- Today I illustrated how to use Fit[] to approximate exponential data. If we have exponentially looking data, then applying Log[] to the second components of data will make this data linear or close to linear. Then we can use Fit[] to find best line approximation. Composing this line with Exp[] will provide a good approximation to the original data, see 20131203.nb.
- The rest of the class I spend constructing the Niels Fabian Helge von Koch curve from scratch, see 20131203.nb.
- Today I illustrated the functions Fit[] and FindFit[] which are essential for Problem 2 (b-3) on Assignment 3, see 20131202.nb. In this file you will also find my short program to illustrate Monty Hall problem. More probability related programs are in Probabilities.nb.
- I added the file DataSets.nb in which I illustrate the use of and differences among Interpolation[], FindFit[] and Fit[].
- Recall that on/in 20131122.nb we discussed the Google E problem which is relevant for Problem 1. In this file we also discussed nesting of functions which is important for Problem 4.
- The Assignment 3 has been posted today.
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In Problem 1 on Assignment 3 you need to use the Calendar package: Needs["Calendar`"]. This package contains several functions related to date arithmetic: DaysBetween[], DaysPlus[], DateQ[].
An important warning: Before you use the commands from the Calendar package you must load the Calendar package. If you by mistake try to use one of the Calendar package commands without having the package loaded, you will have to quit the kernel (Evaluation→Quit Kernel→Local) before loading the package. - The commands Fit[] and FindFit[] are essential for Problem 2 (b-3) on Assignment 3.
- The file Probabilities.nb is dealing with a problem which is similar to questions in Problem 3 on Assignment 3.
- The files vonKoch_curve.nb and PythagorasTree.nb are "guides" for the construction of fractals in Problem 4 on Assignment 3.
- In the first part of Problem 4 on Assignment 3 you are asked to create a function that would produce iterations of the quadratic type 2 curve. In the first picture below I show the 0th iteration in blue and the 1st iteration in red. I emphasize the points that are used. You do not need to do this on your plots. In the second picture below I show the 1st iteration in blue and the 2nd iteration in red. The large picture is the fourth iteration of the quadratic type 2 curve.
- In the second part of Problem 4 on Assignment 3 you are asked to create a function that would produce iterations of the Cesaro fractal which depends on angle $\alpha$. The four pictures below show the 0th, 1st, 2nd and the 3rd iteration of the Cesaro fractal with the small angle $\alpha = \pi/16$.
- Below is an animated gif file that cycles through the 1st, 2nd, 3rd, 4th, 5th and the 6th iteration of the Cesaro fractal and, within each of the iterations cycles through all the angles starting from $\pi$, proceeding towards $0$ and then back to $\pi$ in steps of $\pi/50$. The animation starts by the first iteration and cycles through angles from $\pi$ to $0$ and back to $\pi$. This is repeated for 2nd, 3rd, 4th 5th and 6th iteration. For each iteration there are 101 pictures.
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Place the cursor over the image to start the animation.
- The second assignment has been posted today.
- Very important tools in Mathematica are Module[] and Pure Function. I will illustrate these tools in class.
- Problem 1. This problem is quite similar to the problem solved in ExpfTanl.nb. In the notebook ExpfTanl.nb there are many hints on what to do in this problem.
- Problem 2 is an exploration of a surprising function. You should use what you learned in Calculus combined with the power of Mathematica. The function given in this problem has some surprising features that you need to discover. When you discover these features, formulate them in clear answers to my questions. Support your answers with calculations and illustrations. To explore the function use the functions D[] to find the derivative and FullSimplify[] to simplify it. To find special points you can use Solve[], Reduce[] or FindRoot[] where appropriate. However, Mathematica needs a lot of human help in this problem. It is a very good example of human-machine interaction.
- Problem 3.
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In the first part of this problem you need to unify, as explained in the problem, three animations given below.
Place the cursor over the image to start the animation.
Place the cursor over the image to start the animation.
Place the cursor over the image to start the animation.
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In the second part you need to reproduce two pictures from Wikipidia's
Cardioid page
I made an animation that unifying these two pictures. You should be able to produce something like this. If not you can present one animation and one picture.
Place the cursor over the image to start the animation.
- In the third part of Problem 3 you need to produce generalized cardioids. The animations below are large. On a slow internet connection it takes a while for them to load.
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Below is the cardioid generated by a wheel with radius 1/2 rolling on a circle with radius 1.
Place the cursor over the image to start the animation.
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Below is the cardioid generated by a wheel with radius 1/3 rolling on a circle with radius 1.
Place the cursor over the image to start the animation.
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Below is the cardioid generated by a wheel with radius 2 rolling on a circle with radius 1.
Place the cursor over the image to start the animation.
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Below is the cardioid generated by a wheel with radius 3/2 rolling on a circle with radius 1.
Place the cursor over the image to start the animation.
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In the first part of this problem you need to unify, as explained in the problem, three animations given below.
- I will comment on Problem 4 in class.
- Remember that each definition of a function in Mathematica should be preceded by Clear[];. Inside Clear[] you place the name of your function and all the variables that you are using. Please let me know if I did not follow my own rule in some of my files. I call this rule PPP for Prudent Programming Practice.
- Today, see 20131105.nb we briefly discussed how to place special points on graphs. This is important in the square sin and square cos in Problem 1. Another useful hint for Problem 1: explore the function Clip[].
- Today we also reviewed the file Nothing_Lucky_about_13.nb which give you some ideas that you can use in Problem 3. Please be aware of the differences between Problem 3 and the problem discussed in the above mentioned file. In Problem 3 you will use just three functions FullSimplify[], Sum[] and Table[]. Problem 3 is inspired by the identity \[ \frac{\bigl(\sin(3\pi/7)\bigr)^2}{\bigl(\sin(\pi/7)\bigr)^4} +\frac{\bigl(\sin(\pi/7)\bigr)^2}{\bigl(\sin(2\pi/7)\bigr)^4} +\frac{\bigl(\sin(2\pi/7)\bigr)^2}{\bigl(\sin(3\pi/7)\bigr)^4}=28. \] In part A-3 I ask you to explore analogous expressions when 7 is replaced by 5 and the analogous expressions when 7 is replaced by 9. An analogous expression involving 5 will have fewer summands than the given expression (in fact it will have two summands). An analogous expression involving 9 will have more summands than the given expression (in fact it will have four summands).
- Today we also discussing a problem relevant to Problem 4: given three noncollinear points, find the center of the circle that passes through each of the given points. As we have seen today, Mathematica easily solves this problem symbolically. Given points $P_1 = (a_1, b_1)$, $P_2 = (a_2, b_2)$, $P_3 = (a_3, b_3)$, we seek the point $C = (x,y)$ which is at the same distance, say $r$, from each of the points $P_1$, $P_2$, $P_3$. That is, we need to solve for $x, y, r$ the following system of equations: \begin{align*} \sqrt{(a_1 - x)^2 + (b_1-y)^2} & = r \\ \sqrt{(a_2 - x)^2 + (b_2-y)^2} & = r \\ \sqrt{(a_3 - x)^2 + (b_3-y)^2} & = r \end{align*} Mathematica is very efficient in solving this system. It finds: \begin{align*} x & = \frac{a_2^2 \left(b_3-b_1\right)+a_3^2 \left(b_1-b_2\right)+\left(b_2-b_3\right) \left(a_1^2+\left(b_1-b_2\right) \left(b_1-b_3\right)\right)} {2\left(a_3 \left(b_1-b_2\right)+ a_1\left(b_2-b_3\right)+ a_2 \left(b_3-b_1\right)\right)} \\[10pt] y & = \frac{ a_1\left(a_2^2-a_3^2+b_2^2-b_3^2\right) +a_3 \left(b_1^2-b_2^2\right) +a_2 \left(a_3^2-b_1^2+b_3^2\right) +\left(a_3-a_2\right) a_1^2 -a_2^2 a_3}{2\left(a_3 \left(b_1-b_2\right) +a_1\left(b_2-b_3\right) +a_2\left(b_3-b_1\right)\right)} \\[10pt] r & =\frac{1}{2} \sqrt{ \frac{\left(\left(a_1-a_2\right)^2+\left(b_1-b_2\right)^2\right) \left(\left(a_1-a_3\right)^2+\left(b_1-b_3\right)^2\right) \left(\left(a_2-a_3\right)^2+\left(b_2-b_3\right)^2\right)} {\left(a_3 \left(b_2-b_1\right)+a_2 \left(b_1-b_3\right)+a_1 \left(b_3-b_2\right)\right)^2}} \end{align*} With a little bit of human help these expressions can be brought to a more symmetric form: \begin{align*} x & =\phantom{-} \frac{\left(b_1-b_2\right) \left(b_1-b_3\right) \left(b_2-b_3\right) + a_1^2 \left(b_2-b_3\right)+a_3^2 \left(b_1-b_2\right)+a_2^2 \left(b_3-b_1\right)}{2 \bigl(\left(a_1-a_2\right) \left(b_2-b_3\right)-\left(a_2-a_3\right) \left(b_1-b_2\right)\bigr)} \\[10pt] y & = -\frac{\left(a_1-a_2\right) \left(a_1-a_3\right) \left(a_2-a_3\right)+b_1^2\left(a_2-a_3\right) + b_2^2 \left(a_3-a_1\right) + b_3^2 \left(a_1-a_2\right) }{2 \bigl(\left(a_1-a_2\right) \left(b_2-b_3\right)-\left(a_2-a_3\right) \left(b_1-b_2\right)\bigr)} \\[10pt] r & = \frac{\sqrt{ \left(\left(a_1-a_2\right)^2+\left(b_1-b_2\right)^2\right) \left(\left(a_1-a_3\right)^2+\left(b_1-b_3\right)^2\right) \left(\left(a_2-a_3\right)^2+\left(b_2-b_3\right)^2\right)}} {2\bigl| \left(a_1-a_2\right) \left(b_2-b_3\right)-\left(a_2-a_3\right) \left(b_1-b_2\right) \bigr|} \end{align*} This circle is known as the circumscribed circle of the triangle $P_1P_2P_3$.
- Assignment 1 is due before class on Tuesday, November 12, 2013. You should name your assignment YourlastnameA1.nb. This file should be placed in your Dropbox directory Dropbox\307_Yourlastname that you shared with me. Please do not write anything else except your assignments in this directory.
- Your homework notebooks should be organized neatly. A notebook should start with a title cell. In a separate cell should be your name. Individual assigned problems should be presented as sections. More about organization of your notebook you can find in the information sheet. One of the posted movies at my Mathematica page explains how to organize your homework notebooks. I pointed that out in my comments.
- Problem 1. Today I showed how to define funny trigonometric function based on a parabola in two different ways: using Mod[], see 20131104.nb. There is another way to define such a function using ArcSin[Sin[x]], see 20130729.nb. More about this you can find in More_on_Trig.nb. As we saw in class, you might experience problems with Plot[]; the graph of a funny trig function being not connected. These problems are caused by the plot option Exclusions->Automatic. Changing Exclusions->Automatic to Exclusions->None might fix the problem.
- For Problem 1 you will need to carefully read the file TheBeautyOfTrigonometry.nb. Please pay attention to tricks that I introduce in that file. Reading this file should be a learning experience.
- When you adopt the content of The beauty of trigonometry to your funny Cos, Sin it is essential to pay attention to the proper domains for the variables involved. Here proper means that there should be no overlap in the parametric plots. In 3-d parametric plots overlaps can slow down plotting considerably. Your notebook should evaluate in less than 60 seconds. If it is slower, then comment out the slow parts. That is enclose the slow parts in (* *).
- Today I showed how to define a recursive function in Mathematica, see the beginning of 20131104.nb. More useful stuff for Problem 2 is at my Mathematica page, Section: Recursively defined functions. There are two kinds of functions in this problem: recursively defined functions and functions given by closed form expressions, that are defined in terms of the variable and known functions. These different kind of functions are defined differently in Mathematica. Do not mix two kinds of definitions. This is pointed out in Section: Recursively defined functions.
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Your sources for this class are:
- This webpage.
- My Mathematica page.
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Your Dropbox directory Dropbox\307_Files and the directory Math\Curgus\307 on school's K-drive. The contents of these two directories should be identical.
- The file Primer2013.nb has a lot of useful general Mathematica information.
- For some assignment problems there are specific related files that you will need to read.
- Each class period I will use Mathematica for various demonstrations. I will save these files with the current date. For example, next Thursday's file will be 20130724.nb.
- Mathematica's help file. I find this help file truly helpful. Use it.
- Remember that only you and I have access to your Dropbox directory Dropbox\307_Yourlastname. This directory is exclusively for submitting your homework assignments.
- Remember that the entire class has access to the directory Dropbox\307_Files. Please do not write anything to this directory and do not delete any files from there. You can copy them in your private directory.
- Mathematica part of the class will start on Thursday, October 31, 2013.
- The information sheet
- We will use
- To get started with Mathematica see my Mathematica page. Please watch the videos that are on my Mathematica page before the first class.
- We also have
