Let y(t) = (ekt cost, ekt sint), where - < t < and k is...

 

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Let y(t) = (ekt cost, ekt sint), where - < t < ∞ and k is a non-zero constant. Show that there is a unique unit-speed parameter s on such that s> 0 for all t and s 0 as t→ Foo if ±k > 0, and express s as a function of t. Show that the signed curvature of y is 1/ks. Conversely, describe every curve whose signed curvature, as a function of arc-length s, is 1/ks for some non-zero constant k. Let y(t) = (ekt cost, ekt sint), where - < t < ∞ and k is a non-zero constant. Show that there is a unique unit-speed parameter s on such that s> 0 for all t and s 0 as t→ Foo if ±k > 0, and express s as a function of t. Show that the signed curvature of y is 1/ks. Conversely, describe every curve whose signed curvature, as a function of arc-length s, is 1/ks for some non-zero constant k.