16.0.1 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)-≡ Th.
|r(t+ h)− r (t)|
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) ≡ lih→m0 h = lih→m0 h |r(t+ h)− r(t)|
|r(t+ h)− r(t)| ′
= lih→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 (16.1)
(16.1)
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 15.8.5 on Page 933, which says that the term in 16.1 gets close to a
vector v if and only if all the coordinate functions of the term in 16.1 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 16.0.7Let r
(t)
=
(sint,t2,t +1)
for t ∈
[0,5]
. Find a tangent line tothe curve parameterized 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