# 2250 Lecture Record Week 1 F2009

**Last Modified**: September 12, 2009, 10:41 MDT. **Today**: March 24, 2018, 21:23 MDT.

### Week 1, Aug 24 to 28: Sections 1.1,1.2,1.3,1.4.

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

**Topics**
- Fundamental theorem of calculus.
- Method of quadrature [integration method in Edwards-Penney].
- Three Fundamental Examples introduced: growth-decay, Newton Cooling, Verhulst population.
- Background from precalculus, logs and exponentials.
- Decay Equation Derivation.
- 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)

Week 1 references (documents, slides)

**Slides**: Fundamental Theorem of Calculus, Method of quadrature, Example. (56.0 K, pdf, 23 Jul 2009)

**Slide**: Three Examples (11.8 K, pdf, 28 Aug 2006)

**Transparencies**: Three Examples. Solved problems 1.2-1,2,5 by Tyson Black, Jennifer Lahti, GBG (292.0 K, pdf, 23 Jul 2009)

**Transparencies**: Background Log+exponential. Problem 1.2-2 by J.Lahti. Decay law derivation. (213.3 K, pdf, 23 Jul 2009)

**Manuscript**: The method of quadrature (with drill problems). (125.8 K, pdf, 26 Aug 2008)

**Manuscript**: Direction fields (540.9 K, pdf, 05 Jan 2010)

**Transparencies**: Direction field examples, isocline method. (219.5 K, pdf, 23 Jul 2009)

**Transparency**: Zoomed copy of Edwards-Penney exercise 1.3-8, to be used for homework (102.2 K, jpg, 29 Aug 2008)

**Slides**: Summary of Peano, Picard, Direction Fields. (247.3 K, pdf, 26 Jul 2009)

**Manuscript**: Picard-Lindelof and Peano Existence theory. (125.9 K, pdf, 18 Jan 2006)

**Slides**: Peano and Picard Theory (22.5 K, pdf, 17 Jan 2007)

**Slide**: Picard-Lindelof and Peano Existence Example, similar to 1.3-14. (40.5 K, pdf, 20 Jan 2006)

**Text**: Background material functions and continuity (1.3-14). (4.9 K, txt, 30 Aug 2007)

**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]:
- Solution curves don't cross, and
- 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)The Picard-Lindelof theorem and the Peano theorem are
found in these web references.

**Slides**: Summary of Peano, Picard, Direction Fields. (247.3 K, pdf, 26 Jul 2009)

**Manuscript**: Picard-Lindelof and Peano Existence theory. (125.9 K, pdf, 18 Jan 2006)

**Slides**: Peano and Picard Theory (22.5 K, pdf, 17 Jan 2007)

**Transparencies**: Picard-Lindelof and Peano Existence [1.3-14]. (40.5 K, pdf, 20 Jan 2006)

**Text**: Background material functions and continuity (1.3-14). (4.9 K, txt, 30 Aug 2007)

**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)References for separable DE.

**Slides**: Separable DE method. Tests I, II, III. Equilibrium solutions (101.2 K, pdf, 01 Sep 2009)

**Manuscript**: Method of quadrature (125.8 K, pdf, 26 Aug 2008)

**Manuscript**: Separable Equations (171.3 K, pdf, 31 Aug 2009)

**Slides**: Partial fraction theory (121.5 K, pdf, 30 Aug 2009)

**Manuscript**: Heaviside's coverup method partial fractions (186.8 K, pdf, 20 Oct 2009)

**Text**: How to do a maple answer check for y'=y+2x (0.2 K, txt, 27 Jan 2005)

**Transparencies**: Section 1.4 Exercises (465.0 K, pdf, 26 Aug 2003)