Calculus: Analytic Geometry and Calculus, with Vectors

(^150) Functions, limits, derivatives
where x is the distance from the center of the earth to the particle and R is the
radius of the earth. Supposing that W is a continuous function of x and that
W = 100 when x = R, calculate k, and k2 and sketch a graph of W versus x.
9 Prove that if f is continuous at xo, then so also is the function g having
values defined by g(x) = If(x)l.
(^10) It is never too soon to start becoming acquainted with the idea that if,
during some time interval t, < t < t2, a bumblebee or molecule or rocket buzzes
around, then at each time t in the interval it is surely someplace and that if we
let f,(t), f2(t), f3(t) denote its x, y, z coordinates at time t, then fi, f2, f3 are con-
tinuous functions of t. Since wholesome comprehension of mathematics is salu-
brious, we recognize that we do not quite know how to prove that bumblebees
never fly out of our E3 for a minute or two. Moreover, we do not know how to
devise a mathematical proof that a bumblebee cannot gather honey all morning
in Pennsylvania, be in Chicago at noon, and hunt clover in Los Angeles all after-
noon. The best we can do is make the physical assumption that fi, f2, fa are
continuous and know what the assumption means. What does the assumption
mean? dns.: If t, < t < t2, then
urn fk(t + At) = fk(t)
when k = 1, when k = 2, and when k = 3.
(^11) Abandoning some of the notation of the preceding problem, we suppose
that x, y, z are given functions that are continuous
*+A0 -l"W.-A., over some interval in which t is supposed to lie.
"/- ``.1 Let P(t)denote the pointinE 3 having coordinates
x, y, z for which x = x(t), y = y(t), and z = z(t).
r(tP(t) While the fact will be considered later with more

_-1,,/ details, we can pause tolearn that the ordered set

of points P(t), ordered so that P(t') precedes P(t")

Figure 3.492 when t' < t", is called a curve C. The point P(t)

is then said to move along or traverse the curve C as t

increases. Figure 3.492 may be helpful. For each t, let r(t) be the vector run-

ning from the origin 0 to P(t). This determines a vector function r for which

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

Conversely, if r is a given vector function, then it (and the givencoordinate

system) determines its scalar components. From (1) and

(2) r(t +,6a) = x(t + At)i + y(t + At) j + z(t + At)k

we obtain

(3) r(t + At) - r(t) _ [x(t +,&t)- x(t)]i + [y(t + At) - y(t)]j

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

and

(4) jr(i + At) - r(t)I= [Ix(t + At) - x(t)I2 + (y(t + At) - y(t)IZ

+ Iz(t + ot) - z(t)IT.