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MA 36600 LECTURE NOTES: WEDNESDAY, JANUARY 14
Basic Mathematical Models (cont’d)
Slope Fields.
It is not necessary to fnd the exact solution oF a di±erential equation in order to understand
how the solution behaves. We discuss this concept through several examples.
Consider a plot oF
x
(
t
)versus
t
i.e. we plot the position
x
on the vertical axis and time
t
on the horizontal
axis. Through any two points on the curve we can draw a line segment. Recall that given two distinct points
(
t
0
,x
0
) and (
t
1
,x
1
), the line through them has slope
slope oF secant line =
∆
x
∆
t
=
x
(
t
1
)
−
x
(
t
0
)
t
1
−
t
0
.
OF course, iF the points are close to each other, we fnd the slope oF the tangent line:
slope oF tangent line = lim
t
1
→
t
0
x
(
t
1
)
−
x
(
t
0
)
t
1
−
t
0
=
dx
dt
(
t
0
)
.
In particular, the slope oF the secant line is an approximation oF the slope oF the tangent line:
∆
x
∆
t
≈
dx
dt
=
⇒
∆
x
≈
dx
dt
∆
t.
Conversely, iF we knew the slope
x
°
(
t
) at every point (
t, x
), we could do a reasonable job oF sketching the
path
x
(
t
) vs. t. This is called the
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 Spring '09
 EdrayGoins
 Basic Math, Slope

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