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r(t)= t-1/t+1i+sin(t)j+ln(9-t^2)k
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- Find the two-unit vectors that are parallel to the tangent line to the curve y =x2sin(x) at the point (π/2, π2/4).Find the open interval(s) on which the curve given by the vector-valued function is smooth. (Enter your answers using interval notation.)an object is moving in 3-dimensional space with velocity vector v(t)=(sin(t),t,t^2) starting at point (0,0,1). At what point will it be after 5 times units?
- Evaluate along the curve y=x2 from (-1,1) to (2,4). First find the vector valued function r(t) defining the curve.What is special about the derivatives of vector functions of constant length? Give an example.Find the directional derivative of f(x,y,z) = z3-x2y at the point (4, -3, 4) in the direction of the vector v = <-1,-3,-2>.
- Find the gradient vector field for the scalar function f(x,y)=sin(2x)cos(6y). Enter the exact answer in component form. ∇(x,y)=Identify the minimum for the function using the vectoral application of Newton’s method while applying dynamicSuppose ƒ is differentiable at (9, 9), ∇ƒ(9, 9) = ⟨3, 1⟩, and w = (1, -1). Compute the directional derivative of ƒ at ⟨9, 9⟩ in the direction of the vector w.