1. Consider the graph of r(t) = 2t i+ tj +t° k Determine parametric equations of the tangent line to the curve at the point where the curve intersects the plane x-2y-z = 8.
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- 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 = 8.1. If the parametric curve $x=f(t), y=g(t)$ satisfies $g^{\prime}(1)=0,$ then it has a horizontal tangent when $t=1$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)
- determine the parametric coordinate of the critical point of the curve x=t^2+3t=2 and y=t^2-1.Solve for the critical points of parametric equation x=t^3-1, y=t^2+tFind parametric equations for the tangent line to the curve with the given parametric equations at the specified point ->(5,0,0). x=5e^t, y=te^4t,z=te^(t^5)
- Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. x = t2 + 24 , y = ln(t2 + 24), z = t; (5, ln(25), 1)Show that the tangent line at a point P = (x0, y0) on the hyperbola(x/a)2−(y/b)2= 1 has equation Ax − By = 1 where A = x0 a2 and B = y0b2 .Write the parametric equations for the tangent line to the curve of intersection of surfaces x=10x^2+2y^2 and z=x+y+10 at the point (1,1,12)
- 4. Determine the length of the parametric curve given by the following parametric equations. (a) x = t2 - 1, y = t3 + 1, 0 ≤ t ≤ 1 (b) x = cos(5t), y = sin(5t), 0 ≤ t ≤ πBy sketching the surface that has equation x2 + y2 - z2 = 0, show that the parametric curve r(t) = (3t cos(3t), 3t sin(3t), 3t) lies on that surface.A curve in the plane is defined parametrically by the equations written below. An equation of the line tangent to the curve at t=1 is