Cycloid Consider one arch of the cycloid
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Multivariable Calculus (looseleaf)
- Parametrize the sine curve y = sinx using the parametrization r(t)=(t,sint), t ∈ R. (a) show that this curve is smooth. (b) Compute the curvature, and find all points where the curvature is zero. What geometric property do all those points share? (c) Without doing any computations, explain why the torsion of this curve must be identically zero. (d) Rotating the plane 30◦ is an isometry, which transforms the original sine curve into the curve parametrized by r(t) = √3t −sint 2 , t + √3 sint 2 , t ∈ R. Compute the speed, curvature, and torsion of this curve, and compare them to those of the original curve. (e) By graphing the curve, decide if it is the graph of some function y = f(x). (f) In single-variable calculus, you studied the qualitative properties of the graphs of functions y = f(x). In particular, you characterized the maxima, minima, and inflection points in terms of the vanishing of certain derivatives of the function f(x). Using the earlier parts of this problem to supply…arrow_forwardx(t) = acos((a + b)t) y(t) = acos((a − b)t) Use the GeoGebra tool to graph the parametric equations on the domain [-π, 0] and include the orientation for the following cases: a = 2 and b = 1 a = 3 and b = 2 a = 4 and b = 3 a = 5 and b = 4 Discuss the role of a and b in the parametric equations shown above. Discuss why the domain is given as [-π, 0]. What would change if the domain were given as [0, π] instead?arrow_forwardCurve C is any curvearrow_forward
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