Recent content by fan96

  1. fan96

    Maths neatness

    If your fraction stack is growing out of control, you can just get rid of the fraction entirely - write \frac{\frac{1}{x}+e^x}{1+\frac{1}{\sin^{-1} x}} as \left( \frac{1}{x}+e^x \right) \left( 1+\frac{1}{\sin^{-1} x} \right) ^{-1}. Another useful trick is e^{\frac{1}{x} + \sin x} \to...
  2. fan96


    There isn't really a systematic way (that I'm aware of) of obtaining every asymptote for a given function, but here are a few general rules that will work most of the time. There are three types of asymptotes we're interested in: vertical, horizontal and oblique (or slanted). (Oblique...
  3. fan96

    Challenging (?) Proof Question

    If one wanted to actually find examples of M , here is an approach that works for M \in \mathbb R . I'll use the hyperbolic function \cosh and its inverse, which aren't in the MX2 syllabus but their definitions are quite simple to understand: \cosh x = \frac 12 \left(e^x+e^{-x}\right) and...
  4. fan96

    Challenging (?) Proof Question

    Is the condition M \notin \mathbb Z really necessary? The only case of this I can see is 1 + 1/1 = 2 \in \mathbb Z, so it would be simpler to leave it unmentioned.
  5. fan96

    Perms & Coms

    If the queues are distinct, then we should have 17280 = \underbrace{\left( \binom{8}{4} \times 4! \times 4!\right)}_{\text{no restrictions}} - \underbrace{\left(\binom{6}{3} \times 4!\ \times 4!\right)\times 2}_{\text{Sean and Liam in diff. queues}}, or 17280 = \left( \binom{6}{4} \times 4...
  6. fan96

    Tricky projectile motion question, Thanks

    For both balls we have \begin{cases} y_1(t) &= -\frac{9.8}{2}t^2+100\sin(\pi/6)t+h \\ x_1(t) &=100\cos(\pi/6)t\end{cases} \begin{cases} y_2(t) &= -\frac{9.8}{2}t^2+h \\ x_2(t) &=100t\end{cases} Solve y_1(t) = 0 and y_2(t) = 0 to get the time of flight for each ball (call these t_1 and...
  7. fan96

    What do I transfer to?

    Computer Engineering requires only standard Phys 1B and it's not a prerequisite for anything else. There are quite a few ELEC courses required though.
  8. fan96


    Recall that for two vectors \bold x, \bold y we have \cos \theta = \frac{\bold x \cdot \bold y}{|\bold x||\bold y|}, where \theta is the angle between \bold x and \bold y .
  9. fan96

    4 types of relations

    Unfortunately it's not that simple. You can't "break up" the absolute value function like that. If y = |x-3| , then y = \pm(x - 3). You can try graphing these to visualise the effect of the absolute value function.
  10. fan96

    4 types of relations

    That's called the absolute value function, and it's defined by |x| = \begin{cases} x, &\text{if }x \ge 0, \\-x, &\text{if }x < 0.\end{cases} Basically, you only want the magnitude of the number, not its sign. For example, |-1| = 1, |-2| = 2, |1| = 1, |0| = 0. If x = |y| , then...
  11. fan96

    Help with complex

    Say you have a statement you're trying to prove by induction. A normal induction proof would go like "if this statement is true for k then it must also be true for k + 1." A strong induction proof would be more like "if this statement is true for all numbers less than or equal to k then it...
  12. fan96

    Common Mistakes in differentiation and trigonometry

    This is one of the more common differentiation mistakes. \frac{d}{dx} x^x \overset{?}=x \cdot x^{x-1} For trigonometry in general, people make a lot of mistakes. \cot x \overset{?}= \frac{1}{\tan x} \sin^{-1}( \sin x ) \overset{?}= x , or equivalently, \sin x = y \overset{?} \implies x...
  13. fan96

    Hardest geometry question in history answered by student trivially............ How?

    The quantity \frac{k^2-RX^2}{2\cdot PX \cdot QX} can probably be simplified so as to remove k . The answer I gave holds numerically for all such equilateral and isosceles triangles formed and most likely for the rest of them too. The other solutions are definitely much nicer though. I...
  14. fan96

    Hardest geometry question in history answered by student trivially............ How?

    Must the angles be expressed only in terms of s and t ? The best I could do is s' = \arccos\left( \cos s + \frac{k^2-RX^2}{2\cdot PX \cdot QX}\right), t' = \arccos\left( \cos t + \frac{k^2-PX^2}{2\cdot QX \cdot RX}\right), where k is the side length of the equilateral triangle and...
  15. fan96

    COMP1511 vs COMP1911

    HSC SDD has very little in common with the introductory programming courses. The latter focuses solely on being able to code properly. There's nothing about ethics, dev approaches or that sort of stuff. 1911 is a cut down version of 1511. 1811 is significantly different to both of those. If...