Consider the curve Cwith parametric equations Sa(t) = t3 – 3t +1 lu(t) = t(t + 1) | lefined for t ER. ? Set up an integral for the arc length of C from (1,0) to (3,6).
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- Compute the arc length of the parametric curve given X=1/3t^3-t,y=t^2+3 From t=0 to t=3Consider the parametric equations x = t2 - 1 and y = t2 + 2t. (a) Find (dy)/(dx) and (d2y)/(dx2). (b) Set up, but do not evaluate, an integral representing the arc length over the interval 2 ≤ t ≤ 4.Consider the curve r=[(e^(t))*cos(3t), (e^(t))*sin(3t), e^(t)]Compute the arclength function s(t): (with initial point t=0).
- Consider the parametric equation x = t2 - 1 and y = t2 + 2t. (a) find (dy)/(dx) and (d2y)/(dx2). (b) Set up, but do not evaluate, and integral representing the arc length over the interval 2 ≤ t ≤ 4.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)1. If the parametric curve $x=f(t), y=g(t)$ satisfies $g^{\prime}(1)=0,$ then it has a horizontal tangent when $t=1$
- A particle moves in the xy-plane in such a way that its path is defined by x = e^t cos t and y = e^t sin 2t. Find the speed of the particle when t = /2.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.Consider the spiral given parametrically by x( t ) = 4 e−0.2 t sin( 2 t ) y( t ) = 4 e−0.2 t cos( 2 t ) on the interval 0 ≤ t ≤ 8. Fill in the expression which would complete the integral determining the arc length of this spiral on 0 ≤ t ≤ 8.∫ 08 dt and determine the arc length of the given spiral on 0 ≤ t ≤ 8.
- Show that the curve r(t) = ( t3 /4 - 2)i +(4/t-3)j+cos(t-2)k is tangent to the surface x3 + y3 + z3 - xyz = 0 at (0, -1, 1).(3) Find the arc length of the parametric curve(x, y, z) = (2 − 2 cos(t) − sin(t), 2 + cos(t) + 2 sin(t), 1 + 2 cos(t) − 2 sin(t)) for 0 ≤ t ≤ 5.Consider the curve C parametrized byx = −1 + 6 sin t and y =(1/2)− 6 cos t for −π ≤ t ≤ 7π. What is the radius, and the center (x,y), how many times is it transversed and in what direction?