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Implicit Function Theorem
IMPLICIT FUNCTION THEOREM Suppose that at the point (a, b), z = F(x, y) is smooth, F(a, b) = 0, and ∂z/∂y ≠ 0. Then the curve F(x, y) = 0 at (a, b) has an implicit function and the slope There are three things to prove:
Instead of proving the whole theorem, we give an intuitive argument for (1) and (2) and then prove (3). The surface z = F(x, y) has a tangent plane at (a, b, 0). If we intersect the surface and tangent plane with the plane z = 0 we get the curve 0 = F(x, y) and a line L. Through an infinitesimal microscope aimed at the point (a, b), the curve looks like the graph of a function y = g(x) which has the tangent line L and thus has a slope at (a, b) (Figure 11.6.4). Figure 11.6.4 PROOF OF (3) Given that the slope exists, we compute its value. By the Chain Rule, But F(x, g(x)) is identically zero, so dz/dx = 0 and Since
The best way to remember the minus sign in the above equation is to derive the equation yourself. Start with the Chain Rule for dz/dx = 0 and solve for dy/dx. One way to understand the minus sign is as follows: if ∂z/∂x and ∂z/∂y are positive, an increase in x must be offset by a decrease in y to keep z constant, so dy/dx should be negative. Warning: The two ∂z's have different meanings and cannot be cancelled.
The Implicit Function Theorem gives us a convenient equation for the tangent line to the curve F(x, y) = 0 at (a, b). and finally Tangent Line:


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