6. Proof II r'(x) II = √1 + [f '(x)]² K = II r'(x) x r''(x) II / [II r'(x) II]³ K = I f''(x) I / { 1 + [f '(x)]²}³∕² K = I y'' I / [ 1 + (y') ²]³∕² Polar equation Rectangular equation
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11. Sybase’s Logo We will: - Find the equation of this curve - Calculate the function that represents the curvature of this logo
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13. Curvature of the Sybase Logo Archimedes’ Spiral Parametric Equation: x = t cos t y = t sin t K = ( Iy''I )/ [1 + (y')²]³∕² y' = [( t cos t) + sin t] / [ cos t – t sin t] y'' = (d/dt [ (sin t + t cos t) / ( cos t – t sin t)]) · (1 / dx/dt ) y'' = [ (cos t + cos t – t sin t)(cos t – t sin t) – ( sin t + t cos t)(-sin t – sin t – t cos t) ] / (cos t – t sin t)³ y'' = [ ( 2 cos t – t sin t)(cos t – t sin t) + ( sin t + t cos t)(2 sin t + t cos t)] / (cos t – t sin t)³ y'' = [ (2 cos²t – 2t sin t cos t – t sin t cos t + t² sin²t) + (2 sin²t + 2t sin t cos t + t sin t cos t + t² cos²t) ] / (cos t – t sin t)³ y'' = (2 + t²) / ( cos t – t sin t)³
14. Curvature of the Sybase Logo cont. Finding the curvature: K = I y'' I / [ 1 + (y')²]³∕² K = I (2 + t²) / ( cos t – t sin t)³ I / ([ 1 + ([( t cos t) + sin t] / [ cos t – t sin t])²]³∕² K = (2 + t²) / [ 1 + (t cos t + sin t)²]³∕² Graph the curvature: y x
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