sketch the astroid in curves also calculate its tangent vector at each point .at which point is the tangent vector zero:
(i) γ(t) = (cos^2 t, sin^2 t)
(ii) γ(t) = (e^t, t^2)
"x\\left( t \\right) =\\cos ^2t,y\\left( t \\right) =\\sin ^2\\left( t \\right) \\Rightarrow x+y=1,x,y\\in \\left[ 0,1 \\right]"
The tangent vector is 0 if "\\left\\{ \\begin{array}{c} x'\\left( t \\right) =0\\\\ y'\\left( t \\right) =0\\\\\\end{array} \\right. \\Rightarrow \\left\\{ \\begin{array}{c} 2\\sin t\\cos t=0\\\\ 2\\sin t\\cos t=0\\\\\\end{array} \\right. \\Rightarrow t=\\frac{\\pi n}{2}"
"x\\left( t \\right) =e^t,y\\left( t \\right) =t^2\\Rightarrow y\\left( x \\right) =\\ln ^2x"
The tangent vector is 0 if "\\left\\{ \\begin{array}{c} x'\\left( t \\right) =0\\\\ y'\\left( t \\right) =0\\\\\\end{array} \\right. \\Rightarrow \\left\\{ \\begin{array}{c} e^t=0\\\\ 2t=0\\\\\\end{array} \\right. \\Rightarrow t\\in \\emptyset"
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