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

http://www-math.mit.edu/daimp/EulerMethod.html

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

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