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Example 5: Same of Hyperbolic Paraboloid

Find the tangent line and slope of the level curve of the hyperbolic paraboloid

z = x2 - y2

at the point (a, b) (where b ≠ 0) (Figure 11.6.7).

11_partial_differentiation-386.gif

Figure 11.6.7: Level curves of z = x2 - y2

The level curve has the equation

x2 - y2 = a2 - b2, x2 - y2 - (a2 - b2) = 0.

Put

w = x2 - y2 - (a2 - b2) = 0.

Then

11_partial_differentiation-387.gif

At (a, b),

11_partial_differentiation-388.gif

Tangent Line:

2a(x - a) - 2b(y - b) = 0.

Slope:

11_partial_differentiation-389.gif


Last Update: 2006-11-15