082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

424 Chapter 12 Applications of differentiation
y
r
P
C
r(t)
P
PQ
Q
Figure12.16
Position vector ofapoint Pona
curve C.
x
r(t + dt)
Figure12.17
VectorPQrepresents ⃗ the displacement of
the object during the time interval δt.
Now,sincethexandycoordinatesoftheobjectdependuponthetime,wecanwritethe
position vectorras
r(t) =x(t)i +y(t)j
Therefore,
so that
r(t +δt) =x(t +δt)i+y(t +δt)j
x(t +δt)i+y(t +δt)j−x(t)i−y(t)j
v(t) = lim
δt→0 δt
{ }
x(t +δt)−x(t) y(t +δt)−y(t)
= lim i + j
δt→0 δt
δt
= dx
dt i + dy
dt j
often abbreviated to v =ṙ =ẋi+ẏj. Recall the dot notation for derivatives w.r.t.
time which is commonly used when differentiating vectors. So the velocity vector is
the derivative of the position vector with respect to time. This result generalizes in an
obvious way tothree dimensions. If
then
r(t) =x(t)i +y(t)j +z(t)k
ṙ(t) =ẋ(t)i+ẏ(t)j+ż(t)k
The magnitude of the velocity vector gives the speed of the object. We can define the
acceleration inasimilarway:
a = dv
dt = d2 r
= ¨r =ẍi+ÿj+¨zk
dt2 Inmoregeneralsituations,wewillnotbedealingwithpositionvectorsbutotherphysical
quantities such as time-dependent electricor magnetic fields.
Example12.9 Ifa = 3t 2 i +cos2tj, find
(a) da
(b)
da
dt ∣dt
∣
(c) d2 a
dt 2