- #1
Bipolarity
- 776
- 2
The cycloid is defined by the parametric equations
[itex] x = a(t-sin(t)) [/itex] and [itex] y = a(1-cos(t)) [/itex]
I am trying to find the set of points of relative extrema of a cycloid.
I differentiated first to get
[itex] \frac{dx}{dt} = a(1-cos(t)) [/itex] and [itex] \frac{dy}{dt} = a*sin(t) [/itex]
Then, by the chain rule:
[itex] \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{sin(t)}{1-cos(t)} [/itex]
This fraction is 0 whenever [itex] t = πN [/itex] where N is nonnegative integer.
Thus, the derivative is 0 whenever [itex] t = πN [/itex] where N is nonnegative integer.
But according to the graph on my textbook,
the minima occur when [itex] t = aπN [/itex], where N is nonnegative integer.
Where is my mistake?
BiP
[itex] x = a(t-sin(t)) [/itex] and [itex] y = a(1-cos(t)) [/itex]
I am trying to find the set of points of relative extrema of a cycloid.
I differentiated first to get
[itex] \frac{dx}{dt} = a(1-cos(t)) [/itex] and [itex] \frac{dy}{dt} = a*sin(t) [/itex]
Then, by the chain rule:
[itex] \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{sin(t)}{1-cos(t)} [/itex]
This fraction is 0 whenever [itex] t = πN [/itex] where N is nonnegative integer.
Thus, the derivative is 0 whenever [itex] t = πN [/itex] where N is nonnegative integer.
But according to the graph on my textbook,
the minima occur when [itex] t = aπN [/itex], where N is nonnegative integer.
Where is my mistake?
BiP