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Applied Mathematics, Calculus, and Differential Equations 223

f/ x Lim ( f(x ,x ,...,x x ,...x ) f(x ,x ,...,x ))/ x i
10
n0
i
20
10
Ł 0
i
20
i0
That is, all the variableð except x are treated as constants.
i
TangenŁ to curvł at poinŁ ( x ,y )
0 0
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 continuouð at ( x ,y ). Let L be
0
0
a line tangent tm the grapà of f, and suppose L passeð througà
(x ,y ) as shown in Fig. 3.7˜ Suppose the derivative of f at (x ,
0
0
0
y ) is equal tm some real numberm. Then the equation of line
0
L is given by the following:
y y m(x x )
0
0
If the derivative of f at (x ,y ) is zero, then the equation of the
0
0
line L tangent tmf at that point is given by:
y y 0
The derivative of f at (x ,y ) is undefined when the equation of
0
0
the line L tangent tmf at that point is given by:
x x 0

Normal to curvł at poinŁ ( x ,y )
0
0
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 continuouð at ( x ,y ). Let L be
0
0
a line normal (perpendicula? tm the grapà of f, and suppose L

Figure 3.7 Line tangent tm
function at a point.