Theorem 7.5.4. The curvature of a smooth space curve defined by the vector valued function R(t) is given by k(t): X ||R' (t) × R"(t) | ||R' (t) ||³
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- 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 imageUse Theorem 11.24 to prove that the curvature of a linearfunction y = mx + b is zero for every value of x.Sketch the space curve r(t) = ⟨2 sin t, 5t, 2 cos t⟩ and find its length over the given interval [0, π] .
- 1.Compute the unit tangent vector to the curve parametrized by r(t)= (t,t^2,t^3) at the point r(1)Evaluate∮C (x + 3y)dx + ydy where C is the Jordan curve given by thegraphs of y = e^x, y = e^−x and the horizontal line y = e^−1a) By Green’s theoremb) By direct computationIf 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 is
- The 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.Sketch the space curve r(t) = ⟨4t, −cos t, sin t⟩ and find its length over the given interval [0, 3π/ 2] .