2009 四月 « Mathematics in HKCEE, HKAL, and Beyond

Archive for 四月, 2009

Math 324 Mid-Term 1 Solution

四月 18, 2009

1) (a) $V(S) = \int_{-1}^{1}\int_{-\sqrt{1 - x^2}}^{\sqrt{1 - x^2}} \int_{x^2 + y^2}^{\sqrt{x^2 + y^2}}1 \, dz \, dy \, dx.$

(b and c) $\begin{aligned} V(S) &= \int_{0}^{2\pi}\int_{0}^{1}\int_{r^2}^{r}r \, dz \, dr \, d\theta\\ &= 2\pi \int_{0}^{1} r^2 - r^3 dr \\ &= 2\pi (\frac{1}{3} -\frac{1}{4}) = \frac{\pi}{6}.\end{aligned}$ Read the rest of this entry ?

Posted in Calculus, Teaching | Leave a Comment »

Tricks of Integration (1) – The Koop Subs

四月 8, 2009

In this article, I will talk about some of my favorite substitutions for integration, which I will named them as the “Koop subs”. They are particularly useful in mathematics competitions such as the Putnam competition.

Example 1) Compute the integral: $I = \int_{0}^{1} \frac{x^3}{1 - 3x + 3x^2} dx.$ Read the rest of this entry ?

Posted in Calculus, Teaching | 9 意見»

Solution to “Possible HKAL Pure Problems”

四月 1, 2009

1(a) Expanding along the last row, we have:

$f(x) = 1 \det{\begin{pmatrix}x & x^3\\y & y^3 \end{pmatrix}} - z \det{\begin{pmatrix} 1 & x^3 \\1 & y^3 \end{pmatrix}} + z^3 \det{\begin{pmatrix} 1 & x \\1 & y \end{pmatrix}}.$

Therefore, $f(z)$ is a cubic in $z$ with leading coefficient $\det \begin{pmatrix} 1 & x \\1 & y \end{pmatrix} = y - x.$ Read the rest of this entry ?