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CHAPTER 3. REALVALUED FUNCTIONS: DIFFERENTIATION
by completing the square in each variable
ξ
i
for which
λ
i
n
= 0 we get an
equation in which each variable appears either in the form
λ
i
(
ξ
i
−
ξ
i
0
)
2
(if
λ
i
n
= 0) or
α
i
(
ξ
i
−
ξ
i
0
) (if
λ
i
= 0). We shall not attempt an exhaustive
catalogue of the possible cases, but will consider Fve “model equations”
which cover all the important possibilities. In all of these, we will assume
that
ξ
10
=
ξ
20
=
ξ
30
= 0 (which amounts to displacing the origin); in many
cases we will also assume that the coe±cient of each term is
±
1. The latter
amounts to changing the scale of each coordinate, but not the general
shapeclassiFcation of the surface.
1. The easiest scenario to analyze is when
z
appears only to the frst
power
: we can then move everything except the “
z
” term to the right
side of the equation, and divide by the coe±cient of
z
, to write our
equation as the expression for the graph of a function of
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Spring '08 term at University of Florida.
 Spring '08
 ALL
 Calculus, Completing The Square

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