- Fundamental theorem of calculus. Method of quadrature [integration method in Edwards-Penney].
: Fundamental Theorem of Calculus, Method of quadrature, Example. (87.3 K, pdf, 06 Jan 2010)**Slides** - Three Fundamental Examples introduced: growth-decay, Newton Cooling, Verhulst population.
: Three Examples (11.8 K, pdf, 28 Aug 2006)**Slide** - Background from precalculus, logs and exponentials. Decay Equation Derivation.
: Background Log+exponential. Problem 1.2-2 by J.Lahti. Decay law derivation. (213.3 K, pdf, 23 Jul 2009)**Transparencies** - Black and Lahti presentations of problem 1.2-1 and 1.2-2.
: Three Examples. Solved problems 1.2-1,2,5 by Tyson Black, Jennifer Lahti, GBG (292.0 K, pdf, 23 Jul 2009)**Transparencies**

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.

Maple tutorials start next week. Maple lab 1 is due soon, please print it from the link

More on the method of quadrature:

- Week 1 Syllabus, Format Suggestions, GradeSheet, Maple Lab 1

- Week 1 references (documents, slides)

Topics on QuadratureExercises 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. Integral table methods. Integration theory examples. Method of quadrature: Using Parts, tables, maple. Discuss exercise 1.2-8.

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. 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. Use this prepared copy:

Direction field reference:

Topics on Direction fieldsThreading 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.

- The Picard-Lindelof theorem and the Peano theorem are
found in these web references.

Collected in class Page 16, 1.2: 4, 6. 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. Exercise 1.3-14: Justifications in exercise 1.3-14 are made from background material in the calculus:

Distribution of maple lab 1. No printed maple lab 1? Print a copy from here:

Drill, examples, questions.

Discuss problems sections 1.2, 1.3, 1.4.

Discussion of Exam review plan for the semester.

Review TopicsDrill: Direction fields. Two Threading Rules. Picard and Peano Theorems. We draw threaded solutions from some dot in the graphic. How do we choose the dots? What do they represent? What does dy/dx=f(x,y), y(x0)=y0 have to do with threaded curves? Drill: Quadrature integral of du/(1+u^2), 2u du/(1+u^2). True and false trig formulas: arctan(tan(theta))=theta [false], tan(arctan(x))=x [true].

Definition of separable DE.Examples: 1.4-6,12,18. See the web site Problem Notes 1.4 for complete answers and methods.

Some separability tests.

- References for separable DE.

Theory of separable equations section 1.4.Separation test:Define F(x)=f(x,y0)/f(x0,y0), G(y)=f(x0,y), then FG=f if and only if y'=f(x,y) is separable.Non-Separable Testf_x/f depends on y ==> y'=f(x,y) not separable f_y/f depends on x ==> y'=f(x,y) not separableBasic theory of y'=F(x)G(y):y(x) = H^(-1)( C1 + int(F)), H(u)=int(1/G,u0..u). Solutions y=constant are called equilibrium solutions. Find them using G(c)=0. Non-equilibrium solutions arise from y'/G(y)=F(x) and a quadrature step.

Implicit and explicit solutions.Discussion of answer checks for implicit solutions and also explicit solutions. Troubles with explicit solutions of y'= 3 sqrt(xy) [1.4-6]. Separable DE with no equilibrium solutions. Separable DE with infinitely many equilibrium solutions. The list of answers to a separable DE. Influence of an initial condition to extract just one solution formula from the list. Examples for Midterm 1 problem 2: y'=x+y, y'=x+y^2, y'=x^2+y^2 Example 1: Show that y'=x+y is not separable using TEST I or II (partial derivative tests). Example 2: Find the factorization f=F(x)G(y) for y'=f(x,y), given (1) f(x,y)=2xy+4y+3x+6 [ans: F=x+2, G=2y+3]. (2) f(x,y)=(1-x^2+y^2-x^2y^2)/x^2 [ans: F=(1-x^2)/x^2, G=1+y^2].