Avon High School
AP Calculus AB
Name ___________________________________________
HW 6.1 – Slope Fields and Euler’s Method
dy
dy x
Fill in the chart for
.

dx
dx y
(x, y)
-3
-2
-1
0
1
3
2
1
0
-1
-2
-3
1.)
dy
 sin x
dx
(x, y)
-3
3
2
1
0
-1
-2
-3
Fill in the chart for
-2
-1
0
2
dy
.
dx
1
2
Find the solution graph going through
a.) (0, 1)
b.) (1, 0)
dy
 x y
dx
(x, y)
-3
3
2
1
0
-1
-2
-3
2.)
3
Score ______ / 10
Fill in the chart for
-2
-1
0
dy
.
dx
1
2
3
Find the solution graph going through
a.) (0, 0)
b.) (0, -1)
c.) (1, -2)
Find the solution graph going through
a.) (0, 1)
b.) (1, 0)
c.) (0, 3
3.)
Period _____
dy
dy
 x(1  y )(2  y ) Fill in the chart for
.
dx
dx
(x, y)
-3
-2
-1
0
1
2
3
3
2
1
0
-1
-2
-3
4.)
3
Find the solution graph going through
a.) (0, 0)
b.) (1, 0)
dy
 sin t and y (0)  0 .
dt
a.) Carry out Euler’s Method with 4 steps to estimate y (1). Do not use a program on your calculator.
5.) Consider the differential equation
t
y
dy
dt
0
0.25
.50
.75
1.00
b.) Solve the differentiable equation above. Remember to find the constant of integration.
c.) Let Y (t ) be the exact solution function and let y (t ) be the approximate solution function constructed
by Euler’s Method above. Complete the following table.
t
Y (t )
0
y (t )
d.) Plot y (t ) and Y (t ) on the same axes.
0.25
.50
.75
1.00
dy
 et and y (0)  0 .
dt
a.) Carry out Euler’s Method with 4 steps to estimate y (1). Do not use a program on your calculator.
6.) Consider the differential equation
t
y
0
0.25
.50
.75
1.00
dy
dt
b.) Solve the differentiable equation above. Remember to find the constant of integration.
c.) Let Y (t ) be the exact solution function and let y (t ) be the approximate solution function constructed
by Euler’s Method above. Complete the following table.
t
Y (t )
0
y (t )
d.) Plot y (t ) and Y (t ) on the same axes.
0.25
.50
.75
1.00
7.) Consider the differential equation
dy
 1  t  y, and y (0)  0 .
dt
a.) Carry out Euler’s Method with 5 steps of size 0.4 to estimate y (2).
t
y
0
0.4
0.8
1.2
1.6
2.0
dy
dt
b.) The equation y(t )  Cet  t is a solution to the differentiable equation. Take the derivative of y and
show that it is the solution to the differential equation above.
c.) Find the value of C for this particular equation that fits y (0)  0 .
8.) Consider the differential equation
dy
 xy , and y (0)  1 .
dx
a.) Carry out Euler’s Method with 5 steps of size 0.2 to estimate y (1).
t
y
dy
dx
0
0.2
0.4
0.6
0.8
1.0
b.) Solve the DEQ
dy
 xy for y (0)  1 .
dx
c.) Find the difference between the exact value of y (1) with the estimate you computed in part a.
9.) Consider the differential equation
dy
 1  3 x  2 y, and y (0)  2 .
dx
a.) Carry out Euler’s Method with 5 steps of size 0.2 to estimate y (1).
t
y
0
0.2
0.4
0.6
0.8
1.0
dy
dx
b.) The solution to the DEQ
y  Ce 2 x 
dy
3x 1
 1  3x  2 y for y (0)  2 is y  Ce 2 x   . Differentiate
dx
2 4
3x 1
 to show that this is true.
2 4
d.) Find the value of C for this particular equation that fits y (0)  2 . Find the difference between the
estimate and the actual values of y (1) .