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Notes for 11.5

# Notes for 11.5 - The sharper the curve the greater the...

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Calculus III – Section 11.5 – Arc Length and Curvature Theorem 11.6 If C is a smooth curve given by k t z j t y i t x t r ) ( ) ( ) ( ) ( + + = on the interval [ ] b a , , then the arc length of C on the interval is [ ] [ ] [ ] = + + = b a b a dt t r dt t z t y t x x 2 2 2 2 ) ( ' ) ( ) ( ) ( . Example: #4 #8

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Curvature Let C be a smooth curve (in the plane or in space) given by r (s), where s is the arc length parameter. The curvature at s is given by ) ( ' s T ds dT K = = OR for a general parameter t 3 ) ( ' ) ( " ) ( ' ) ( ' ) ( ' t r t r t r t r t T K × = = . Curvature is basically a measure of how sharply a curve bends.
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Unformatted text preview: The sharper the curve, the greater the curvature. • Notice that you are just calculating the magnitude of the rate of change of the unit tangent vector with respect to the arc length s. • Use the version of the curvature formula that is most convenient for your problem. Example: #24 #34...
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Notes for 11.5 - The sharper the curve the greater the...

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