Week 1: 24 Aug, 25 Aug, 26 Aug, 27 Aug. 28 Aug.

# 2250 Lecture Record Week 1 F2009

Last Modified: September 12, 2009, 10:41 MDT.    Today: July 17, 2018, 17:24 MDT.

## 24 Aug: Details about exams and dailies. Intro to DE, section 1.1.

Topics
1. Fundamental theorem of calculus.
2. Method of quadrature [integration method in Edwards-Penney].
3. Three Fundamental Examples introduced: growth-decay, Newton Cooling, Verhulst population.
4. Background from precalculus, logs and exponentials.
5. Decay Equation Derivation.
6. Black and Lahti presentations of problem 1.2-1 and 1.2-2.
Week 1 Syllabus, Format Suggestions, GradeSheet, Maple Lab 1
2250: Syllabus F2009 (146.2 K, pdf, 09 Nov 2009)
2250: How to improve written work (25.0 K, pdf, 01 Jan 2009)
2250: Gradesheet F2009. Record-keeping system. (63.5 K, pdf, 09 Nov 2009)
Maple: Lab 1, Introduction (111.9 K, pdf, 19 Jul 2009)

## 25 Aug: Method of Quadrature. Section 1.2.

Collected exercise 1.2-2.
Exercises 1.2-4, 1.2-6, 1.2-10 discussion.
Projection: Tyson Black 1.2-1, Jennifer Lahti 1.2-2, 1.2-5,
Textbook reference: Sections 1.1, 1.2.
Example for problem 1.2-1, similar to 1.2-2.
Started topic of partial fractions, to be applied again in 2.1-2.2. Heaviside's method. Fail-safe method.
Panels 1 and 2 in the answer check for an initial value problem like 1.2-2: y'=(x-2)^2, y(2)=1.
Answer checks. Proof that "0=1" and logic errors in presentations.
Maple tutorials start next week. Maple lab 1 is due soon, please print it from the link
Maple: Lab 1, Introduction (111.9 K, pdf, 19 Jul 2009)
More on the method of quadrature:
Manuscript: The method of quadrature (with drill problems). (125.8 K, pdf, 26 Aug 2008)
Integration details and how to document them using handwritten calculations like u-substitution, parts, tabular.
Maple integration methods.
Integral table methods.

## 26Aug: Direction fields and Existence-Uniqueness. Section 1.3.

Discuss problem 1.3-8.
Collected in class Page 16, 1.2: 4, 6.
Integration theory examples. Method of quadrature. Parts, tables, maple.
Euler's directional field visualization. Tools for using Euler's idea, which reduces an initial value problem to infinitely many graphics.
The Idea: Display the behavior of all solutions, without solving the differential equation.
Lecture on exercise 1.2-8.
Direction field reference:
Manuscript: Direction fields (540.9 K, pdf, 05 Jan 2010)
Threading edge-to-edge solutions is based upon two rules [explained in the manuscript]:
1. Solution curves don't cross, and
2. Threaded solution curves nearly match tangents of nearby direction field arrows.
For problem 1.3-8, xerox at 200 percent the textbook exercise page, then cut and and paste the figure. Draw threaded curves on this figure according to the rules in the direction field document above. Use this prepared copy:
Transparency: Zoomed copy of Edwards-Penney exercise 1.3-8, to be used for homework (102.2 K, jpg, 29 Aug 2008)

## 27 Aug: Intro by TA staff, Richins and Fusi.

Discuss submitted work presentation ideas.
Drill, examples, questions.
Discuss problems 1.2-1.3.
Discussion of Exam review plan for the semester.

## 28 Aug: Picard and Peano theorems. Intro Separable DE. Section 1.4.

Collected nothing.
Drill: How to thread curves on a direction field. Exercise 1.3-8.
Drill: Picard-Lindelof Theorem, Peano Theorem.
Picard-Peano Example y'=3(y-1)^(2/3), y(0)=1, similar to 1.3-14.
. Justifications in exercise 1.3-14 are made from background material in the calculus:
Text: Background material functions and continuity (1.3-14). (4.9 K, txt, 30 Aug 2007)
Distribution of maple lab 1. No printed maple lab? Print a copy from here:
Maple: Lab 1, Introduction (111.9 K, pdf, 19 Jul 2009) Definition of separable DE.
Some separability tests.
Slides: Separable DE method. Tests I, II, III. Equilibrium solutions (101.2 K, pdf, 01 Sep 2009)
Examples: 1.4-6,12,18. See the web site Problem Notes 1.4 for complete answers and methods.
html: Problem notes F2009 (4.0 K, html, 22 Sep 2009)