Engineering Calculus Notes 410

Engineering Calculus Notes 410 - 398 CHAPTER 3. REAL-VALUED...

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Unformatted text preview: 398 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION while 1 √ 2 √ =4 2 1 (3) + − √ 2 α= 1 √ (3) + 2 √ = − 2. β= 1 √ 2 (−5) (−5) This leads to the equation (in ξ1 and ξ2 ) 2 ξ1 + √ 2 2 − √ 2ξ2 = −5 + 4 = −1; we can rewrite this as √ √ 1 ξ2 − √ = 2(ξ1 + 2)2 2 which we recognize as a parabola with vertex at (ξ1 , ξ2 ) = √1 − 2, √ 2 that is, (x, y ) = 1 − ,1 , 2 and opening along the line √ ξ1 = − 2 in the direction of ξ2 increasing, which in terms of x and y is the line x − y = −1 i.e., y =x+1 in the direction of y increasing. (See Figure 3.31.) ...
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