I posted this problem yesterday, but I want to make some changes regarding the questions I asked. Therefore I post it again.
So here are my questions:
- What does $C:[0,2\pi]\rightarrow R^2$ mean. Does the interval $[0, 2\pi]$ denote x-values, y-values or t-values?
- How do I find the unittangentvector? I have done the following:
But this is not a vector. This is just another parametric equation for the tangent of the curve, and only gives represents points. So how do I find the unittangentvector? 3. I'm also supposed to find the length of the curve. I know that I'm supposed to use this formula
But I don't know what my $a$ and $b$ values should be do to question number 1.
I really appreciate some help! :)
For 1. it represents $t$ values and $\gamma(t)=(x(t), y(t))$ is the curve cycloid!
For 2. Find $\gamma'(t)=(x'(t), y'(t))=(b(1-\cos t), b\sin t)$ is the tangent vector and unit tangent vector is $\gamma'(t)/\|\gamma'(t)\|=(b(1-\cos t), b\sin t)/(2b\sin(t/2)).$ Note that $\|\gamma'(t)\|=\sqrt{b^2(1-\cos t)^2+b^2\sin^2t}=b\sqrt{1+\cos^t-2\cos t+\sin^2t}=b\sqrt{2(1-\cos t)}=b\sqrt{2.2\sin^2t/2}=2b\sin t/2.$
Calculate the value at $t=\pi.$ You get the tangent vector at $t=\pi.$