Engineering Calculus Notes 291

Engineering Calculus Notes 291 - 279 3.5. SURFACES AND...

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Unformatted text preview: 279 3.5. SURFACES AND THEIR TANGENT PLANES (a) Show that for y = 0 ∂f 1 1 (x, y ) = x + 2y sin − cos . ∂y y y (b) Show that ∂f (x, 0) = x. ∂y (c) Show that ∂f /∂y is not continuous at (1, 0). (d) Show that if f (x, y ) = 0 and y = 0, then 1 x = −y sin . y (e) Show that the only solutions of f (x, y ) = 0 in the rectangle 11 24 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.

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