Engineering Calculus Notes 275

Engineering Calculus Notes 275 - 263 3.4. LEVEL CURVES Now,...

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Unformatted text preview: 263 3.4. LEVEL CURVES Now, since ε → 0, the ratio on the right converges to the ratio of the partials, and so is bounded by, say that ratio plus one, for △x sufficiently near zero: ∂ f /∂x △y ≤ +1 . △x ∂f /∂y This in turn says that the term multiplying |ε| in Equation (3.20) is bounded, so ε → 0 implies the desired equation ∂f /∂x △y =− . △x→ 0 △x ∂f /∂y φ′ (x) = lim This shows that φ is differentiable, with partials given by Equation (3.18), and since the right hand side is a continuous function of x, φ is continuously differentiable. We note some features of this theorem: • The hypothesis that f (x, y ) is continuously differentiable is crucial; there are examples of differentiable (but not continuously differentiable) functions for which the conclusion is false: the function φ(x) required by Equation (3.17) does not exist (Exercise 5). • The statement that L(f, c) ∩ R is the graph of φ(x) means that the function φ(x) is uniquely determined by Equation (3.17). • Equation (3.18) is simply implicit differentiation: using y = φ(x) and setting z = f (x, y ) we can differentiate the relation z = f (x, φx) using the Chain Rule and the fact that z is constant to get 0= dz ∂f ∂f dy = + dx ∂x ∂y dx ∂f ′ ∂f + φ (x) = ∂x ∂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.

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