Slope Fields: Motivation, Activity, APcalcAB04#6, DE matching, TI89 Matching slope fields handout with accompanying TI-89 script will make graphing and teaching about differential equations for the AP Calculus exam so much easier. StudyCards™ for AP Physics and Calculus will also be shared. 194 ■ Slope fields the Easy Way on the TI-89 and StudyCard™ Sharing 60-Minute Hands-On ■ IntermediateTI-89 ■ Calculus, Physics Slope Fields: Motivation, Activity, APcalcAB04#6, DE matching, TI89 Sean Bird Beginning with the 2004AB Calc Exam “Geometric interpretation of differential equations via slope fields and the relationship between slope fields and solutions curves for differential equations.” This topic has been part of the topic outline for Calc BC since the 1998 exam AP Physics & Calculus Covenant Christian High School 7525 West 21st Street Indianapolis, IN 46214 Phone: 317/390.0202 x104 Email: seanbird@covenantchristian.org Website: http://covenantchristian.org/bird Psalm 111:2

Three/Four Pronged approach We now know how to algebraically solve separable differential equations. Today we will graphically find the particular solution for any DE using SLOPE FIELDS. In calcBC we will then numerically solve DEs using Euler’s Method Verbally (Recall new sign chart policy)

Slope Fields If you enjoyed connecting the dots, you’ll love slope fields It is a graphical method to find a particular solution to any differential equation. a DE is nothing but a slope equation -

Given: Sketch the slope field …

But how? Substitute the x and y into the differential equation for each of the points. Plot this slope on the graph.

1st the slopes will be positive when the differential equation is positive, i.e. And x2 is always positive (except when x=0) y – 1 > 0 when y > 1 ANSWER: y > 1, but x ≠ 1

Separate the variables Exponentiate both sides Let K

Form B 2004#5 was the same question but

The thought process… E C A G Isoclines along horizon. lines → DE depends only on y Same slope along vert. line → DE depends only on x Sinusoidal Consider a specific point – what is the slope there. What makes the slope zero? What values of x and y make the slope 1? Note y'(x) = x+y has same slopes along the diagonal. Solve the separable DE B D I H F K J

Running the DEmatch script Select TEXT EDITOR under APPS Open… Folder: calculus, Variable: dematch Green (chartreuse for Titanium) diamond, Up arrow will quickly jump cursor to the top. 2nd, up arrow scrolls up a page. Press F4 all the way down Note what each step accomplishes and READ important rules at the end.

StudyCard Stacks Trig/PreCalc: study of functions Calculus: identity ** limits83 ** limits89 ** derivative ** integral ** Algebra: slope, slope/intercept for 89 for 83/84 new! Physics: AP Physics for the TI89 ** 83* Prefix for the 83/84 ** History of Physics for the 89 ** or 83/84 **new!