Quod Erat Demonstrandum

2020/05/21

Basic question of differentiation

Filed under: Additional / Applied Mathematics,mathematics,NSS — johnmayhk @ 7:20 下午
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HKDSE 2020 M2 Q.9 (b), a 2-mark question:

Given $\displaystyle f(x)=\frac{(x+4)^3}{(x-4)^2}$, find $f''(x)$.

How fast can you finish this part and obtain the correct answer, especially when you are under the pressure during the public examination?

3 minutes? (2/100 * total time allowed = 2/100 * 150 minutes)

(more…)

2019/05/05

What’s wrong?

Filed under: Additional / Applied Mathematics,NSS — johnmayhk @ 7:05 下午
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Here is a basic level M2 question:

Given that $\sqrt{xy}=7+2y$, find $\frac{dy}{dx}$ at ($-\frac{1}{3}$,$-3$).

Student 1 gave

$\frac{1}{2\sqrt{xy}}(x\frac{dy}{dx}+y)=2\frac{dy}{dx}$

$\frac{1}{2}\sqrt{\frac{x}{y}}\frac{dy}{dx}+\frac{1}{2}\sqrt{\frac{y}{x}}=2\frac{dy}{dx}$

$\frac{dy}{dx}=\sqrt{\frac{y}{x}}\cdot\frac{1}{4-\sqrt{\frac{x}{y}}}$

Thus, at ($-\frac{1}{3}$,$-3$),

$\frac{dy}{dx}=\sqrt{\frac{-3}{-1/3}}\cdot\frac{1}{4-\sqrt{\frac{-1/3}{-3}}}=\frac{9}{11}$

Student 2 gave (more…)

2018/07/18

畫 y=x^(1/n)

Filed under: Additional / Applied Mathematics,mathematics,NSS — johnmayhk @ 12:57 下午
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$\sqrt[3]{4}-\sqrt[3]{3}$$\sqrt[3]{3}-\sqrt[3]{2}$

2018/03/22

a question about inequality with derivatives

Filed under: Fun,mathematics,NSS,Pure Mathematics — johnmayhk @ 3:46 下午
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Question

Let $p(x)$ be a polynomial with real coefficients. Prove that if $p(x)-p'(x)-p''(x)+p'''(x)\ge 0$ for any real $x$, then $p(x) \ge 0$ for any real $x$.

Solution (elementary) (more…)

2018/03/21

某求導題

Filed under: Additional / Applied Mathematics,NSS — johnmayhk @ 3:29 下午
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If $\displaystyle \sqrt{x^3+y^3}=6(xy+1)$, find $\displaystyle \frac{dy}{dx}$ at $(1,-1)$.

$\displaystyle x^3+y^3=36(xy+1)^2$

$\displaystyle \Rightarrow \frac{d}{dx}(x^3+y^3)=\frac{d}{dx}36(xy+1)^2$

$\displaystyle \frac{dy}{dx}=\frac{24xy^2+24y-x^2}{y^2-24x^2y-24x}$

$\displaystyle \frac{dy}{dx}|_{(1,-1)}=-1$

2018/03/05

度數弧度微積分

Filed under: Additional / Applied Mathematics,NSS — johnmayhk @ 12:10 下午
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（免插聲明：本篇頗無聊，高手見諒）

$\displaystyle \frac{d}{dx}\sin x$　at　$x=0^o$

M2 學生應知

$\displaystyle \frac{d}{dx}\sin x=\cos x$

$\displaystyle \frac{d}{dx}\sin x=\cos 0^o=1$

2016/03/21

用Ｄ證trigo

Filed under: Additional / Applied Mathematics,NSS — johnmayhk @ 11:13 上午
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Prove the following identity

$\cos^2x+\cos^2(x+y)-2\cos y\cos x\cos(x+y)=\sin^2y$.

2014/12/19

某些M2題

Filed under: NSS — johnmayhk @ 9:30 下午
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Q.1

$\sin\theta+\cos\theta=\frac{7}{3}$　時，同學已指出：冇可能！因為 $\sin\theta$$\cos\theta$ 的最大值不過是 1。

Q.2

2014/11/14

M2 堂偶拾

Filed under: NSS,Pure Mathematics — johnmayhk @ 8:19 下午
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1.

$\cos^{-1}\frac{4}{5}+2\tan^{-1}\frac{1}{2}=90^o$

2014/08/08

利用比較係數做積分

Filed under: NSS,Pure Mathematics — johnmayhk @ 12:16 上午
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https://johnmayhk.wordpress.com/2012/02/07/integration-by-parts/

$\frac{d}{dx}P(x)e^x=(P(x)+P'(x))e^x$

$P(x)$$P(x)+P'(x)$ 皆是多項式，且它們的次數（degree）相同，

2014/06/11

M2 某題

Filed under: Additional / Applied Mathematics,NSS — johnmayhk @ 9:06 上午
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M2 學生問以下一題

The slope at any point (x,y) of a curve is given by

$\frac{dy}{dx}=y(2x+9)$.

If the curve lies above the x-axis, and it passes (0,8), find the equation of the curve.
(more…)

2014/05/16

證明某級數

Filed under: Fun,Pure Mathematics — johnmayhk @ 11:00 下午
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2013/09/27

D兩次是零的非拐點一定是極大或極小點嗎

Filed under: NSS — johnmayhk @ 4:55 下午
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$y=x^4$

$\frac{d^2y}{dx^2}=12x^2$

$\frac{d^2y}{dx^2}|_{x=0}=0$

2013/06/16

f4 m2 revision

Filed under: NSS — johnmayhk @ 6:39 下午
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（免插聲明：純為中四同學溫習用，高手見諒。）

$\frac{d}{dx}(x^n)=nx^{n-1}$

$\frac{d}{dx}(n^x)$

$\frac{d}{dx}((\ln x)^x)$

2012/03/07

答網友：求導

Filed under: NSS — johnmayhk @ 10:05 上午
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Find $\frac{dy}{dx}$ if

$3x^3+2y^2-4x+\frac{5}{y}=0$

$\frac{d}{dx}(3x^3+2y^2-4x+\frac{5}{y})=0$