Category: Assignments (Page 2 of 3)

OpenLab #3: Numerical Methods PROJECT – Part 2

March 12, 2017 / Jonas Reitz / 42 Comments

This post contains additional information related to the Numerical Methods Project posted last week, due March 23rd.

Submission guidelines. Your completed project will consist of three elements:

A working numerical methods calculator using your choice of technology (spreadsheet, programming, mathematical software), as described in the previous post.
HOW TO SUBMIT:
- If your project is a spreadsheet, either share it with me (if it is in Google Docs or a similar cloud-based platform), or email the file to me as an attachment.
- If your project is code, please submit it using an online coding site like ideone.com – once your code is working on the site, you can simply submit a link. If you are using a programming language not supported by ideone.com, you can email the source code to me.
- If your project uses mathematical software, either share it with me (if it is in MatLab Online or a similar cloud-based platform), or email the file to me as an attachment.
Solutions to the following example using each of the three methods studied in class (Euler’s Method, Improved Euler’s Method, and Runge-Kutta), generated by your numerical methods calculator. Submit using the same method as in part 1.

Project Example. Given the differential equation $dy/dx = \frac{xy}{x-y}$ and initial condition $y(6)=0.8$ , approximate the value of $y(7.1)$ using Euler’s Method, Improved Euler’s Method, and Runge-Kutta.

For each method, choose a step size that gives four correct digits following the decimal point. How many steps are required to obtain this level of precision?

NOTE: The actual solution is $y(7.1)=3.700936$ …

Writing assignment. Write one or two paragraphs (minimum 300 words) responding to the following. Leave your response as a comment on this post.
1. Describe your project and how it works.
2. Describe the process of building your numerical methods calculator. What kind of technology did you decide to use, and why? Did you encounter any unexpected challenges in completing this project?
3. Why do we need numerical methods in addition to the other methods studied in the class?
4. Why is this assignment included in the class (instead of just computing the various methods using a calculator)?
5. Please include a link to your project (if it is online), or clearly state that you will be sending me the files by email (and don’t forget to do it!).

OpenLab #3: Numerical methods PROJECT – Example Data

March 10, 2017 / Jonas Reitz / 0 Comments

Below you will find complete solutions for the Example given the previous post – you can use this to test your project.

Example. Given the differential equation and initial condition , approximate the value of using step size

ACTUAL ANSWER: y(2.5) = 3.49201…

Euler’s Method
i	h	x_i	y_i	k = f(x_i,y_i)	y_(i+1)
0	0.05	1.5	2.2	0.6	2.23
1	0.05	1.55	2.23	0.67425	2.2637125
2	0.05	1.6	2.2637125	0.74903	2.301164
3	0.05	1.65	2.301164	0.8240397	2.342365985
4	0.05	1.7	2.342365985	0.8989889128	2.387315431
5	0.05	1.75	2.387315431	0.9735989982	2.435995381
6	0.05	1.8	2.435995381	1.047604158	2.488375588
7	0.05	1.85	2.488375588	1.120752581	2.544413217
8	0.05	1.9	2.544413217	1.192807443	2.60405359
9	0.05	1.95	2.60405359	1.26354775	2.667230977
10	0.05	2	2.667230977	1.332769023	2.733869428
11	0.05	2.05	2.733869428	1.400283836	2.80388362
12	0.05	2.1	2.80388362	1.465922199	2.87717973
13	0.05	2.15	2.87717973	1.52953179	2.95365632
14	0.05	2.2	2.95365632	1.590978049	3.033205222
15	0.05	2.25	3.033205222	1.650144125	3.115712428
16	0.05	2.3	3.115712428	1.706930708	3.201058964
17	0.05	2.35	3.201058964	1.761255718	3.289121749
18	0.05	2.4	3.289121749	1.813053901	3.379774445
19	0.05	2.45	3.379774445	1.862276305	3.47288826
20	0.05	2.5	y(2.5) = 3.47288826

Improved Euler’s Method
i	h	x_i	y_i	k1	z_(i+1)	k2	y_(i+1)
0	0.05	1.5	2.2	0.6	2.23	0.67425	2.23185625
1	0.05	1.55	2.23185625	0.6728114063	2.26549682	0.7476025438	2.267366599
2	0.05	1.6	2.267366599	0.746106721	2.304671935	0.8211456538	2.306547908
3	0.05	1.65	2.306547908	0.8195979758	2.347527807	0.8946013641	2.349402892
4	0.05	1.7	2.349402892	0.8930075421	2.394053269	0.9677033899	2.395920665
5	0.05	1.75	2.395920665	0.9660694182	2.444224136	1.040198278	2.446077357
6	0.05	1.8	2.446077357	1.038530378	2.498003876	1.111846414	2.499836777
7	0.05	1.85	2.499836777	1.110150981	2.555344326	1.18242289	2.557151124
8	0.05	1.9	2.557151124	1.180706432	2.616186446	1.251718216	2.61796174
9	0.05	1.95	2.61796174	1.249987303	2.680461105	1.319538895	2.682199895
10	0.05	2	2.682199895	1.317800105	2.7480899	1.385707852	2.749787594
11	0.05	2.05	2.749787594	1.383967716	2.81898598	1.450064721	2.820638405
12	0.05	2.1	2.820638405	1.448329675	2.893054889	1.512465995	2.894658297
13	0.05	2.15	2.894658297	1.510742331	2.970195413	1.572785045	2.971746481
14	0.05	2.2	2.971746481	1.571078871	3.050300425	1.630912022	3.051796253
15	0.05	2.25	3.051796253	1.629229215	3.133257714	1.686753629	3.134695825
16	0.05	2.3	3.134695825	1.685099802	3.218950815	1.740232793	3.220329139
17	0.05	2.35	3.220329139	1.738613261	3.307259802	1.791288237	3.308576677
18	0.05	2.4	3.308576677	1.789707988	3.398062076	1.839873957	3.399316225
19	0.05	2.45	3.399316225	1.838337624	3.491233107	1.885958617	3.492423631
20	0.05	2.5	y(2) = 3.492423631

Runge-Kutta
i	h	x_i	y_i	k1 = f(x_i,y_i)	k2 = f(x_i+.5h,y_i+.5hk1)	k3 = f(x_i+.5h, y_i+.5hk2)	k4 = f(x+h,y+hk3)	Runge-Kutta y_(i+1) = y_i + h*(k1+2k2+2k3+k4)/6
0	0.05	1.5	2.2	0.6	0.6366875	0.6359881445	0.6728554594	2.23181839
1	0.05	1.55	2.23181839	0.6728407481	0.709821466	0.7090934081	0.746181552	2.267292157
2	0.05	1.6	2.267292157	0.7461662747	0.7832936203	0.7825394711	0.8197042176	2.306438296
3	0.05	1.65	2.306438296	0.8196884061	0.8568207014	0.856043244	0.8931456109	2.349259645
4	0.05	1.7	2.349259645	0.8931293019	0.9301304558	0.9293326184	0.9662395087	2.395745436
5	0.05	1.75	2.395745436	0.9662227434	1.002962858	1.002147687	1.038732462	2.445871905
6	0.05	1.8	2.445871905	1.038715285	1.075071194	1.074241825	1.110384803	2.499602956
7	0.05	1.85	2.499602956	1.110367265	1.146222996	1.145382627	1.180971517	2.556890873
8	0.05	1.9	2.556890873	1.18095367	1.216200837	1.215352702	1.250282954	2.617677071
9	0.05	1.95	2.617677071	1.250264856	1.284802979	1.283950319	1.318125413	2.681892878
10	0.05	2	2.681892878	1.318107122	1.351843875	1.350989913	1.384321567	2.749460347
11	0.05	2.05	2.749460347	1.384303145	1.417154527	1.416302445	1.448710757	2.820293079
12	0.05	2.1	2.820293079	1.448692267	1.480582715	1.479735625	1.51114915	2.894297063
13	0.05	2.15	2.894297063	1.511130657	1.541993079	1.541154007	1.57150976	2.971371518
14	0.05	2.2	2.971371518	1.57149133	1.601267084	1.600438946	1.629682352	3.051409732
15	0.05	2.25	3.051409732	1.629664051	1.658302858	1.657488442	1.685573222	3.134299898
16	0.05	2.3	3.134299898	1.685555117	1.713014923	1.712216872	1.739104879	3.219925928
17	0.05	2.35	3.219925928	1.739087035	1.765333814	1.764554613	1.79021561	3.308168257
18	0.05	2.4	3.308168257	1.790198091	1.815205609	1.814447568	1.838858971	3.398904619
19	0.05	2.45	3.398904619	1.838841842	1.862591365	1.861856614	1.885003188	3.492010794
20	0.05	2.5	y(2.5) = 3.492010794

OpenLab #3: Numerical methods PROJECT – Part 1

March 8, 2017 / Jonas Reitz / 1 Comment

Numerical methods provide a way to compute (approximate) values of solutions to differential equations, even when we cannot solve the equations exactly. The drawback is the large number of numerical calculations required to obtain a desired value and level of precision. In this project, you will use technology to implement the various numerical methods and use your technological solution to solve differential equations problems.

Assignment (Due Thursday, March 23). Create a numerical methods calculator. You can choose your technology tool for this job – use any one of the following:

a spreadsheet (Excel, Google Sheets, or other spreadsheet)
- if you choose to create a spreadsheet, you should have columns for $x, y, f(x,y)$ , and so on, and each stage should appear in its own row
a programming language (Java, Perl, or other programming language)
- if you choose to write code, your program should output the values of $x, y, f(x,y)$ and so on at each stage
mathematical software (MatLab, Maple, Mathematica, or other mathematical software)
- if you choose to use mathematical software, your program should output the values of $x, y, f(x,y)$ and so on at each stage

How to submit. Part 2 of this project will talk about how to submit your project – you will be asked to upload your solution (spreadsheet, or code, or mathematical software document) and to write about what you did. For now, focus on getting your solution working.

Requirements:

Your solution should allow you to solve problems of this type:

Example. Given the differential equation and initial condition , approximate the value of using step size

Your solution must be able to carry out Euler’s Method, Improved Euler’s Method, and Runge-Kutta (you may implement these as three separate spreadsheets or programs if you wish).
Your solution should display all the points $(x,y)$ found along the way, not just the final point.
Your solution should also display other values found while carrying out each method:
1. Euler’s Method: display the slope $f(x,y)$ at each stage
2. Improved Euler’s: display the values of $k1, k2$ at each stage
3. Runge-Kutta: display the values of $k1, k2, k3, k4$ at each stage
4. You can display other values as well, if you wish (for example, the intermediate y-value in the Improved Euler method that we refer to as $z$ ).
Your solution may NOT use any built-in version of these methods (for example, most mathematical software contains a built-in command for Euler’s Method – you can use this to check your work, but you need to create your own solution).
You should be able to relatively easily change the initial condition, step size, and target value.
You should be able to relatively easily change the differential equation.

Test your project. Solution data for the above example using Euler, Improved Euler, and Runge-Kutta will be posted this weekend so you can test your project. You can also use examples from class to test your work, since you know what the solutions are.

Assignments Week 6

March 6, 2017 / Jonas Reitz / 0 Comments

WeBWorK: Assignments 7-Trench-EulersMethod and 7b-Trench-ImprovedEulersMethod will be due on Tuesday 3/14 at midnight.
OpenLab: OpenLab #3: Numerical methods PROJECT - Part 1 due March 23rd

Exam 1 Grades are posted, and SPECIAL OFFER

March 5, 2017 / Jonas Reitz / 0 Comments

Hi everyone,

The grades for Exam 1 are posted on the Grades page (email me if you have forgotten the password).

With some exceptions, you will notice that the scores are not as high as you might have liked! This exam covered a lot of material, and relied on a great deal of prior knowledge and skills (especially Calculus and Algebra). With that in mind, I am giving you the option to improve your score through the ONE-TIME SPECIAL OFFER below (note: this offer will almost certainly *not* be repeated on future exams), due in two weeks on Tuesday 3/21.

Let me know if you have any questions,
Prof. Reitz

Exam 1 Special Offer – earn bonus points. You can improve your grade on the exam, by doing the following:

Choose ONE OR TWO problems in which you did NOT earn full points. You are working to earn back (some of ) the points you missed on this problem. For each problem you chose:
Do the problem over, neatly and completely, start to finish, on a separate sheet of paper.
Include the following information on the sheet:
1. Your name
2. The date
3. The problem number
4. Your original score on the problem (out of 25)
On the same sheet, write a short paragraph (at least two complete sentences) explaining why you lost points.
1. If you lost points due to a mistake or mistakes, explain what you did wrong (this is to let me know that you understand your error).
2. If you lost points due to not completing the problem, then explain how to solve it (this is to let me know that you have learned how to solve it).
Hand in your original exam and your corrected problem(s) and explanation(s), stapled together, in class on Tuesday, 3/21.
Bonus points will be added to your Exam score based on the number of points you missed on the chosen problem(s), the accuracy of your corrections and explanation, and your overall grade on the exam. Bonus points are limited as follows:
1. If you received less than 50% on the exam, you can earn a maximum of 30 bonus points.
2. If you received between 50% – 59% on the exam, you can earn a maximum of 20 bonus points.
3. If you received between 60% – 69% on the exam, you can earn a maximum of 15 bonus points.
4. If you received between 70% – 79% on the exam, you can earn a maximum of 10 bonus points.
5. If you received between 80% – 89% on the exam, you can earn a maximum of 5 bonus points.
6. If you received 90% or more on the exam, you may not earn any bonus points.

Assignments Week 5

February 26, 2017 / Jonas Reitz / 0 Comments

WeBWorK: Assignments 6-Modeling1-PopulationAndCooling and 6b-Modeling2-FallingObjects will be due on Tuesday 3/7 at midnight.
OpenLab: OpenLab #2 (technology survey) must be completed by this Thursday, March 2nd.

Assignments Week 4

February 19, 2017 / Jonas Reitz / 0 Comments

UPDATE 2/21/17: Because Cooling problems were not covered in class today (Tuesday 2/21), I have changed the due date for Assignment 6-Modeling1-PopulationAndCooling to the following week (Tuesday, 3/7). However, I strongly suggest that you complete problems 1, 2 and 3 prior to the exam (population problems).

Exam #1 will take place on Tuesday, 2/28, covering material through Sec 4.2. A review sheet will be distributed on Tuesday, 2/21, and posted to the OpenLab.

WeBWorK: none
NOTE: Assignments 6-Modeling1-PopulationAndCooling and 6b-Modeling2-FallingObjects will be due one week after the exam, on Tuesday 3/7 at midnight.
OpenLab: none

Assignments Week 3

February 13, 2017 / Jonas Reitz / 0 Comments

WeBWorK: Assignments 4-Bernoulli-nosub, 4b-Homogeneous-YoverX and 5-Exact Due Tuesday 2/21 at midnight
OpenLab: none

OpenLab #1: Advice from the Past

February 6, 2017 / Jonas Reitz / 60 Comments

In Fall 2014 I taught this same course. At the end of the semester, I gave my students the following assignment:

Imagine that you are invited to speak on the first day of MAT 2680, to give advice to entering students. Write at least three sentences … describing what you would tell them.

To see the assignment and the students’ responses, follow this link.

Your assignment, due at the beginning of class next Thursday, February 16th, is to:

Read through ALL the responses (there are 57 of them, but many of these are short replies to other comments).
Write a reply to this post (1 paragraph) responding to all of the following:
1. What advice seemed most relevant to you personally? Why? (you can copy/paste a short statement, or put it in your own words)
2. Based on this advice, what changes can you make right now to help you succeed in this course?

Extra Credit. For extra credit, write a response to one of your classmates’ comments. Do you have any advice? Be kind.

Assignments Week 2

February 6, 2017 / Jonas Reitz / 0 Comments

WeBWorK: 3-Separable and 4-Bernoulli-nosub, Due Tuesday 2/14 at midnight
UPDATE: As classes were cancelled on Thursday, 2/9, due to snow, WeBWorK assignment 4-Bernoulli-nosub is extended to the following week, 2/21, at midnight.
OpenLab: Complete your first OpenLab Assignment “Advice from the Past” by Thursday, 2/16, at the beginning of class.