- Assignment 3 has been posted Tuesday last week. We already discussed Problem 1. See the file 20190530_A3P1.nb.
-
In Problem 1 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. - I will comment on Problem 2 and Problem 3 in class.
- The files vonKoch_curve.nb and PythagorasTree.nb are "guides" for the construction of fractals in Problem 4.
- In the first part of Problem 4 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 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.
|
|
|---|---|
|
|
|
|
|---|---|
|
|
Place the cursor over the image to start the animation.
- Assignment 2 is has been posted today. Comments about the problems are below.
- Very important tools in Mathematica and on this assignment are Module[] and Pure Function. We have seen examples of pure functions when we used FindSequenceFunction[] in Problem 3 on Assignment 1. I will talk about these two objects more in class.
- Problem 1 is an exploration of a surprising function. The point of the problem is to find accurate answers to the questions that are asked and support them with as rigorous explanations as you can. You should use what you learned in Calculus combined with the power of Mathematica. 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.
- Problem 2 consists of three parts: cycloids, the cardioid and generalized cardioids.
- Cycloids In this part of the problem you need to reproduce three animations presented below. Some hints on how to derive the parametric equation for the standard cycloid you can find at my Math 225 page
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.
-
Cardioid In this part of the problem you need to reproduce two pictures from Wikipidia's
Cardioid page
I made an animation that unifies these two pictures. You should be able to produce something like this. If not, you can present one animation and one picture. For the derivation of the parametric equation of a cardioid you can loog at my Math 225 page
Place the cursor over the image to start the animation.
- Generalized Cardioids In this part of Problem 2 you need to produce generalized cardioids as shown in the animations below. These animated gif files are large. On a slow internet connection it takes a while for them to load.
-
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.
-
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.
-
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.
-
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.
-
Below is the cardioid generated by a wheel with radius 1/2 rolling on a circle with radius 1.
- Cycloids In this part of the problem you need to reproduce three animations presented below. Some hints on how to derive the parametric equation for the standard cycloid you can find at my Math 225 page
- The file Probabilities.nb deals with questions which are similar to questions in Problem 3.
- I will comment on Problem 4 in class.
- Assignment 1 is has been posted on today. Your notebook with solutions should be named Yourlastname_A1.nb. This file should be placed in your Dropbox directory Dropbox\307_Yourlastname that you shared with me. Please do not save anything else except your assignments in this directory.
- The due date for Assignment 1 is Wednesday, May 15, 2019 at noon. Assignment 2 will be posted on Monday, May 13, 2019 and discussed in class the following Tuesday.
- Your homework notebooks should be organized neatly. The organization should follow the format of my assignment notebook 201920_A1.nb. Your homework 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.
- The movie Hands-on start to Mathematica: Notebooks which is also posted movies at my Mathematica page explains how to organize your homework notebooks. I pointed that out in my comments.
- For Problem 1 you will need to carefully read the file TheBeautyOfTrigonometry_8.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 (* *). For example, in my Primer.nb notebook I commented out several parts that are slow to evaluate.
- 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.
- Mathematica part of the class starts on Thursday, May 2, 2019.
- 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. Watching the movies is essential for being able to organize your homework notebooks well! I will not discuss the basics of notebook structuring.