Transcript Slope Fields - FreibergMath

Warm Up
NO CALCULATOR
Homework Answers
pg366
x2
1) y 
 2x  C
2
3) y  Ce x  2
1 2 5 2
5) y  x  C
2
2
7) y  Ce
(2/3)x3/2
2
9) y  C (1  x )
x2
2) y  4x 
C
2
4) y  4  Ce  x
3 2 2 32
6)
y  x C
2
3
8) y  Ce
10) y  Ce
x2 /2
 x2 /2
1
 100
Pg378
2
x
55) y  2e  14
60) 1  y2  1  x2  1
1
58) y  (ln x)2  2
2
64) y  70e kt  70
Slope Fields
A slope field is a graphical picture of a
derivative that projects the curve within
the picture.
Or
a bunch of little line segments that
show the slope of the curve (y = ) at
different points.
dy 1
 x 1
dx 2
The slope field for a certain differential equation is shown.
Which of the following could be a particular solution to
that differential equation?
y = x2
y = ex
y = e –x
y = cos x
y = ln x
Make your own slope field…
dy
x

dx
2y
x
-2
-2
-1
y
0
1
2
-1
0
1
2
Make your own slope field…
dy
x

dx
2y
Sketch a short line segment with the slope at each point.
Where are the tangent lines
horizontal? Why?
Where are the tangent lines
vertical? Why?
dy
x

dx
2y
Solve the differential equation to find the particular
solution with initial condition y(1) = 2. Sketch the
particular solution on your slope field.
Group Activity
Make your own…
Student 1
dy
 1 y
dx
Student 3
dy
2
y
dx
Student 2
dy
 1 x
dx
Student 4
dy
2
 x
dx
• What do you notice about the slope fields
whose differential equation had only x’s?
• What do you notice about the slope fields
whose differential equation had only y’s?
How to pick out a multiple choice answer
(equation) for a slope field…
• pay attention to whether you need just the x- or
y-values or both
• Look for places where the slope is 0
• Look at the slopes along the x-axis
(where y = 0)
• Look for slopes along the y-axis (where x = 0)
• Notice where the slopes are positive and where
they are negative
Shown is a slope field for which of the following
differential equations?
A)
B)
dy
 1 y
dx
dy 2 y 2

dx
x
dy
C)
 y2
dx
dy
x
D)

dx
y
dy
E)
 yx
dx
Shown is a slope field for which of
the following differential equations?
dy
 x y
dx
dy
 x2
dx
dy
 ey
dx
dy
 x y
dx
dy
2x

dx
y
Which of the following differential equations matches
the slope field given?
dy
 1 x
dx
dy
 x y
dx
dy
 ln y
dx
dy
2
x
dx
dy x

dx y
Which of the following differential equations matches
the slope field given?
dy
y
dx
dy 1
 x 1
dx 2
dy
 2 xy
dx
dy y

dx x
dy
x
dx