수학 모음 (Maths collection)

• Technical stuff: 수학적 지식을 요구하는 글 (A - Exploring ideas, B - Problem solving)
• Non-technical stuff: 이야기가 중심이 되는 글

1. Technical stuff A - Exploring ideas

 

(1) History of the Integral of the Secant (sec x)

 

(2) 시컨트/세칸트(sec x)의 적분법 | Integration of sec x

$$ \begin{align} \int \sec x \, \textrm{d}x &= \ln \left\vert \tan \left( \frac{x}{2} + \frac{\pi}{4} \right) \right\vert + C \\ &= \ln \vert \sec x + \tan x \vert + C \\ &= \frac12 \ln \left\vert \frac{ 1 + \sin x }{ 1 - \sin x} \right\vert + C \\ &= \ln \left\vert \frac{ 1 + \tan \frac{x}{2} }{ 1 - \tan \frac{x}{2} } \right\vert + C \end{align} $$

 

(3) 코시컨트/코세칸트(cosec x)의 적분법 | Integration of cosec x

$$ \begin{align} \int \csc x \, \textrm{d}x &= - \ln \vert \csc x + \cot x \vert + C \\ &= \frac12 \ln \left\vert \frac{ 1 - \cos x }{ 1 + \cos x} \right\vert + C \\ &= \ln \left\vert \tan \frac{x}{2} \right\vert + C \\ &= \ln \left\vert \tan \left( \frac{x}{2} + \frac{\pi}{4} \right) \right\vert + C \end{align} $$

 

(4) 쌍곡시컨트(sech x)의 적분법 | Integration of sech x

$$ \begin{align} \int \textrm{sech} \, x \, \textrm{d}x &= 2 \arctan \left( \textrm{e}^x \right) + C \\ &= \arctan \left( \sinh x \right) + C \\ &= 2\arctan \left( \tanh \frac{x}{2} \right) + C \end{align} $$

 

(5) 쌍곡코시컨트(cosech x)의 적분법 | Integration of cosech x

$$ \begin{align} \int \textrm{csch} \, x \, \textrm{d}x &= - \ln \vert \textrm{csch} \, x + \textrm{coth} \, x \vert + C \\ &= \ln \left\vert \frac{ \textrm{e}^x - 1 }{ \textrm{e}^x + 1 } \right\vert + C \\ &= \frac12 \ln \left\vert \frac{ \cosh x - 1 }{ \cosh x + 1 } \right\vert + C = - \textrm{arcoth} (\cosh x) + C \\ &= \ln \left\vert \tanh \frac{x}{2} \right\vert + C \end{align} $$

 

(6) 바이어슈트라스 t-치환 1 삼각함수 | Weierstrass t-substitution for circular (trigonometric) functions (under construction)

 

(7) 바이어슈트라스 t-치환 2 쌍곡함수 | Weierstrass t-substitution for hyperbolic functions (under construction)

 

(9) RSA cryptography (under construction)

 

(10) 대출금 계산기 | Mortgage calculator (under construction)

$$ \begin{align} \end{align} $$

 

(11) Interest rate vs Tax rate

 

(12) Area of regular octagon, n-gon and circle

 

(13) Surface area of a circular cone

 

(14) Integration of $\sqrt{\tan x}$

$$ \begin{align} \int \sqrt{\tan x}\,\textrm{d}x &= \frac1{\sqrt{2}}\arcsin \left( \sin x - \cos x \right) - \frac1{\sqrt{2}} \textrm{arcosh} (\sin x + \cos x) + C \\ &= \frac1{\sqrt{2}}\arcsin \left( \sin x - \cos x \right) - \frac1{\sqrt{2}} \ln \left( \sin x + \cos x + \sqrt{ \sin 2x } \right) + C \\ &= \frac1{2\sqrt{2}} \ln \left\vert \frac{ \sqrt{\tan x} + \sqrt{\cot x} - \sqrt{2} }{ \sqrt{\tan x} + \sqrt{\cot x} + \sqrt{2} } \right\vert + \frac1{\sqrt{2}}\arctan \left( \frac{\sqrt{\tan x} - \sqrt{\cot x}}{\sqrt{2}} \right) + C \end{align} $$

 

(15) Integration of $\sqrt{ \cot x }$

$$ \begin{align} \int \sqrt{\cot x}\,\textrm{d}x &= \frac1{\sqrt{2}}\arcsin \left( \sin x - \cos x \right) + \frac1{\sqrt{2}} \, \textrm{arcosh}(\sin x + \cos x) + C \\ &= \frac1{\sqrt{2}}\arcsin \left( \sin x - \cos x \right) + \frac1{\sqrt{2}} \ln \left( \sin x + \cos x + \sqrt{ \sin 2x } \right) + C \\ &= \frac1{2\sqrt{2}} \ln \left\vert \frac{ \sqrt{\cot x} + \sqrt{\tan x} + \sqrt{2} }{ \sqrt{\cot x} + \sqrt{\tan x} - \sqrt{2} } \right\vert + \frac1{\sqrt{2}}\textrm{arccot} \left( \frac{\sqrt{\cot x} - \sqrt{\tan x}}{\sqrt{2}} \right) + C \end{align} $$

 

(16) Integration of powers of the sine function (Wallis integral) $$ \begin{align} J_{2m}&=\int_0^{\frac{\pi}{2}}\sin^{2m}x\,\textrm{d}x=\frac{(2m)!}{2^{2m}(m!)^2}\frac{\pi}{2} \\ J_{2m+1}&=\int_0^{\frac{\pi}{2}}\sin^{2m+1}x\,\textrm{d}x=\frac{2^{2m}(m!)^2}{(2m+1)!} \end{align} $$

 

(17) Integration of powers of the cosine function (Wallis integral) $$ \begin{align} J_{2m}&=\int_0^{\frac{\pi}{2}}\cos^{2m}x\,\textrm{d}x=\frac{(2m)!}{2^{2m}(m!)^2}\frac{\pi}{2} \\ J_{2m+1}&=\int_0^{\frac{\pi}{2}}\cos^{2m+1}x\,\textrm{d}x=\frac{2^{2m}(m!)^2}{(2m+1)!} \end{align} $$

 

(18) Integration of generalised powers of the sine function (A generalisation of the Wallis integral) $$ \begin{align} \int_0^{\frac{\pi}{2}} \sin^p x \, \textrm{d}x = \frac12 B\left( \frac{p+1}{2}, \frac12 \right) = \frac12 \frac{ \Gamma \left( \frac{p}{2} + \frac12 \right) \Gamma \left( \frac12 \right) }{ \Gamma \left( \frac{p}{2} + 1 \right) } \end{align} $$

 

(19) Integration of generalised powers of the cosine function (A generalisation of the Wallis integral) $$ \begin{align} \int_0^{\frac{\pi}{2}} \cos^q x \, \textrm{d}x = \frac12 B\left( \frac{q+1}{2}, \frac12 \right) = \frac12 \frac{ \Gamma \left( \frac{q}{2} + \frac12 \right) \Gamma \left( \frac12 \right) }{ \Gamma \left( \frac{q}{2} + 1 \right) } \end{align} $$

 

(20) Integration of products of the sine and cosine functions $$ \begin{align} \int_0^{\frac{\pi}{2}} \sin^p x \cos^q x \, \textrm{d}x = \frac12 B\left( \frac{p+1}{2}, \frac{q+1}{2} \right) = \frac12 \frac{ \Gamma \left( \frac{p}{2} + \frac12 \right) \Gamma \left( \frac{q}{2} + \frac12 \right) }{ \Gamma \left( \frac{p+q}{2} + 1 \right) } \end{align} $$

 

(21) The Euler beta and gamma functions (under construction)

 

(22) A first-order differential equation with complex coefficients

$$ \begin{align} & (a_1 + ia_2) \frac{ \textrm{d} y }{ \textrm{d}x } + (b_1 + ib_2) y = 0 \\ & y = \psi + i \chi = C \textrm e^{ - \frac{ \textbf a \cdot \textbf b }{ | \textbf a |^2 } x } \left[ \cos \left( \frac{ ( \textbf a \times \textbf b ) \cdot \textbf k }{ | \textbf a |^2 } x \right) - i \sin \left( \frac{ ( \textbf a \times \textbf b ) \cdot \textbf k }{ | \textbf a |^2 } x \right) \right] \quad \textrm{with} \quad C \in \mathbb C \end{align}$$

 

(23) Integration of a product of an exponential and a trigonometric/hyperbolic function $$ \begin{align}     \textrm{(a)} &&& I_1(a,b) = \int \textrm e^{ ax } \cos b x \, \textrm dx = \frac{ \textrm e^{ ax } ( a \cos bx + b \sin bx ) }{ a^2 + b^2 } + C \\   \textrm{(b)} &&& I_2(a,b) = \int \textrm e^{ ax } \sin b x \, \textrm dx = \frac{ \textrm e^{ ax } ( a \sin bx - b \cos bx ) }{ a^2 + b^2 } + C \\   \textrm{(c)} &&& I_3(a,b) = \int \textrm e^{ ax } \cosh b x \, \textrm dx = \frac{ \textrm e^{ ax } ( a \cosh bx - b \sinh bx ) }{ a^2 - b^2 } + C, \quad a^2 \ne b^2 \\ &&& I_3(a,a) = \int \textrm e^{ ax } \cosh a x \, \textrm dx = \frac1{ 4a }\textrm e^{ 2ax } + \frac12 x + C \\   \textrm{(d)} &&& I_4(a,b) = \int \textrm e^{ ax } \sinh b x \, \textrm dx = \frac{ \textrm e^{ ax } ( a \sinh bx - b \cosh bx ) }{ a^2 - b^2 } + C, \quad a^2 \ne b^2 \\ &&& I_4(a,a) = \int \textrm e^{ ax } \sinh a x \, \textrm dx = \frac1{ 4a }\textrm e^{ 2ax } - \frac12 x + C \end{align} $$

 

(24) A classification of critical points with the Hessian matrix $$ \begin{align} \begin{array}{|c|c|c|c|} \hline & \det H = \textrm f_{xx} \textrm f_{yy} - \textrm f_{xy}^2 > 0 & \det H = \textrm f_{xx} \textrm f_{yy} - \textrm f_{xy}^2 = 0 & \det H = \textrm f_{xx} \textrm f_{yy} - \textrm f_{xy}^2 < 0 \\\hline \textrm{tr} \, H = \textrm f_{xx} + \textrm f_{yy} > 0 & \textrm{A lcoal minimum} & \textrm{Degenerate} & \textrm{A saddle point} \\\hline \textrm{tr} \, H = \textrm f_{xx} + \textrm f_{yy} < 0 & \textrm{A lcoal maximum} & \textrm{Degenerate} & \textrm{A saddle point} \\\hline \end{array} \end{align} $$

 

2. Technical stuff B - Problem solving

 

(1) 헤론의 공식 유도 (삼각형의 넓이) | A derivation of Heron's formula (Area of a triangle) $$ \begin{align} A = \sqrt{ s (s-a) (s-b) (s-c) } \qquad \textrm{where} \qquad s = \frac{a+b+c}{2} \end{align} $$

 

(2) A question on exponential decay (Eton College 01C_MT1 Q14)

$$ \begin{align} T = A \textrm{e}^{-kt} \end{align} $$

 

(3) A question on trigonometry (Eton College 01C_MT1 Q15)

$$ \begin{align} \textrm{(a)}&\qquad \cos( 2 \theta ) + \sin( 2 \theta ) = R \sin (2 \theta + \alpha) \\ \textrm{(b)}&\qquad \frac{ 1 - \sqrt{2} }{2} \le \cos\theta(\cos\theta+\sin\theta) \le \frac{ 1 + \sqrt{2} }{2} \end{align} $$

 

(4) A question on logarithm (Eton College 01C_MT1 Q16)

$$ \begin{align} \log_{10}\left(3^x-5^{2-x}\right)&=2+\log_{10}(2)-x\log_{10}(5) \\ \left(5\sqrt{5}\right)^{3x}-31&=30\left(\frac1{\sqrt[4]{5}}\right)^{9x} \end{align} $$

 

(5) A counting problem - Lucy & Anton in the photo (GCSE)

Question. Lucy and Anton are partying with 8 friends. A photo of 6 people in a line is taken. How many different photos are possible if:
(a) Lucy is always in the photo;
(b) Lucy and Anton are always in the photo;
(c) Only one of Lucy or Anton are in the photo.

 

(6) Probability - Jumping goldfish (GCSE)

Question. There are goldfish in two bowls $P$ and $Q$. During one minute the probability of one goldfish jumping from bowl $P$ to $Q$ is $\frac25$. During one minute the porbability of one goldfish jumping from bown $Q$ to $P$ is $\frac13$. Calculate the probability that: (a) the number of goldfish in each bowl at the end of 1 minute is equal to the number of goldfish in each bowl at the start of the minute; (b) the number of goldfish in each bowl at the end of 2 minutes is equal to the number of goldfish in each bowl at the start of this two minute period.

 

(7) Differentiation - Some inverse trigonometric functions

Question. Find $\frac{dy}{dx}$ for: $$ \begin{align} \textrm{(a)}&\qquad y = \arcsin(1-2x) \\ \textrm{(b)}&\qquad y = \arctan\left(x^2+1\right) \end{align} $$

 

(8) Some integrals

$$ \begin{align} {\rm (a)}&& &\int\frac{1}{\sqrt{1-3x^2}}\,\textrm{d}x \\ {\rm (b)}&& &\int\frac{x}{4x^2+8x+13}\,\textrm{d}x \\ {\rm (c)}&& &\int_0^1\arcsin x\,\textrm{d}x \end{align} $$

 

(9) Three cards with an even score

Question. Rebecca has 9 cards, each with a number on them. The numbers are: 2, 2, 3, 4, 5, 5, 6, 6, 7, 9. She picks three cards at random without replacement. Rebecca multiplies three numbers to get a score. Calculate the probability that the score is an even number.

 

(10) A challenging geometry question (STEP level)

Question. The line $L$ has equation $y = c − mx$, with $m > 0$ and $c > 0$. It passes through the point $R (a, b)$ and cuts the axes at the points $P(p, 0)$ and $Q(0, q)$, where $a, b, p$ and $q$ are all positive. Find $p$ and $q$ in terms of $a, b$ and $m$. As $L$ varies with $R$ remaining fixed, show that the minimum value of the sum of the distances of $P$ and $Q$ from the origin is $\left(a^{\frac12}+b^{\frac12}\right)^2$ and find in a similar form the minimum distance between $P$ and $Q$.

 

(11) A problem on coordinate geometry

Question. Triangle $HJK$ is isosceles with $HJ=HK$ and $JK=\sqrt{80}$. (i) $H$ is the point with coordinates $(-4,1)$. (ii) $J$ is the point with coordinates $(j,15)$ where $j < 0$. (iii) $K$ is the point with coordinates $(6,k)$. (iv) $M$ is the midpoint of $JK$. (v) The gradient of $HM$ is 2. Find the value of $j$ and the value of $k$.

 

(12) Another problem on coordinate geometry (GCSE)

Question. In the diagram, $ABC$ is the line with equation $ y=-\frac12x+5 $. It is also given that $AB=BC$ and that $D$ is the point with coordinates $(-13,0)$. Find an equation of the line through $A$ and $D$.

 

(13) Implicit differentiation and the second order derivatives (7 problems)

Question. Given $x^2+y^2=r^2$, show that $ \frac{{\rm d}y}{{\rm d}x}=-\frac{x}{y} $ and $ \frac{{\rm d}^2y}{{\rm d}x^2}=-\frac{r^2}{y^3} $ and so on.

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3. Non-technical stuff

 

(1) 0의 유래 - A brief history of zero

 

(2) 원주율($\pi$)의 간략한 역사 - A brief history of $\pi$

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