- Fundamental theorem of calculus. Method of quadrature [integration method in Edwards-Penney].
: Fundamental Theorem of Calculus, Method of quadrature, Example. (133.9 K, pdf, 03 Mar 2012)**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. Lahti presentations problems 1.2-5,8,10.
: 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)**Transparencies**: Fundamentals, exponential modeling, applications, differential equations, direction fields, phase line, bifurcation, computing, existence (1122.3 K, pdf, 05 May 2011)**Manuscript**

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

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 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:

Maple lab 1Quadratic 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

Integration tablesThe 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:

Direction field references:

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.

Picard and Peano TheoremsThe 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.

Remarks on Exercises 1.3How 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".

Intro to Maple lab 1Theory of equations review quadratic equations, Factor and root theorem, division algorithm, recovery of the quadratic from its roots.

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.

Review TopicsDrill: 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 FractionsStart 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 methods.

Some separability tests.Read the first slide link below, Tests I, II, III.

Separable equations depend on partial fraction theory, reading below.