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Graphical Understanding of Derivatives |
CALCULUS
Higher Concepts of Mathematics
The instantaneous velocity again equals the
Figure 5 Graph of Distance vs. Time
value of the derivative ds/dt. This value is
changing with time.
However, the
instantaneous velocity at any specified time
can be determined. First, small changes in
s and t are considered.
Ds
Dt
(s
Ds)
s
(t
Dt)
t
The values of (s + Ds) and s in terms of
(t + Dt) and t using Equation 5-5, can then
be substituted into this expression. At time
t, s = 10t2; at time t + Dt, s + Ds = 10(t +
Dt)2. The value of (t + Dt)2 equals t2 +
2t(Dt) + (Dt)2; however, for incremental
values of Dt, the term (Dt)2 is so small, it
can be neglected. Thus, (t + Dt)2 = t2 +
2t(Dt).
Ds
Dt
10[t2
2t(Dt)]
10t2
(t
Dt)
t
Ds
Dt
10t2
20t(Dt)]
10t2
t
Dt
t
Figure 6 Slope of a Curve
Ds
Dt
20t
The value of the derivative ds/dt in the case
plotted in Figure 5 equals 20t. Thus, at time
t = 1 s, the instantaneous velocity equals 20
ft/s; at time t = 2 s, the velocity equals 40
ft/s, and so on.
When the graph of a function is not a straight
line, the slope of the plot is different at
different points. The slope of a curve at any
point is defined as the slope of a line drawn
tangent to the curve at that point. Figure 6
shows a line drawn tangent to a curve. A
tangent line is a line that touches the curve at
only one point. The line AB is tangent to the
MA-05
Page 36
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