5013 - Slope Fields and Euler’s Method

AP Calculus

Introduction.Anti-derivatives find families of Accumulation (position) functions from

given Rate of Change (velocity) functions.

However, 97.8% of Rate of Change functions do not have elementary Accumulation functions.

NEED A METHOD TO APPROXIMATE THE

ACCUMULATION FUNCTION

A) Slope Fields or Direction Fields – graphical (gives the impression of the family of curves)

B) Euler’s Method – numerical (finds the approximate next value on a particular curve)

A) Slope Fields or Direction Fields – graphical (gives the impression of the family of curves)

Slope Fields

Slope Fields: SketchTo Sketch:

Evaluate each point in and sketch a small slope segment at that point.

dy

dxdy

x ydx

( 0 , -1 )

( 0 , 0)

( 0 , 1 )

( 0 , 2 )

( 1 , 0)

x

y

Slope Fields: SketchTo Sketch:

Choose an initial position and sketch the curve trapped by the slope segments This is a Solution Curve or Integral Curve.

dyx y

dx

Slope Fields: SketchTo Sketch:

Choose an initial position and sketch the curve trapped by the slope segments This is a Solution Curve or Integral Curve.

cos( )dy

xdx

x

y

x

y

x

y

…………………………

Slope Fields: IdentifyA Family of Curves

To Identify a Solution Function:

if vertically parallel, f(x,y) is in terms of x only

if horizontally parallel, f (x,y) is in terms of y only.

if not parallel, f(x,y) is in terms of both x and y.

* II.

May have to test the slope at points to differentiate between possibilities. Choose an extreme point.

I.

Slope Fields : IdentifyA Family of Curves

To Identify a solution function:

if vertically parallel, f(x,y) is in terms of x only

if horizontally parallel, f (x,y) is in terms of y only.

if not parallel, f(x,y) is in terms of both x and y.

2dyx

dx

Slope Fields : IdentifyA Family of Curves

To Identify a solution function:

if vertically parallel, f(x,y) is in terms of x only

if horizontally parallel, f (x,y) is in terms of y only.

if not parallel, f(x,y) is in terms of both x and y.

dyx y

dx

Slope Fields : IdentifyA Family of Curves

To Identify a solution function:

if vertically parallel, f(x,y) is in terms sof x only

if horizontally parallel, f (x,y) is in termw of y only.

if not parallel, f(x,y) is in terms of both x and y.

2 2dy

y ydx

Slope Fields : Identify

End Behavior : For some functions in terms of BOTH x and y you must look at the local and end behaviors:large x / small x large y / small y

dyxy

dx

x

y

x

y

dyx y

dx

dyx

dx

Sample 1:

1dy

dx x

Sample 2:

2dyx

dx

Sample 3:

dyy

dx

Sample 4:

2 2dy

ydx

Sample 5:

2dy

x ydx

Sample 6:

dy x

dx y

Sample 7:

dyxy

dx

Sample 8:

EULER’S Method

• Euler’s Method – numerical (finds the approximate next value on a particular curve)

x

y

EULER’S Method

• Euler’s Method – numerical (finds the approximate next value on a particular curve)

Euler’s method is

TANGENT LINE APPROXIMATION

1 1

2 1

( )

( ) ( )

y y m x x

y y f a x

Euler’s

2 1dy

xdx

Given and initial condition ( 0 , 1 ),

Use Euler’s Method with step size to estimate the value of y at x = 2.

.5x

Euler’s Method: Approximate a value

Given and initial condition ( 0 , 1 ),

Use Euler’s Method with step size to estimate the value of y at x = 2.

2 1dy

xdx

.5x

x f x f x ( )f x f x x f x x      

  

       

  

     

  

         

  

       

  

         

  

         

  

         

  

         

0 1

.5

1

1.5

At x = 2, y

Euler’s Method: Graph

Given and initial condition ( 1 , 1 ),

Use Euler’s Method with step size to approximate f (1.3)

dyx y

dx

.1x

Last Update

• 2/16/10

p.328 41-47 odd