1. Let R(t) be a smooth vector-valued function such that Ř(0) = (-1,2, – 1), ' (0) : (0, 1, –3), Ř" (0) = (4,0, 1). (a) Determine the curvature of the graph of R at t = 0. (b) Find an equation of the osculating plane at t = 0.
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- Consider a curve C in 3D space given by the vector function r(t) = <a sin(t), b cos(t), ct> where a, b, and c are constants. Determine the curvature κ(t) of the curve C at any point t.If an object travels in the xy-plane along the curve traced out by the vector function r(t) = (t 3/2,-t) for t ≥ 0, then the total distance traveled by the object from t = 0 to t = 4 is1.Compute the unit tangent vector to the curve parametrized by r(t)= (t,t^2,t^3) at the point r(1)
- Find the unit tangent vector T(t) and a set of parametric equations for the tangent line r(t) = e2ti + cos tj − sin 3tk to the space curve at point P (1, 1, 0).Consider the curve C parametrized byx = (t^2+1)/(t^2-1) and y = (2t)/(t^2-1) for all t in (−∞, −1 ) ∪ (−1, 1) ∪ (1, ∞) .By squaring both x and y, find an equation of C in terms of just x and y (no t):For vector field F = (-2y, y), y > 0. Find all points P such that the amount of fluid flowing into P equals the amount of fluid flowing out of P. Write down the equation these points satisfy.
- Consider the curve C in R3 whose parameterization is given by: eq. in image If is there a point on C P( 3/2, 1/2, √2) A vector tangent to C in P corresponds to options in imageThe graph y = f(x) in the x plane automatically has the parameterization x = x, y = f(x) and the vector formula r(x) = xi + f(x)j. Use this formula to demonstrate that if f is a function of x twice differentiable, then, B) Use the kappa formula in subsection a) to determine the Curvature of y = In(cos x), -pi/2 < x < pi/2. Compare your Answer with that of exercise 1. C) Demonstrate that the curvature is zero at a turning point.Find an equation of the tangent plane to the surface z=−1x^2+1y^2+3x−1y+3 at the point (5, 5, 13).z=