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Unformatted text preview: 279 3.5. SURFACES AND THEIR TANGENT PLANES
(a) Show that for y = 0
(x, y ) = x + 2y sin − cos .
(b) Show that
(x, 0) = x.
(c) Show that ∂f /∂y is not continuous at (1, 0).
(d) Show that if f (x, y ) = 0 and y = 0, then
x = −y sin .
(e) Show that the only solutions of f (x, y ) = 0 in the rectangle
3 , 3 × − 3 , 3 are on the y -axis.
(f) Conclude that there is no function φ deﬁned on I =
that f (x, φ(x)) = 0 for all x ∈ I . 3.5 24
3, 3 such Surfaces and their Tangent Planes In this section, we study various ways of specifying a surface, and ﬁnding
its tangent plane (when it exists) at a point. As a starting point, we deal
ﬁrst with surfaces deﬁned as graphs of functions of two variables. Graph of a Function
The graph of a real-valued function f (x) of one real variable is the subset
of the plane deﬁned by the equation
y = f (x) ,
which is of course a curve—in fact an arc (at least if f (x) is continuous,
and deﬁned on an interval). Similarly, the graph of a function f (x, y ) of
two real variables is the locus of the equation
z = f (x, y ) ,
which is a surface in R3 , at least if f (x, y ) is continuous and deﬁned on a
reasonable region in the plane. ...
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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