9.2.1 Geometric And Physical Significance Of The Derivative
Suppose r is a vector valued function of a parameter t not necessarily time and consider the following
picture of the points traced out by r.
PICT
In this picture there are unit vectors in the direction of the vector from r
(t)
to r
(t+ h)
. You can see
that it is reasonable to suppose these unit vectors, if they converge, converge to a unit vector
T which is tangent to the curve at the point r
(t)
. Now each of these unit vectors is of the
form
r(t+ h)− r(t)
|r(t+-h)−-r(t)| ≡ Th.
Thus T_{h}→T, a unit tangent vector to the curve at the point r
(t)
. Therefore,
′ r(t+-h)−-r(t) |r(t+-h)−-r(t)|-r(t+-h)−-r(t)-
r (t) ≡ hli→m0 h = hli→m0 h |r(t+ h)− r(t)|
|r(t+ h)− r(t)|
= hli→m0 ------h-------Th = |r′(t)|T.
In the case that t is time, the expression
|r(t+ h)− r(t)|
is a good approximation for the distance
traveled by the object on the time interval
[t,t+ h]
. The real distance would be the length of the curve
joining the two points but if h is very small, this is essentially equal to
|r(t+ h)− r (t)|
as suggested by the
picture below.
PICT
Therefore,
|r(t+h)−-r(t)|
h
gives for small h, the approximate distance travelled on the time interval
[t,t+ h]
divided by the length of time h. Therefore, this expression is really the average speed of the object
on this small time interval and so the limit as h → 0, deserves to be called the instantaneous speed of the
object. Thus
′
|r (t)|
T represents the speed times a unit direction vector T which defines the direction in
which the object is moving. Thus r^{′}
(t)
is the velocity of the object. This is the physical significance of the
derivative when t is time. In general, r^{′}(t) and T(t) are vectors tangent to the curve which point in the
direction of motion.
How do you go about computing r^{′}
(t)
? Letting r
(t)
=
(r1(t),⋅⋅⋅,rq(t))
, the expression
r(t0 + h)− r(t0)
-------h------- (9.2)
(9.2)
is equal to
(r (t + h)− r (t ) r (t + h)− r (t))
-1--0-------1--0 ,⋅⋅⋅,-q-0--------q-0- .
h h
. This is because of Theorem
8.5.5 on Page 419, which says that the term in (9.2) gets close to a vector v if and only if all the
coordinate functions of the term in (9.2) get close to the corresponding coordinate functions of
v.
In the case where t is time, this simply says the velocity vector equals the vector whose components are
the derivatives of the components of the displacement vector r
(t)
.
Example 9.2.7Let r
(t)
=
( )
sint,t2,t+ 1
for t ∈
[0,5]
. Find a tangent line to the curveparameterized by r at the point r
(2)
.
From the above discussion, a direction vector has the same direction as r^{′}
(2)
. Therefore, it suffices to
simply use r^{′}
(2)
as a direction vector for the line. r^{′}
(2)
=
(cos2,4,1)
. Therefore, a parametric equation for
the tangent line is