MAT 2680 Differential Equations - Reitz

"...how it differs from the rocks"

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UPDATED Midsemester Grades are posted

Hi everyone,

Updated Midsemester Grades incorporating your project, exams 1&2, WeBWorK and OpenLab assignments have been posted on the “Grades” page.  Send me an email if you want to be reminded of the password.   Exam #2 will be returned in class on Tuesday, 4/25.

Heads up: The last day to drop the class with a “W” grade is April 19th.  If you decide to drop, it is *much* better to do so before that date (but you should be aware of any financial aid impact that dropping a class may have).  Please let me know if you wish to discuss it.

Regards,
Prof. Reitz

OpenLab #4: Flipping the class – Taylor Series

Hi everyone,

After Spring Break we will be launching into a new and incredibly useful topic – using Taylor Series to solve differential equations.  Of course, using this method requires you to remember something about Taylor Series (uh oh!).  You should have studied them in Calculus II.  Don’t remember? This is your chance to refresh your memory.

Assignment (Due Tuesday, April 25th, BEFORE CLASS):  Watch all of the following videos carefully.  As you are watching, make a note of any questions you have.  When you are done, complete the following exercise:

Exercise:  Find at least the first 15 terms of the Taylor Series for y=e^{2x} at the point x=0.

Finally, post a comment here confirming that you watched the videos, and including the following:

  1. Confirm that you watched the videos (say something like “I watched all the videos.”)
  2. Post ONE of the terms you found while solving the exercise above (for example, you might say “One of the terms is \frac{4x^5}{15}“).  CHALLENGE: Don’t use the same term as anyone else.
  3. Ask a question about Taylor Series, either about one of the videos (tell me which one), or about Taylor Series in general. Make it an honest question – something you are confused about, or stuck on, or genuinely interested in.

Extra Credit.  For extra credit, answer someone else’s question.

Here are the videos:

Video 1: Taylor/Maclauren Series intuition

 

Video 2: Taylor Series for Cosine at x=0 (Maclaurin Series)

Video 3 (OPTIONAL): Taylor Series for Sine at x=0  (NOTE: This video is OPTIONAL, but it might help you better understand the next video)

Video 4: Visualizing Taylor Series Approximation

Exam 2 Review – CORRECTION to problem #11

Hi everyone,

Based on your feedback in class, I have taken a careful look at problem 11 on the review sheet (numerical methods). It turns out there was an error in the original differential equation – the correct version is:  y\prime = xy-x.

Once this correction is made, the answer key should be correct.  My apologies!

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

Preliminary Midsemester Grades are posted

PRELIMINARY Midsemester Grades are posted on the Grades page (send me an email if you have forgotten the password).

However, these grades are based only on two criteria: WeBWorK performance (weighted 40%) and Exam 1 (weighted 60%, includes the results of the SPECIAL OFFER), and  may not give the most accurate picture of “how you are doing in the class”.

I will be posting updated Midsemester grades during the Spring Break which include more data (namely, your Project and other OpenLab assignments, and Exam #2).  These grades will give you a better picture of your overall performance in the class so far.

Heads up: The last day to drop the class with a “W” grade is April 19th.  If you decide to drop, it is *much* better to do so before that date (but you should be aware of any financial aid impact that dropping a class may have).  Please let me know if you wish to discuss it.

-Prof. Reitz

OpenLab #3: Numerical Methods PROJECT – Scoring Guide and Solutions to Project Example

Hi everyone,

I wanted to share the scoring guide that I will use when grading your project – this is based on the Part 1 and Part 2 project assignments.  I also included my own solutions to the Project Example given in Part 2, if you’d like to compare.

Project Scoring Guide

This project is worth 15 points towards the “OpenLab” portion of your grade, equivalent to three OpenLab assignments.  These points are assigned as follows:

  • (5 points) A working numerical methods calculator using your choice of technology.
    • 3 points – solution correctly implements each method (Euler’s Method, Improved Euler’s Method, Runge-Kutta Method)
    • 1 point – solution displays intermediate points and other important values (k1, k2, etc.)
    • 1 point – ease of implementation (can easily change initial condition, step size, target value)
  • (5 points) Solutions to the Project Example
    • 2 points – value of y(7.1) is approximated by each method
    • 3 points – appropriate step size is chosen for each method
  • (5 points) Writing Assignment
    • 2 points – meet minimum length requirement (300 words)
    • 1 point – Describe your project
    • 1 point – Describe the process
    • 1 point – Why do we need numerical methods? Why is this assignment included in the class?

 

 

Project Example Solutions

The example given in the “Project Part 2” post was:

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

In my own solution, implemented first in Google Sheets and later (when I needed more computing power) in Microsoft Excel, I obtained the following results:

Euler’s Method

To obtain four correct digits after the decimal point required:

  • 275,000 steps
  • step size h = 0.000004
  • final approximation y(7.1) = 3.70090152430816

 

Improved Euler’s Method

To obtain four correct digits after the decimal point required:

  • 390 steps
  • step size h = 0.002820513
  • final approximation y(7.1) = 3.70090010561837

 

Runge-Kutta Method

To obtain four correct digits after the decimal point required:

  • 8 steps
  • step size h = 0.1375
  • final approximation y(7.1)=3.70091299278611

 

Making up for snow days

UPDATE: As decided in class on Tuesday, 3/28, we will make up for the two snow days as follows:

  1.  Class will end 8 minutes later each day (10:00am – 11:23am)
  2. There will be an assignment over spring break with the goal of getting everyone back up to speed on Taylor Series (as this will be our first topic after the break).

Regards,
Prof. Reitz 


Hi everyone,

We will need to make up some material because of the days we have lost to snowstorms this semester.  Let’s talk about it in class today.

-Prof. Reitz

 

From the Mathematics Department Chair:

In order to make up the class time, you may spread the number of minutes over the remaining 13 classes (excluding the last class for final exam) based on the following recommendation:
To make up one 75-min (3-cr) class, add 6 minutes to each of the remaining 13 classes;
To make up one 100-min (4-cr) class, add 8 minutes to each of the remaining 13 classes;
To make up one 125-min (5-cr) class, add 10 minutes to each of the remaining 13 classes;
To make up one 175-min (7-cr) class, add 13 minutes to each of the remaining 13 classes;
However, if you teach Tuesday/Thursday classes, you would have missed two classes, thus, you must double the number of minutes to be added to each of the remaining classes.  In most cases, that is not possible. I’d suggest you incorporate a combination of adding minutes to each of the remaining classes, and giving additional assignment.  

Exam 1 SPECIAL OFFER Scoring Guide

For your reference, here is the scoring guide I will be using for the EXAM 1 Special Offer

-Prof. Reitz


 

Name: ____________________

 

____ Includes Name, Date, Problem #s, original scores (up to 6 points deduction)

____ Presentation is neat, well-organized, readable (up to 4 points deduction)

____ Includes Original Exam

____ Max bonus (30 points for <50%, 20 points 50%-59%, 15 points 60%-69%, 10 points 70%-79%, 5 points 80%-89%)

 

First problem #:  ____

____ Original Score (out of 25)

____ Revised Score

____ (up to 5 points deduction if incomplete) Written explanation,  2 sentences,  what you did wrong OR how to solve the problem.

____ Bonus points earned for problem 1

 

Second problem #: ____

____ Original Score (out of 25)

____ Revised Score

____ (up to 5 points deduction if incomplete) Written explanation,  2 sentences,  what you did wrong OR how to solve the problem.

____ Bonus points earned for problem 2

 

EXAM 1 SPECIAL OFFER BONUS POINTS:   _______

(Bonus points for problems 1 and 2, with maximum bonus based on original exam score, minus any deductions, ).

EXAM 1 REVISED SCORE: ______

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