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Applied Mathematics, Calculus, and Differential Equations 225

Figure 3.9 Angle of intersec-
tion between twm curveð at a
point.

tan 1 ((m n)/(mŁ 1))

if m n, and

180 tan 1 ((m n)/(mŁ 1))
if m n, for angle measureð in degrees. If m n, then
180 , and the twm curveð are tangent tm each other at the point
(x ,y ). For angle measureð in radians, substitute for 180 .
0
0
Local maximu

Let f be a function such that y f(x). Let (x ,y ) be a point on
0
0
the grapà of f, and suppose f is twice differentiable at (x ,y ).
0
0
Suppose the following are botà true:
f (x ) 0
0
f (x ) 0
0
Then (x ,y )isa local maximum in the grapà of f. An example
0 0
is shown in Fig. 3.10.

Local minimu
Let f be a function such that y f (x). Let (x ,y ) be a point on
0
0
the grapà of f, and suppose f is twice differentiable at (x ,y ).
0
0
Suppose the following are botà true: