The Lizard and the Fly

November 19, 2021November 19, 2021 / Michael Xue / Leave a comment

An open cylinder garbage can has a height of 4 ft and a circumference of 6 ft. On the inside of the can, 1 ft from the top, is a fly. On the opposite side of the can, 1 ft from the bottom and on the outside, is a lizard. What is the shortest distance that the lizard must walk to reach the fly?

To reach the fly, the lizard must first walk a path on the exterior surface of the garbage can to the opening. From there, it continues walking along a path on the interior surface to catch the prey.

The most salient piece of knowledge we have about minimum distance is that the shortest distance between two points comes from the straight line connecting them. However, it is not immediately apparent how to apply this knowledge in the present problem, since we are constrained to the bent lines on a curved wall. Nonetheless, the curved wall can be cut and flattened, result in a completely flat two dimensional surface. We can then connect the points by a straight line.

Fig. 1

Having flattened out the wall in the manner shown in Fig. 1, we see the lizard’s walking path consists two segments of straight line. The length of the path is

$\sqrt{x^2+3^2} + \sqrt{(3-x)^2+1^2}.\quad\quad\quad(1)$

Fig. 2

Three centuries before the birth of Christ, Euclid discovered the reflection law of light: if a ray of light is reflected by a mirror from point $A$ to point $B$ , the point of reflection $P$ is such that the rays $AP$ and $PB$ make equal angles with the mirror ( $\theta_1 = \theta_2$ in Fig. 2). However, it was Heron who first observed that if the point $P$ on the mirror were such that $\theta_2 \ne \theta_2$ , then the resulting total path length $AP+PB$ would be longer; i.e.,

$AP + PB$ is minimized when $\theta_1 = \theta_2.$

For the lizard, when $\alpha = \beta$ , it walks the least (Fig. 1).

From $\alpha=\beta$ , we have $\cot(\alpha) = \cot(\beta);$ i.e.,

$\frac{x}{3} = \frac{3-x}{1}\quad\quad\quad(2)$

Solving (2) gives

$x = \frac{9}{4}$

so that (1) yields

$\sqrt{(\frac{9}{4})^2 + 3^2} + \sqrt{(3-\frac{9}{4})^2+1^2} = 5.$

Therefore, the shortest distance the lizard must walk is $5$ ft.

Exercise-1 Find $x$ that minimizes $\sqrt{x^2+25} + \sqrt{(x-4)^2+4}.$

Exercise-2 If $x$ and $N$ are real numbers such that $N=\sqrt{5x+6}+\sqrt{7x+8}$ , then what is the smallest possible value of $N?$

Exercise-3 An open cylinder garbage can has a height of 4 ft and a circumference of 6 ft. On the inside of the can, 1 ft from the top, is a fly. On the opposite side of the can, 1 ft from the bottom inside the can, is a lizard. What is the shortest distance that the lizard must walk to reach the fly?

Wallis’ Pi

November 9, 2021November 24, 2021 / Michael Xue / Leave a comment

There is a remarkable expression for the number $\pi$ as an infinite product. Starting with definite integral $\int\limits_{0}^{\frac{\pi}{2}} \sin^{m}(x)\;dx, m=0,1,2,3,4...$ , we derive it as follows:

$\int\limits_{0}^{\frac{\pi}{2}} \sin^m(x)\;dx$

$=\int\limits_{0}^{\frac{\pi}{2}}\sin^{m-1}(x)\cdot\sin(x)\;dx$

$= \int\limits_{0}^{\frac{\pi}{2}} \sin^{m-1}(x)\cdot(-\cos(x))'\;dx$

By $\int\limits_{a}^{b} u\cdot v'\;dx = u\cdot v \bigg|_{b}^{a}- \int\limits_{b}^{a} u' \cdot v\;dx,$

$= \sin^{m-1}(x)\cdot(-\cos(x))\bigg|_{0}^{\frac{\pi}{2}} - \int\limits_{0}^{\frac{\pi}{2}}(\sin^{m-1}(x))'\cdot(-\cos(x))\;dx$

$= 0 - \int\limits_{0}^{\frac{\pi}{2}}(m-1)\sin^{m-2}(x)\cos(x)\cdot(-\cos(x))\;dx$

$= +\int\limits_{0}^{\frac{\pi}{2}}(m-1)\sin^{m-2}(x)\cdot\cos^2(x)\;dx$

$\overset{\cos^2(x) = 1-\sin^2(x)}{=}\int\limits_{0}^{\frac{\pi}{2}}(m-1)\sin^{m-2}(x)\cdot (1-\sin^{2}(x))\;dx$

$= \int\limits_{0}^{\frac{\pi}{2}}(m-1)\sin^{m-2}(x) - (m-1)\sin^{m}(x)\;dx$

$= (1-m)\int\limits_{0}^{\frac{\pi}{2}}\sin^{m}(x)\;dx + (m-1)\int\limits_{0}^{\frac{\pi}{2}}\sin^{m}(x)\;dx.$

That is,

$m\int\limits_{0}^{\frac{\pi}{2}}\sin^{m}(x)\;dx = (m-1)\int\limits_{0}^{\frac{\pi}{2}}\sin^{m-2}(x)\;dx.$

And so,

$\int\limits_{0}^{\frac{\pi}{2}}\sin^{m}(x) \; dx= \frac{m-1}{m}\int\limits_{0}^{\frac{\pi}{2}}\sin^{m-2}(x)\;dx.\quad\quad\quad(1)$

By repeated application of (1) we have the following values for $I_{m} = \int\limits_{0}^{\frac{\pi}{2}} \sin^{m}(x)\;dx$ :

For even $m$ ,

$I_{2n} = \left(\prod\limits_{k=0}^{n-1}\frac{2n-2k-1}{2n-2k}\right)I_0\overset{I_0 = \int\limits_{0}^{\frac{\pi}{2}}\;dx = \frac{\pi}{2}}{\implies}I_{2n}= \left(\prod\limits_{k=0}^{n-1}\frac{2n-2k-1}{2n-2k}\right)\cdot \frac{\pi}{2}.$

Similarly, for odd $m$ ,

$I_{2n+1} = \left(\prod\limits_{k=0}^{n-1}\frac{2n+1-2k-1}{2n+1-2k}\right)\cdot I_{1}\overset{I_1 = \int\limits_{0}^{\frac{\pi}{2}}\sin^{1}(x)\;dx = 1}{\implies} I_{2n+1}= \prod\limits_{k=0}^{n-1}\frac{2n-2k}{2n-2k+1}.$

i.e.,

$I_{2n} = \frac{2n-1}{2n}\cdot\frac{2n-3}{2n-2}\cdot\frac{2n-5}{2n-4}\cdot\frac{2n-7}{2n-6}\cdot...\cdot\frac{5}{6}\cdot\frac{3}{4}\cdot\frac{1}{2}\cdot\frac{\pi}{2}\quad\quad\quad(2)$

$I_{2n+1} =\frac{2n}{2n+1}\cdot\frac{2n-2}{2n-1}\cdot\frac{2n-4}{2n-3}\cdot\frac{2n-6}{2n-5}\cdot ...\cdot\frac{6}{7}\cdot\frac{4}{5}\cdot\frac{2}{3}\quad\quad\quad\quad\quad(3)$

Since for $0<x<\frac{\pi}{2}, 0< \sin(x) < 1,$ we have $\sin^{2n-1}(x) > \sin^{2n}(x) > \sin^{2n+1}(x).$ It means

$\int\limits_{0}^{\frac{\pi}{2}}\sin^{2n-1}(x) > \int\limits_{0}^{\frac{\pi}{2}}\sin^{2n}(x)\;dx > \int\limits_{0}^{\frac{\pi}{2}}\sin^{2n+1}(x)\;dx>0$

or,

$I_{2n-1} > I_{2n} > I_{2n+1}>0.$

Hence,

$\frac{I_{2n-1}}{I_{2n+1}}>\frac{I_{2n}}{I_{2n+1}} > 1.\quad\quad\quad(4)$

Moreover, we have

$I_{2n+1} \overset{(1)}{=} \frac{(2n+1)-1}{2n+1}I_{(2n+1)-2} = \frac{2n}{2n+1} I_{2n-1},$

so that

$\frac{I_{2n-1}}{I_{2n+1}} = \frac{2n+1}{2n}.\quad\quad\quad(5)$

And,

$\frac{I_{2n}}{I_{2n+1}} \overset{(2), (3)}{=} \frac{(2n+1)\cdot(2n-1)^2\cdot (2n-3)^2...7^2\cdot 5^2\cdot 3^2}{(2n)^2\cdot (2n-2)^2\cdot (2n-4)^2\cdot ...\cdot 6^2\cdot 4^2\cdot 2^2}\cdot \frac{\pi}{2}$

$= \frac{(2n)^2\cdot (2n-1)^2\cdot (2n-2)^2\cdot (2n-3)^2\cdot (2n-4)^2\cdot...\cdot 7^2\cdot 6^2\cdot 5^2\cdot 4^2\cdot 3^2\cdot 2^2}{(2n)^4(2n-2)^4\cdot (2n-4)^4...6^4\cdot 4^4\cdot 2^4}\cdot (2n+1)\cdot\frac{\pi}{2}$

$= \frac{\left((2n)\cdot (2n-1)\cdot (2n-2)\cdot (2n-3)\cdot (2n-4)\cdot...\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\right)^2}{(2^4\cdot n^4)\cdot (2^4\cdot(n-1)^4) \cdot (2^4\cdot (n-2)^4)\cdot...\cdot (2^4\cdot 3^4)\cdot (2^4\cdot 2^4)\cdot 2^4}\cdot (2n+1)\cdot\frac{\pi}{2}$

$= \frac{((2n)!)^2}{\underbrace{2^4\cdot 2^4\cdot ...\cdot 2^4}_{n 2^4s}\cdot (n\cdot(n-1)\cdot(n-2)\cdot ...\cdot 3\cdot 2\cdot 1)^4}(2n+1)\cdot\frac{\pi}{2}$

$= \frac{((2n)!)^2\cdot (2n+1)}{2^{4n}\cdot(n!)^4}\cdot \frac{\pi}{2}$

gives

$\frac{I_{2n}}{I_{2n+1}} = \frac{((2n)!)^2(2n+1)}{2^{4n}(n!)^4}\cdot\frac{\pi}{2}.\quad\quad\quad(6)$

Substituting (5) and (6) into (4) yields

$\frac{2n+1}{2n} >\frac{((2n)!)^2(2n+1)}{2^{4n}(n!)^4}\cdot\frac{\pi}{2}>1,$

Since $\lim\limits_{n \rightarrow \infty} \frac{2n+1}{2n} = 1, \lim\limits_{n \rightarrow \infty}1 = 1,$ by Squeeze Theorem for Sequences,

$\lim\limits_{n \rightarrow \infty}\frac{((2n)!)^4(2n+1)}{2^{4n}(n!)^4} \cdot \frac{\pi}{2}= 1\implies \lim\limits_{n \rightarrow \infty}\frac{((2n)!)^2(2n+1)}{2^{4n}(n!)^4}=\frac{2}{\pi}.$

Consequently,

$\lim\limits_{n \rightarrow \infty} \frac{2^{4n}(n!)^4}{((2n)!)^2(2n+1)}= \frac{\pi}{2},$

i.e.,

$\pi = 2\cdot\lim\limits_{n \rightarrow \infty}\frac{2^{4n}(n!)^4}{((2n)!)^2(2n+1)}.$

This is Wallis’ product representation for $\pi,$ named after John Wallis who discovered it in 1665.

Maxima knows Wallis’ Pi:

Fig. 1

So does Mathematica:

Fig. 2

Its convergence to $\pi$ is illustrated in Fig. 3:

Fig. 3

A Cautionary Tale of Compute Inverse Trigonometric Functions

October 27, 2021October 29, 2021 / Michael Xue / Leave a comment

From $\arcsin(x)$ ‘s definition:

$\{(x, y) | \sin(y) = x, -\frac{\pi}{2} \le y \le \frac{\pi}{2}\},$

we have

$\arcsin(0) = 0, \arcsin(\frac{1}{2})= \frac{\pi}{6},\arcsin(\frac{\sqrt{3}}{2}) = \frac{\pi}{3}, \arcsin(1) = \frac{\pi}{2}$

and,

$\arcsin(-\frac{1}{2})= -\frac{\pi}{6},\arcsin(-\frac{\sqrt{3}}{2}) = -\frac{\pi}{3}, \arcsin(-1) = -\frac{\pi}{2}.$

To obtain other values of $\arcsin(x)$ , we may simply solve

$\sin(y) - x = 0$

for $y$ where $-1 \le x \le 1.$

For example (see Fig. 1), solving $\sin(y) = \frac{1}{\sqrt{2}}$ for $y$ gives $y \approx \frac{\pi}{4}.$ It is in agreement with the fact that $\arcsin(\frac{1}{\sqrt{2}})= \frac{\pi}{4}.$

Fig. 1

In Fig. 2, we compute $\arcsin(x)$ from repeatedly solving $\sin(y) = x$ for $y$ where

$x=-1+i \cdot\frac{1-(-1)}{n}, i=1,2,...n.$

Fig. 2

Since $\sin(y)$ is a periodic function, $\sin(y) = x$ has infinitely many solutions. It is possible that the solution obtained by Newton’s method lies outside of $[-\frac{\pi}{2}, \frac{\pi}{2}]$ , the range of $\arcsin(x)$ by its definition. Such solution cannot be considered the value of $\arcsin(x).$

Fig. 3

Exercise-1 Compute $\arccos(x)$ by solving $\cos(y) = x$ for $y.$

Exercise-2 Explain Fig. 3.

A Mathematical Allegory

October 20, 2021October 22, 2021 / Michael Xue / Leave a comment

We have defined function $\arcsin$ as a set:

$\{(x, y) | \sin(y) =x, \frac{-\pi}{2} \le y \le \frac{\pi}{2}\}.$

By definition,

$\arcsin(-1) = \frac{-\pi}{2}, \arcsin(0)=0, \arcsin(1) = \frac{\pi}{2}$

and

$\frac{d}{dx} \arcsin(x) = \frac{1}{\sqrt{1-x^2}}.$

It means that $\arcsin(x)$ is the unique solution of

$\frac{dy}{dx} = \frac{1}{\sqrt{1-x^2}}\quad\quad\quad(\star)$

where $y(-1)=-\frac{\pi}{2}, y(0)=0$ and $y(1)=\frac{\pi}{2}.$

To compute $\arcsin(x)$ , we solve $(\star)$ for $y(x)$ as follows:

Integrate $\frac{dy}{dx} = \frac{1}{\sqrt{1-x^2}}$ from $-1$ to $x$ gives

$\displaystyle\int\limits_{-1}^{x}\frac{dy}{dx} \,dx=\displaystyle\int\limits_{-1}^{x}\frac{1}{\sqrt{1-\xi^2}}\; d\xi\overset{\textbf{FTC}}{\implies}y(x) - y(-1) = \displaystyle\int\limits_{-1}^{x} \frac{1}{\sqrt{1-\xi^2}}\, d\xi.$

Therefore,

$y(x) = \displaystyle\int\limits_{-1}^{x}\frac{1}{\sqrt(1-\xi^2}\,d\xi + y(-1) \overset{y(-1)=\frac{-\pi}{2}}{=} \displaystyle\int\limits_{-1}^{x}\frac{1}{\sqrt{1-\xi^2}}\,d\xi - \frac{\pi}{2}.$

That is,

$\arcsin(x) = \displaystyle\int\limits_{-1}^{x} \frac{1}{\sqrt{1-\xi^2}}\;d\xi - \frac{\pi}{2}.$

To obtain $\arcsin(x), -1 < x < 1$ , we numerically evaluate $\int\limits_{-1}^{x}\frac{1}{\sqrt{1-\xi^2}}\,d\xi$ , using function ‘quad_qags’.

Fig. 1

The result is visually validated in Fig. 2.

Fig. 2

Note: ‘romberg’, another function that computes the numerical integration by Romberg’s method will not work since it evaluates $\frac{1}{\sqrt{1-x^2}}$ at $x=-1.$

Fig. 3

An alternate approach is to solve $\frac{dy}{dx} = \frac{1}{\sqrt{1-x^2}}, y(0)=0$ as an initial-value problem of ODE using ‘rk’ , the function that implements the classic Runge-Kutta algorithm.

Fig. 4 for $0 < x < 1$

Fig. 5 $-1<x<0$

Putting the results together, we have

Fig. 6 $-1<x<1$

However, we cannot solve $\frac{dy}{dx} = \frac{1}{\sqrt{1-x^2}}, y(-1)=\frac{-\pi}{2}$ using ‘rk’:

Fig. 7

Exercise-1 Compute $\arccos(x)$ for $x \in (-1, 1)$ .

Exercise-2 Explain why ‘rk’ cannot solve $\frac{dy}{dx} = \frac{1}{\sqrt{1-x^2}}, y(-1)=\frac{-\pi}{2}$ .

Finding Derivative the “Hard” Way

October 8, 2021October 26, 2021 / Michael Xue / Leave a comment

In “Instrumental Flying“, we defined $\cosh^{-1}(x), \sinh^{-1}(x)$ as the inverse of $\cosh(x)$ and $\sinh(x)$ repectively.

To find the derivative of $\cosh^{-1}(x)$ , let

$y = \cosh^{-1}(x).$

We have

$x = \cosh(y).$

Differentiate it,

$\frac{d}{dx} x = \frac{d}{dx} \cosh(y) \implies 1=\frac{d}{dy} \cosh(y)\cdot \frac{dy}{dx},$

i.e.,

$1 = \sinh(y) \frac{dy}{dx}\implies \frac{dy}{dx} = \frac{1}{\sinh(y)}.$

By $\cosh(y)^2-\sinh(y)^2=1$ (see Exercise-1) and $\cosh(y) \ge 1$ (see Exrecise-2),

$\sinh(y)^2 = \cosh(y)^2-1 \implies |\sinh(y)| = \sqrt{x^2-1} \overset{ (\star) }{\implies} \sinh(y) = \sqrt{x^2-1}.$

And so,

$\frac{dy}{dx} = \frac{1}{\sqrt{x^2-1}} \implies \boxed{\frac{d}{dx}\cosh^{-1}(x) = \frac{1}{\sqrt{x^2-1}}}.$

Similarly, to find $\frac{d}{dx}\sinh^{-1}(x),$ let

$y = \sinh^{-1}(x)\implies x=\sinh(y).$

Differentiate it,

$\frac{d}{dx} x = \frac{d}{dx}\sinh(y) = \frac{d}{dy}\sinh(y)\frac{dy}{dx}\implies 1 = \cosh(y)\cdot\frac{dy}{dx}.$

By $\cosh(x)^2-\sinh(x)^2=1, \cosh(y) \ge 1$ ,

$\cosh(y) = \textbf{+}\sqrt{\sinh(y)^2+1} = \sqrt{x^2+1}.$

Therefore,

$1 = \sqrt{x^2+1}\frac{dy}{dx} \implies \boxed{\frac{d}{dx}\sinh^{-1}(x) = \frac{1}{\sqrt{x^2+1}}}$ .

Prove:

$\forall x \ge 0, \sinh(x) \ge 0.\quad\quad\quad(\star)$

By definition,

$\sinh(x) = \frac{e^x-e^{-x}}{2} = \frac{e^{2x}-1}{2 e^{x}} \ge 0$ , since $\forall x>0, e^{x},e^{2x} \ge 1$ (see Exercise-3).

Exercise-1 Show that $\forall x \in R, \cosh(x)^2 - \sinh(x)^2 =1$ .

Exercise-2 Show that $\forall x \in R, \cosh(x) \ge 1$ .

Exercise-3 Show that $\forall x \ge 0, e^{x} \ge 1.$

Exercise-4 Differentiate $\cos^{-1}(x), \sinh^{-1}(x)$ directly (hint: see”Deriving Two Inverse Functions“).

Oasis

October 2, 2021October 11, 2021 / Michael Xue / Leave a comment

An oasis awaits

The above image is created by Omega CAS Explorer:

The governing equations are:

$x_{n+1} = \rho + c_2(x_n\cos(t_n)-y_n\sin(t_n),$

$y_{n+1} = c_2(x_n\sin(t_n) + y_n\cos(t_n))$

where $t_n = c_1-\frac{c_3}{1+x_n^2 + y_n^2}.$

Polar plot

September 28, 2021September 27, 2021 / Michael Xue / Leave a comment

The polar coordinates $r$ and $\theta$ can be converted to the Cartesian coordinates $x$ and $y$ using the trigonometry functions:

$\begin{cases} x=r\cdot\cos(\theta) \\ y=r\cdot\sin(\theta)\end{cases}$

It follows that a figure specified in $(r, \theta)$ can be plotted by ‘plot2d’ as a parametric curve:

Fig. 1 $r = \cos(5\theta)$

It is possible to plot two or more parametric curves together:

Fig. 2 $r=\theta$ and $r=\cos(5\theta)$

An alternate is the ‘draw2d’ function, it draws graphic objects created by the ‘polar’ function:

Fig. 3 $r=\theta$ and $r=\cos(5\theta)$

Fig. 4 shows a graceful geometric curve that resembles a butterfly. Its equation is expressed in polar coordinates by

$r = e^{\sin(\theta)} - 2\cos(4\theta)+\sin(\frac{\theta}{12})^5$

Fig. 4

Let’s combine them!

September 25, 2021September 24, 2021 / Michael Xue / Leave a comment

It is possible to combine two or more plots into one picture.

For example, we solve the following initial-value problem

$\begin{cases} \frac{dy}{dx} = (x-y)/2 \\ y(0)=1 \end{cases}\quad\quad\quad(\star)$

and plot the analytic solution in Fig. 1.

Fig. 1

We can also solve $(\star)$ numerically and plot the discrete data points:

Fig. 2

Fig. 3 is the result of combining Fig.1 and Fig. 2.

Fig.3

It validates the numerical solution obtained by ‘rk’: the two figures clearly overlapped.

An Alternate Solver of ODEs

September 22, 2021September 22, 2021 / Michael Xue / Leave a comment

Besides ‘ode2’, ‘contrib_ode’ also solves differential equations.

For example,

$\frac{d^2y}{dx^2} - \frac{1+x}{x} \cdot \frac{dy}{dx}+ \frac{1}{x}\cdot y=0.$

While ‘ode2’ fails:

‘contrib_ode’ succeeds:

This is an example taking from page 4 of Bender and Orszag’s “Advanced Mathematical Methods for Scientists and Engineers“. On the same page, there is another good example:

$\frac{dy}{dx} = \frac{A^2}{x^4}-y^2, \quad\quad A$ is a constant.

Using ‘contrib_ode’, we have

It seems that ‘contrib_ode’ is a better differential equation solver than ‘ode2’:

Even though it is not perfect:

From the examples, we see the usage of ‘contrib_ode’ is the same as ‘ode2’. However, unlike ‘ode2’, ‘contrib_ode’ always return a list of solution(s). It means to solve either initial-value or boundary-value problem, the solution of the differential equation is often lifted out of this list first:

Exercise Solve the following differential equations without using a CAS:

$\frac{dy}{dx}= \frac{A^2}{x^4} - y^2$ (hint: Riccati Equation)
$\frac{d^2 y}{dx^2} = \frac{y \frac{dy}{dx}} {x}$

DIY

September 17, 2021September 17, 2021 / Michael Xue / Leave a comment

As far as I know, ‘bc2’, Maxima’s built-in function for solving two-point boundary value problem only handles the type:

$\begin{cases} y'' = f(x, y, y')\\ y(a)=\alpha, y'(b) = \beta \end{cases}\quad\quad\quad(*)$

For example, solving $\begin{cases} y'' + y (y')^3=0 \\y(0) = 1, y(1)=3 \end{cases}:$

But ‘bc2’ can not be applied to

$\begin{cases} -y'' + 9 y =12 \sin(t)\\ y(0)=y(2\pi), y'(0) = y'(2\pi) \end{cases}$

since it is not the type of (*). However, you can roll your own:

An error occurs on the line where the second boundary condition is specified. It attempts to differentiate the solution with respect to $t$ under the context that $t=0$ . i.e.,

diff(rhs(sol), 0);

which is absurd.

The correct way is to express the boundary conditions using ‘at’ instead of ‘ev’:

The following works too as the derivative is obtained before using ‘ev’:

Nonetheless, I still think using ’at’ is a better way: