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2250-2 7:30am Lecture Record Week 1 S2012

Last Modified: December 11, 2011, 14:30 MST.    Today: September 20, 2018, 08:22 MDT.

Week 1, Jan 9 to 13: Sections 1.1,1.2,1.3,1.4.

9 Jan: Details about exams and dailies. Intro to DE, section 1.1.

  1. Fundamental theorem of calculus. Method of quadrature [integration method in Edwards-Penney].
    Slides: Fundamental Theorem of Calculus, Method of quadrature, Example. (133.9 K, pdf, 03 Mar 2012)
  2. Three Fundamental Examples introduced: growth-decay, Newton Cooling, Verhulst population.
    Slide: Three Examples (11.8 K, pdf, 28 Aug 2006)
  3. Background from precalculus, logs and exponentials. Decay Equation Derivation.
    Transparencies: Background Log+exponential. Problem 1.2-2 by J.Lahti. Decay law derivation. (213.3 K, pdf, 23 Jul 2009)
  4. Black and Lahti presentations of problem 1.2-1 and 1.2-2. Lahti presentations problems 1.2-5,8,10.
    Transparencies: Three Examples. Solved problems 1.2-1,2 by Tyson Black, Jennifer Lahti, GBG (257.0 K, pdf, 26 Aug 2010)
    Transparencies: Solved problems 1.2-5,8,10 by Jennifer Lahti, GBG (139.1 K, pdf, 26 Aug 2010)
    Manuscript: Fundamentals, exponential modeling, applications, differential equations, direction fields, phase line, bifurcation, computing, existence (1122.3 K, pdf, 05 May 2011)

Textbook reference: Sections 1.1, 1.2.
Example for problem 1.2-1, similar to 1.2-2.
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.
    Syllabus, Writing Suggestions, GradeSheet
    2250-1: Syllabus S2012 (238.8 K, pdf, 26 Dec 2011)
    2250: How to improve written work (26.6 K, pdf, 09 Dec 2011)
    2250-1: Gradesheet S2012. Used as a book-mark. (134.1 K, pdf, 26 Dec 2011)

10 Jan: Quadrature. Section 1.2. Maple lab 1.

Projection: Tyson Black 1.2-1, Jennifer Lahti 1.2-2
Topics on Quadrature
 Exercises 1.2-4, 1.2-6, 1.2-10 discussion.
 Integration details and how to document them using handwritten
   calculations like u-substitution, parts, tabular.
   Maple integration methods are possible [later in the course].
   Integral table methods.
   Integration theory examples.
 Method of quadrature: Using Parts, tables, maple.
   Discuss exercise 1.2-2 and exercise 1.2-10.
 Reference for the method of quadrature:

Manuscript: The method of quadrature (with drill problems). (247.9 K, pdf, 11 Dec 2011)
Maple lab 1
    Quadratic equations.
      Inverse FOIL, complete the square, quadratic formula.
    Theory of Equations.
      Factor and root theorems.
      Division algorithm.
      Rational root theorem.
      Descartes rule of signs.
      Fundamental theorem of algebra.
    Maple tutorials start next week. Maple lab 1 is due soon, 
    please print it from the link

Maple: Lab 1, Introduction (76.7 K, pdf, 09 Dec 2011)

11 Jan: Integral table. Direction fields. Section 1.3

 Integration tables
   The first 20 entries in the front cover of our textbook are
     required background.
   Drill: Quadrature
     Compute the integral of du/(1+u^2), 2u du/(1+u^2).
   Integrals of rational functions have answers:
       polynomial + log + arctan + constant.
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.
 The rules:
   1. Solutions don't cross.
   2. Threaded solutions pass other solutions with tangent line slope
      nearly matching the nearby solutions.
 Discuss problem 1.3-8.
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. To save
the xerox work, please print 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)
Direction field references:
Manuscript: Direction fields (540.9 K, pdf, 05 Jan 2010)
Slides: Summary of Peano, Picard, Direction Fields. (293.7 K, pdf, 03 Mar 2012)
 Topics on Direction fields
 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.
Picard and Peano Theorems
  The Picard-Lindelof theorem and the Peano theorem are
  found in the web references below. The theorems appear in 
  section 1.3 of the textbook, without names.
  PEANO THEOREM [brief statement]
    y'=f(x,y), y(x0)=y0 has at least one solution 
    provided f is continuous.
  PICARD-LINDELOF THEOREM [brief statement]
    y'=f(x,y), y(x0)=y0 has at one and only one solution 
    provided f is continuous differentiable.

Manuscript: Picard-Lindelof and Peano Existence theory. (304.0 K, pdf, 11 Dec 2011)
Slides: Peano and Picard Theory (22.5 K, pdf, 17 Jan 2007)
Transparency: Picard-Lindelof and Peano Existence [1.3-14, Dirichlet]. (40.5 K, pdf, 20 Jan 2006)
Text: Background material functions and continuity (1.3-14). (5.0 K, txt, 05 Jan 2010)
Remarks on Exercises 1.3
  How to thread curves on a direction field: Exercise 1.3-8.
  Picard-Peano Example
    y'=3(y-1)^(2/3), y(0)=1, similar to 1.3-14, from Peano-Picard slide above.
  Exercise 1.3-14:
  Justifications in exercise 1.3-14 are made from background
  material in the calculus, taken from the link above "Background ... continuity".
Distribution of maple lab 1. No printed maple lab 1? Print a copy from here:
Maple: Lab 1, Introduction (76.7 K, pdf, 09 Dec 2011)
Intro to Maple lab 1
  Theory of equations review
    quadratic equations,
    Factor and root theorem,
    division algorithm,
    recovery of the quadratic from its roots.

12 Jan: Intro by Thu teaching assistant.

 Patrick Bardsley
      email address and phone in the syllabus.
 Discuss submitted work presentation ideas.
 Drill, examples, questions.
 Discuss problems sections 1.2, 1.3, 1.4.
 Discussion of Exam review plan for the semester.

13 Jan: Theory and Examples for Separable Equations, sections 1.4, 2.1

Review Topics
  Drill: Direction fields.
          Two Threading Rules.
          Picard and Peano Theorems.
    Question. We draw threaded solutions from some dot in the graphic.
                   How do we choose the dots? What do they represent?
    Question. What does dy/dx=f(x,y), y(x0)=y0 have to do with threaded curves?
    True and false trig formulas:
      arctan(tan(theta))=theta  [false],
      tan(arctan(x))=x [true].
Partial Fractions
    Start topic of partial fractions, to be applied again in 2.1-2.2.
    To be studied: Heaviside's method. Sampling method [a Fail-safe method].
         Please do the reading, listed at the end of this page.
    Definition: A partial fraction is a constant divided by a polynomial
    with exactly one root, that is, c/(x-r)^k. The root can be real or complex.
Definition of separable DE.
  Examples: 1.4-6,12,18.
  See the web site Problem Notes 1.4 for complete answers and
Some separability tests.
  Read the first slide link below, Tests I, II, III.
    Problem notes for separable equations.
    html: Problem notes S2012 (5.2 K, html, 08 Apr 2012)
    References for separable DE.
    Slides: Separable DE method. Tests I, II, III. Equilibrium solutions (148.9 K, pdf, 03 Mar 2012)
    Manuscript: Method of quadrature (247.9 K, pdf, 11 Dec 2011)
    Manuscript: Separable Equations (171.3 K, pdf, 31 Aug 2009)
    Text: How to do a maple answer check for y'=y+2x (0.2 K, txt, 27 Jan 2005)
    Transparencies: Section 1.4 and 1.5 Exercises (465.0 K, pdf, 26 Aug 2003)
    Separable equations depend on partial fraction theory, reading below.
    References on partial fractions. We study (1) sampling, (2) method of atoms, (3) Heaviside cover-up.
    Slides: Partial Fraction Theory (160.7 K, pdf, 03 Mar 2012)
    Manuscript: Heaviside coverup partial fraction method (152.1 K, pdf, 07 Aug 2009)
    Manuscript: Heaviside's method and Laplace theory (186.8 K, pdf, 20 Oct 2009)