Calculus: Analytic Geometry and Calculus, with Vectors

178 Functions, limits, derivatives

buzzes through space E3. While other tactics are both possible and use-
ful, we suppose that we have a rectangular x, y, z coordinate system
bearing unit vectors i, j, k as in Section 2.2. At each time t, the coordi-
nates of the bumblebee can be denoted by x(t), y(t), z(t). Letting r(t)
denote the vector running from the origin to the bumblebee, we obtain
the vector equation

(3.74) r(t) = x(t)i + y(t)j + z(t)k.

This vector r(t) is called the displacement (or displacement vector) of the

bumblebee at time t. Supposing that At 0, we can write

(3.75) r(t + At) = x(t + At)i + y(t + At)j + z(t + At)k

and form the difference quotient

r(t + At) - r(t)_x(t + At) - x(t)i +, y(t + At) - y(t) j

At At At

+z(t + At) - z(t)k

At

which can be written in the abbreviated form

(3.751)

ArAxi+Ayj+Azk.

At At At At

In accordance with general terminology, this difference quotient is called

the average rate of change of the vector r(t) with respect to t over the

interval from the lesser to the greater of t and t + At. It is also called

the average velocity of the bumblebee over this same interval. In case

the above difference quotients have limits as At - 0, the limit of the

average velocity is called the velocity at time t and is denoted by v(t).

Thus

(3.76) v(t) = r1(t) = x'(t)i + y'(t)j + z'(t)k

or

(3.761) v(t) =

dt dt

i

+ dt J + it -

providedprovided the derivatives exist. Figures 3.762, 3.763, and 3.764 show how

the vectors Ar, Ar/At, and v(t) might appear in a particular example.

Figure 3.762 Figure 3.763 Figure 3.764