Cycloids problem?
(a). Let f(t) = t - sin(t) for all t. Show that f is a strictly increasing function of t, and determine the inflection points of the graph of f.
(b). Consider the cycloid C parameterized by x = t - sin(t) and y = 1 - cos(t), for all real t. Find C(0), C(π), and C(2π), and show that the highest point on C occurs for t = π. Tell why this shows that one arch of the cycloid is not a semicircle.
(c). Let P(t) = (t - sin(t), 1 -cos(t)) for all real t. Use (a) to show that if t_1 ≠ t_2, then x(t_1) ≠ x(t_2).
Thank you
Comments
a)
f(t) = t-sin(t)
Graph of f(t) shows it is strictly increasing
http://www.wolframalpha.com/input/?i=graph+t-sin(t...
f'(t) = 1- cos(t)
f''(t) = sin(t) = 0
t = n pi , where n is an integer (inflection points)