=3 + 3\cdot\left (\frac{4}{5} \right ) + 3\cdot\left (\frac{4}{5} \right )^2 + \cdots\end{align*} which is an infinite geometric series, with a = 3 and r = 4/5 Using the limiting sum formula (noting that |a| < 1)...
  16. rolpsy

    Integration

    Mathematica refuses to integrate it so I highly doubt it can be found in terms of elementary functions
  17. rolpsy

    complex number help

    for future reference... note that z^{n} + z^{-n} = 2\cos(n\theta) then \begin{align*}z^6 + z^3 + 1 &= \left (z^2 -2z\cos\left ( \frac{2\pi}{9} \right ) + 1\right )\left (z^2 -2z\cos\left ( \frac{4\pi}{9} \right ) + 1\right )\left (z^2 -2z\cos\left ( \frac{8\pi}{9} \right ) + 1\right...
  18. rolpsy

    Inequality Question

    here's another way \begin{align*}\sqrt{n+1} &> \sqrt{n}\\\frac{n+1}{\sqrt{n+1}} &> \frac{n}{\sqrt{n}}&&\text{rationalising numerator}\\\frac{\sqrt{n}}{\sqrt{n+1}} &> \frac{n}{n+1}\\&= 1 - \frac{1}{n+1}\\\frac{\sqrt{n}}{\sqrt{n+1}} + \frac{1}{n+1} &> 1\\\therefore \sqrt{n} +...
  19. rolpsy

    Proof by Induction - Inequalities

    It helps a lot for the 3rd step if you state what you wish to prove, in this case: 2(k+1) + 2 < 2^{k+1} We then work the RHS (since it will be easier to apply the hypothesis) \begin{align*}2^{k+1} &= 2\times 2^k\\&> 2\Big ( 2k + 2\Big ) &&\text{using the induction hypothesis}\\&= 4k + 4\\&>...
  20. rolpsy

    Proof by Induction - Inequalities

    okay so you're concerned about this step: 2\times3^k < 3\times 3^k I want to emphasise the 'less than'; the 3rd line is less than the 2nd (not equal to), allowing us to 'replace' the 2. \begin{align*}2 &< 3\\2 \times 2^k &< 3 \times 2^k &&\text{multiplying both sides by } 2^k \quad...
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