Assignment 10: Mid-semester Reflections and Euler’s Method

Weeks 11 and 12

Mid-semester Reflections:

You received mid-semester grades and feedback from me about a week ago.  Now it is your turn to give me some feedback.  The most important objectives of this course are to develop your mathematical reasoning and problem solving skills.  In particular, I hope that you are learning to use mathematics to investigate interesting questions, to structure your understanding of the world around you, to formulate and solve problems, and to communicate your solutions to others.  (See the Syllabus for a more detailed articulation of the goals and objectives of this course.)

1.      How do you feel you are progressing in learning to use mathematics to investigate interesting questions, solve problems, and communicate your solutions to others?

2.      Which class activities and assignments are most helpful for you in achieving the goals of this course as outlined in the Syllabus?

3.      What do you need to do to meet the goals of this course as outlined in the Syllabus?

4.      What specific suggestions do you have for the instructor?  What should we continue to do, what could we change about the way the class is organized to help you achieve the goals of this class?

Type out your responses to these questions as a Word document, and submit them by uploading this document to your folder for this course.  Submit this document by noon on Wednesday, [due date].  I will collate your responses and discuss your suggestions at the beginning of class on Friday, [following due date]. 

Monday, [day 29]

·        Project 2 was due. 

·        Introduce Chapter 5

Section 5.1: Raindrops: Three models (Galileo's model, Stokes Law, velocity-squared model) for the speed of a falling raindrop are presented.  In class last Friday, we discussed Galileo’s model, which assumes that the only force acting on the raindrop is gravity (g) and ignores air resistance.  We realized that the predicted speed of a falling raindrop did not correspond with our experiences of walking in the rain.  On Monday, we worked through Stokes’ Law, which is a reasonably good model for small raindrops, and the velocity-squared model, which is a better model for fat raindrops.

Wednesday, [day 30]

·        Section 5.2:  Euler's Method

o       Section 5.2: Euler’s Method – Getting numerical approximations

We walked through the Activities in this section as a whole class.  Euler’s method gives us a strategy for approximating the solution of a differential equation – even if we cannot take the antiderivative by hand.  The idea is to use the point-slope form of the equation for a line, and to walk a very short distance (Dx) along the tangent line.  Then we walk another short distance (Dx) … and another (Dx) … inching our way along an approximation to the curve.  By increasing n, we make Dx smaller, and the approximation gets better. 

Friday, [day 31]

·        Class activity on Euler’s Method (available as a separate handout)

Monday, [day 32]

·        Review of Trigonometry

·        Section 5.3 – 5.4: Periodic Motion and Circular Functions

I will give a lecture over important concepts of these two sections. 

For the time being, I want to focus on the trigonometry.  We will review some basic trigonometry, and motivate the rules for derivatives of the sine and cosine functions.  Thus, we will focus on Sections 5.3.3 – 5.4.1 (doing just enough of the physics to motivate the trigonometry).

Wednesday, [day 33]

·        Overview of Trigonometry in Calculus

·        Derivatives of Sines and Cosines

Friday, [day 34]

·        Complete Sections 5.3 and 5.4