Post on 06-Jan-2016
description
Problem of the Day
If f (x) = sin(e-x), then f '(x) =
A) -cos(e-x)B) cos(e-x) + e-x
C) cos(e-x) - e-x
D) e-xcos(e-x)E) -e-xcos(e-x)
Problem of the Day
If f (x) = sin(e-x), then f '(x) =
A) -cos(e-x)B) cos(e-x) + e-x
C) cos(e-x) - e-x
D) e-xcos(e-x)E) -e-xcos(e-x)
Euler's Method is a numerical approach
to approximating the particular solution of the differential equation y'
Born April 15, 1707 in Basel, Switzerland
Leonhard Euler
Euler's Method is a numerical approach
to approximating the particular solution of the differential equation y'
From this starting point, you proceed in the direction indicated by the slope.
Using a small step h, move along the tangent line until you arrive at the point (x , y ) wherex = x + h
01y = y + hy'(x , y )
01 0 0
11
The graph of the solution passes through the point (x , y ) and has a slope of y'(x , y )
00 0 0
step length of 1.5
step length of 1.5
step length of .75
step length of .75
step length of .25
Use Euler's Method to approximate the particular solution of the differential equation y' = x - y and point (0, 1). Use a step of h = 0.1
h = 0.1x = 0x = 0.1x = 0.2x = 0.3 . . .y = 1y' = x - y
1
2
3
0
y = y + hy'(x , y )
01 0 0
h = 0.1 y = 1 y' = x - y
2
1y = y + hy'(x , y ) = 1 + 0.1(0 - 1) = 0.9
00 0
y = y + hy'(x ,y ) = 0.9 + 0.1(0.1 - 0.9) = 0.822 111
0x = 0
1x = 0.1
x = 0.2
3y = y + hy'(x ,y ) = 0.82 + 0.1(0.2 - 0.82) = 0.758
2 22
y = y + hy'(x , y )
01 0 0
Hot coffee in a 70-degree room cools at a rate proportional to the difference between the coffee temperature and room temperature.
y' (t) = k(y - 70)
At a certain time, a thermometer showed a coffee temperature of 190 degrees, dropping at a rate of 12 degrees per minute.
Hot coffee in a 70-degree room cools at a rate proportional to the difference between the coffee temperature and room temperature.
y' (t) = k(y - 70)At a certain time, a thermometer showed
a coffee temperature of 190 degrees, dropping at a rate of 12 degrees per minute.
y(0) = 190 degrees y'(0) = -12 degrees
y' (t) = k(y - 70)y(0) = 190 degrees y'(0) = -
12 degrees
Find the particular solution (find k)
y' (t) = k(y - 70)y(0) = 190 degrees y'(0) = -
12 degreesFind the particular solution (find k)-12 = k (190 -
70)-12 = 120k-0.1 = ky' = -0.1 (y - 70)
y' = -0.1 (y - 70)
How hot will the coffee be after 10 minutes?
Start (0, 190)use h = 1
y' = -0.1 (y - 70)
How hot will the coffee be after 10 minutes?
Start (0, 190)use h = 1x = 0
y1= y0 + h y'(0, 190) = 190 + 1(-12) = 178
If we continue in this manner we get -
x = 1
y2= y1 + h y'(1, 178) = 178 + 1(-10.8) = 167.2
y' = -0.1 (y - 70)
How hot will the coffee be after 10 minutes?
Start (0, 190)use h = 1
Attachments