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Unformatted text preview: 263 3.4. LEVEL CURVES Now, since ε → 0, the ratio on the right converges to the ratio of the
partials, and so is bounded by, say that ratio plus one, for △x suﬃciently
near zero:
∂ f /∂x
△y
≤
+1 .
△x
∂f /∂y
This in turn says that the term multiplying ε in Equation (3.20) is
bounded, so ε → 0 implies the desired equation
∂f /∂x
△y
=−
.
△x→ 0 △x
∂f /∂y φ′ (x) = lim This shows that φ is diﬀerentiable, with partials given by Equation (3.18),
and since the right hand side is a continuous function of x, φ is
continuously diﬀerentiable.
We note some features of this theorem:
• The hypothesis that f (x, y ) is continuously diﬀerentiable is crucial;
there are examples of diﬀerentiable (but not continuously
diﬀerentiable) functions for which the conclusion is false: the function
φ(x) required by Equation (3.17) does not exist (Exercise 5).
• The statement that L(f, c) ∩ R is the graph of φ(x) means that the
function φ(x) is uniquely determined by Equation (3.17).
• Equation (3.18) is simply implicit diﬀerentiation: using
y = φ(x)
and setting
z = f (x, y )
we can diﬀerentiate the relation
z = f (x, φx)
using the Chain Rule and the fact that z is constant to get
0= dz
∂f
∂f dy
=
+
dx
∂x ∂y dx
∂f ′
∂f
+
φ (x)
=
∂x ∂y ...
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.
 Fall '08
 ALL
 Calculus

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