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Unformatted text preview: 282 CHAPTER 3. REALVALUED FUNCTIONS: DIFFERENTIATION
which we recognize as a restatement of Equation (3.22) identifying this
plane as a graph:
3
1
+ (x − 1) −
8
2
= T(1, 1 ) f (x, y ) . y− z= 1
2 2 These formulas have a geometric interpretation. The parameter s = x − 1
represents a displacement of the input from the base input (1, 1 ) parallel
2
to the xaxis—that is, holding y constant (at the base value y = 1 ). The
2
intersection of the graph z = f (x, y ) with this plane y = 1 is the curve
2
1
2 z = f x,
which is the graph of the function
z= x2 3
−;
2
8 at x = 1, the derivative of this function is
dz
dx =
1 ∂f
∂x 1, 1
2 =1 9
and the line through the point x = 1, z = 4 in this plane with slope 1 lies in
→
→
→−
1
the plane tangent to the graph of f (x, y ) at (1, 2 ); the vector − 1 = − + k
v
ı
is a direction vector for this line: the line itself is parametrized by x=
y=
z= 1 +s
1
2
1
8 +s which can be obtained from the parametrization of the full tangent plane
by ﬁxing t = 0. (see Figure 3.11.)
Similarly, the intersection of the graph z = f (x, y ) with the plane x = 1 is
the curve
z = f (1, 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, Formulas

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