5013 - Slope Fields and Euler’s Method AP Calculus.
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Transcript of 5013 - Slope Fields and Euler’s Method AP Calculus.
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