Download Introducing Slope Fields

Introducing Slope Fields
A slope field is simply a graphical representation of a differential equation. Many real
dy
dy
x

 x2 ,
world phenomena can be modeled by differential equations such as
or
dx y
dx
dy/dx = f(x,y) where f(x,y) is some expression involving x and y. In calculus we use a
slope field to paint a picture of a differential equation. From the slope field it is possible
to make some general statements about the original function whose derivative we know.
This activity will help you begin to observe some of those characteristics.
After observing and learning about the various types of information you can read about
a function from a slope field, you’ll learn techniques for reversing the differentiation
process to reveal the actual function. This will be called anti-differentiation or
integration.
A slope field is a graph of short line segments whose slope is determined by evaluating
the derivative at the midpoint of the segment. But for this activity we’ll begin with
several family of functions, draw short line segments that represent the slope a various
points and step back and just observe and analyze what the picture of short tangent line
segments tells us about the original family of functions.
Let’s create our very first slope field. Figure 1 shows a family of functions of the
form y  kx 2  c and Figure 2 shows a family of functions of the form y  kx 3  c .
Project these figures on a whiteboard or a smartboard. (A smartboard file with this
activity can be found at http://jamesrahn.com/CalculusI/PAGES/slope_fields.htm
(Introduction to Slope Fields)).
1.
Where each parabola intersects a vertical line, draw a short tangent segment that
represents the slope of the parabola at that x-value.
2.
Remove the image of the family of functions and leave only the picture of the
short tangent segments. The picture of these short line segments is called a slope
field. The segments visually describe the steepness or slope of a function at
several x-values.
3.
Describe any patterns you see in the slope of the short tangent line segments.
4.
Describe the set of points where the slopes are equal to each other.
5.
6.
If they exist, describe any points where the slope is undefined.
Describe where the slope of the short tangent line segments are positive;
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negative; or zero.
7.
What do you notice about the slope of the short tangent line segments at any
specific x value?
8.
What do you notice about the slope of the short tangent line segments along any y
value as x increase? (Move from left to right along the slope field.)
9.
Is there a place with the slope of the original function is always zero? How can
you tell?
10.
What graph(s) does the slope field appear to be illustrating?
11.
Describe a form for the original function from the slope field.
12.
Describe a form for the slope of the short tangent line segments.
13.
Describe any symmetry in the slope field. What does this tell you about the
original function represented by the slope of these short tangent segments?
14.
Try to draw a representative curve that starts at x = 0 and y = 3 that follows the
slopes illustrated by the slope field.
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Figure 1
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Figure 2
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