Calculus: Analytic Geometry and Calculus, with Vectors

7.1 Curves and lengths 415

there being one such point for each t for which t -1. One who has consid-

erable time to study the functions defined by (1) and to makesome calculations

can discover the manner in which P wanders as t increases. As t increases over

the interval - oo < t < -1, P runs over the curve C1 extending from the origin

(but not including the origin) down into the fourth quadrant. As t increases

over the interval -1 < t < w, P traverses

the curve C2 which comes from the third y

quadrant to the origin and then runs around

the loop toward the origin but does not

again contain the origin. This problem t> -11 Y X

requires that we obtain some more informa-

tionwithout working so hard. Show that

if (1) holds, then y = tx and

-a x

(3) x3 + y3 = 3axy.

Show thatif( 3 )holds and y = tx, then (1)

holds. Thus (3) is an equation of the I \

folium. If P(x,y) is a point on the folium,

then (3) shows that P(y,x) is also on the

folium. Therefore, the line having the

equation y = x is an axis of symmetry

Figure 7.196

of the folium. This suggests that we introduce the X, Y axes of Figure 7.196

which bear unit vectors I and J. Use the formulas

(4) i= (I-J), i= (I+J)

with (2) to obtain the new equation

5r t t(1-t) l

() r=- 1-t+t2I+ 1+t8 J

of the folium. Treating (5) as an equation of the form r = X1 + YJ, show that

X= 3a t dX= 3a 1t2

1 t+ t2' dt (1 -t+t2)2

and use the result to find the minimum and maximum values of X and obtain
more information about the folium.
13 Let f be continuous over the interval 0 S x 5 1 and let C be the set of
points (x, f(x)) so ordered that (x1, f(x1)) precedes (x2i f(x2)) if x1 < x2. We
may feel that C must have finite length, and we investigate the matter. Theorem
7.17 shows that C must have finite length if f is piecewise monotone, but we cannot
be so sure when C is not piecewise monotone. When p and q are positive integers,
the function sing x4 [or the function g for which g(x) = sing x4] is not piecewise
monotone over the infinite interval x > 1 and the function sing (1/x4) is not
piecewise monotone over the interval 0 < x < 1. Functions f for which f(0) = 0
and

f(x) = x' singQ

x