Engineering Calculus Notes 334

Engineering Calculus Notes 334 - 322 CHAPTER 3. REAL-VALUED...

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Unformatted text preview: 322 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION First, let us find the maximum and minimum of the function f (x, y ) = x2 − 2x + y 2 inside the disc of radius 2 x2 + y 2 ≤ 4. (See Figure 3.22.) x2 + y 2 = 4 f (1, 0) = −1 f (−2, 0) = −1 f (2, 0) = 0 x2 + y 2 < 4 Figure 3.22: Critical Points and Boundary Behavior of f (x, y ) = x2 − 2x + y 2 on {(x, y ) | x2 + y 2 ≤ 4} Critical Points: → − → → ı ∇ f (x, y ) = (2x − 2)− + 2y − this vanishes only at the point x=1 y=0 and the value of f (x, y ) at the critical point (1, 0), which lies inside the disc, is f (1, 0) = 1 − 2 + 0 = −1. ...
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