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EXERCISES The are length of y² = x3 is today a fairly routine exercise with the arc length integral 17.1.1 Show that the arc length of y = x312 between o and X = a is 31(14-291-11 8 27 Likewise, it is easy for us to derive properties of the logarithmic spiral from its polar equation and knowledge of the exponential function. 17.1.2 Show that the logarithmic spiral is self-similar. That is, magnifying r = eht by a factor m to r = meke gives a curve that is congruent to the original (in fact, it results from a rotation of the original). Jakob Bernoulli was so impressed by this property of the logarithmic spi- ral that he arranged to have the spiral engraved on his tombstone, with a motto: Eadem mutata resurgo ("Though changed, I arise again the same"). (See Jakob Bernoulli (1692) p. 213.) 17.1.4 Explain why the constant tangent property implies dy = "as then multiply both sides of this equation = + deduce that dx dy = + y 17.5.1 Are the circles on the pseudosphere, in planes perpendicular to its axis, geodesics? Give a qualitative argument to support your answer. It may be easier to answer this question if one first considers the cone, a surface also obtained by bending the plane. To avoid worrying about the apex, where the cone is not smooth, we omit this point. 17.5.2 Show that the circles on the cone, in planes perpendicular to its axis, are not geodesics.

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