14. Compute fxyzds, where C is the curve given by the parametric equation x = t, y = 2t, z = t² from t = 0 tot = 2.
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- Consider the following parametric curve: x(t) = 3t^2 −6 and y(t) = 3t^4 −3t^2 +12t +10 Find the equation of the tangent line to to the point along the curve at(−3,−2).1. If the parametric curve $x=f(t), y=g(t)$ satisfies $g^{\prime}(1)=0,$ then it has a horizontal tangent when $t=1$Consider the parametric curve segment (t, t2), t ∈ [0, 1]. What is the firstorder derivative of the curve at t = 0? Show that exactly the same curve segment can be re-parameterized so that the first-order derivative at t = 0 is different.
- computes curve length for 0<=t<=1 if r(t)=<sin 2t,2t^3/2,cos 2t>Consider the graph ofr(t) = 2t^2i + t^2j + t^3 kDetermine parametric equations of the tangent line to the curve at the point where the curve intersects the planex − 2y − z = 8.Find the tangent line(s) to the parametric curve given by x=t^5 - 4t^3 , y=t^2 , at (0,4)
- The parametric curve x(t)=t−2t6 and y(t)=t7+2t2 is given. In which case is the tangent line at the point t=1 given to this curve?Find, in the standard y = mx + b form, the tangent line to the parametric curve( 1/(t^2 +1) , 1/t) at the point ( 1/2 , 1)Suppose C is the curve given by r(t) = (3 cos(t), -3 sin(t), 4t). Find an equation for the osculating plane of C at the pt(-3, 0, 4π)