Let f(t) = t - sin t for all t. Show that f is a strictly increasing function of
ID: 2880335 • Letter: L
Question
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 parametrized by x = t - sin t and y = 41 - cos t, for all real t. Find C(0), C(pi), and C(2 pi), and show that the highest point on C occurs for t = pi. 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 notequalto t_2, then x(t_1) notequalto x(t_2). (This means that the cycloid C is the graph of a function!)Explanation / Answer
4 a) f(t) = t-sint
f'(t) = 1-cost >=0 for all values of t by property of cos function.
Since derivative is positive or 0 the funciton is always increasing for all values of t.
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b) x= t-sint, y =1-cost
C(0) = (0,0)
C(pi) = (pi-sin pi, 1-cos pi)
= (pi, 2)
C(2pi) = (2pi-0, 1-1)
=(2pi,0)
dy/dx = (1-cost)/sint
= tan t/2
tan t/2 =0 gives t = 0 or pi
o is minimum point and pi is highest point.
Semicircle parametrics would be x =cost and y =sint
Since x = t-sint and y = 1-cost this cannot be semi circle
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x(t1) = t1-sin t1
x(t2) = t2-sin t2
If x(t1) = x(t2) then we get
t1-sin t1= t2-sin t2
Or t1-t2 = sint1-sin t2
Since left side is not zero right side cannot be zero. So sin t1 cannot be equal to sin t2
It follows that t1-sin t1 cannot be equal to t2-sint2 if t1 and t2 are different.
So cycloid C is the graph of a function.
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