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Differential Calculus Grinshpan Rotated Ellipse The implicit equation x2 − xy + y 2 = 3 describes an ellipse. The following 12 points are on this ellipse: √ √ √ √ √ √ (± 3, 0), (0, ± 3), ( 3, 3), (− 3, − 3), (1, −1), (−1, 1), (1, 2), (2, 1), (−1, −2), (−2, −1). The ellipse is symmetric about the lines y = x and y = −x. It is inscribed into the square [−2, 2] × [−2, 2]. Solving the quadratic equation y 2 − xy + (x2 − 3) = 0 for y we obtain a pair of explicit equations: √ 1 3 √ y= x ± 4 − x2 , 2 2 where the plus corresponds to the top portion (red) and the minus corresponds to the bottom portion (blue). Consequently, we obtain a formula for the slopes: √ √ 1 3 1 1 3 x ′ √ √ ± (−2x) = ∓ . y = 2 2 2 2 4−x 2 2 4 − x2 For instance, √ 1 3 −1 ′ √ y (−1,1) = − = 1. 2 2 4 − (−1)2 Alternatively, we can calculate the slopes using implicit differentiation: 2x − (y + xy ′ ) + 2yy ′ = 0 gives y′ = y − 2x . 2y − x 1 − 2(−1) Thus y ′ (−1,1) = = 1. Note that the slopes at x = ±2 are infinite. 2 · 1 − (−1)