Calculus: Analytic Geometry and Calculus, with Vectors

S.1 Graphs, slopes, and tangents 293

19 It is easy to show that the graph of the equation

(1) x4 + 2x2y2 + y4 - 2x3 - 2x2y - 2xy2 - 2y3
+5x2+5y2-6x-6y+6=0

contains the point (1,1). What have we learned that could make us sure that
the graph contains another point? Ans.: Nothing. Remark: We do not yet
have enough mathematical equipment to enable us to answer basic questions
about natures of graphs of complicated equations. One who has or develops
interest in such matters must continue study of calculus. Problem 7 and the
following problems at the end of Section 11.3 provide reasons why profound
study of graphs should follow (not precede) study of calculus. While the opera-
tion gives no information about the natures of graphs of other equations, one
who cares to do so may show that (1) can be put in the form

(2) [(x-1)2+(y-1)2][x2+y2+3]=0

and hence that the point (1,1) is in fact the only point on the graph.

20 Let f be the function for which f(0) = 0 and

f (x) = x2 sin 12

when x 96 0. Prove that f'(0) = 0 and hence that the x axis is tangent to the

graph off at the origin. Sketch the graph off and tell why the x axis is not a

line of support of the graph. Hint: To calculate f'(0), use the fact that

f(x) - f(0)

x-0 x sin 12x

s IxI (x 0 0)

and apply the sandwich theorem.

21 When we study a science, it is sometimes worthwhile to obtain preliminary

ideas about machinery that we are not yet prepared to understand fully. This

is an example which involves curves and tangents. Let S be a set of points in a

plane (in E2) which is bounded (this means that there is a rectangle which con-

tains S), is convex (this means that if P, and P2 lie in S,

then the whole line segment joining Pl andP2lies in S),

and which contains at least one inner point (this means

that there is a point P in S and a positive number S

such that S contains each point inside the circle with

center at P and radius 3). Figure 5.192 shows an

example. Let r (capital gamma) be the boundary of

S; a point Q is a point of r if each circular disk with A B

center at Q contains at least one point in S and also

at least one point not in S. We can wonder whether

r should be called a curve. We can observe that

Figure 5.192

there may be points, such as .4, B, C, D in the figure, at which r has many

lines of support but has no tangent. We can observe that there may be

points, such as E in the figure, at which r has only one line of support and

has a tangent. We can say that r has a corner at a point B if B is on

r and r has more than one line of support at B. We can wonder whether r