Slope Fields: Motivation, Activity, APcalcAB04#6, DE matching, TI89

Sean Bird

AP Physics & CalculusCovenant Christian High School7525 West 21st StreetIndianapolis, IN 46214Phone: 317/390.0202 x104Email: seanbird@covenantchristian.org Website: http://covenantchristian.org/bird Psalm 111:2

“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

194 ■ Slope fields the Easy Way on the TI-89 and StudyCard™ Sharing

60-Minute Hands-On ■ IntermediateTI-89 ■ Calculus, Physics

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.

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 -

0limx

dy ydx x

Given:

2 ( 1)dy

x ydx

Let’s 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.

2( 1)dy

x ydx

2( 1)dy

x ydx

1

2

0

( 1) ( 1) 10yx

dy

dx

0

2

0

( )0 01)0(xy

dy

dx

1

2

1

( 1) 01)1(xy

dy

dx

1

2

0

( ) ( 11 ) 10yx

dy

dx

1

2

2

( 1) 11)2(xy

dy

dx

1

2

3

( )1 21)3(xy

dy

dx

2( 1)dy

x ydx

i) 1st the slopes will be positive when the differential equation is positive, i.e.

ii) And x2 is always positive (except when x=0)

iii) y – 1 > 0 when y > 1

ANSWER: y > 1, but x ≠ 1

2( 1) 0x y

When are the slopes positive?

2( 1)d

xy

xy

d

Separate the variables

2

1

yd

y

dxx

21

1dy x dx

y

31

ln 13

y x C

Exponentiate both sides3 31 1

3 31 x C x Cy e e e

Let K Ce31

3

313

1 , apply IC

1

(0) 3

2

x

x

y e

y

fK

e

Given f(0)=3, what is the particular solution.

The thought process…

i) Isoclines along horizon. lines → DE depends only on y

ii) Same slope along vert. line → DE depends only on x

iii) Sinusoidal

iv) Consider a specific point – what is the slope there.

v) What makes the slope zero? What values of x and y make the slope 1?

vi) Note y'(x) = x+y has same slopes along the diagonal.

vii) Solve the separable DE

E

C

A

G

B

D

I

H

F

K

J

1. Select TEXT EDITOR under APPS

2. Open…

3. Folder: calculus, Variable: dematch

4. Green (chartreuse for Titanium) diamond, Up arrow will quickly jump cursor to the top. 2nd, up arrow scrolls up a page.

5. Press F4 all the way down

6. Note what each step accomplishes and READ important rules at the end.

Running the DEmatch script

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!