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- Find both the parametric and vector equation of the line segment between (−2, 3) and (5, −1) where 0 ≤ t ≤ 1. Please show all your work.2. Calculate the gradient vector Vf of the function f (x, y) = x² – x + y - x²y - 2y2 at the point (2,1) and sketch it on the attached contour plot (you can save the picture, open in photo editor and use drawing tools). Explain in one paragraph (about 200-300 words) the meaning of the gradient vector Vf(2,1), negative gradient vector -Vf(2,1).Find both the parametric and vector equation of the line segment between (−2, 3) and (5, −1) where 0 ≤ t ≤ 5. Please explain your steps. Thank you.
- Use the gradient vector to find the equation of the tangent plane to the surface x? + y? - at the point (2,2,4). Write your answer in the form Ax + By + Cz = D.Find the directional derivative of the function at the given point in the direction of vector v. f(x, y) = 2 + 6x√y, (4, 16), v = (4, -3) Duf(4,16) =Sketch the gradient vector Vf (4,6) for the function f whose level curves are shown. Explain how you chose the direction and length of this vector.
- Find the rate of change of F at P in the direction of the vector ⟨2,-3,6⟩ and the minimum rate of change of F at P.Evaluate fs xyz dS where S is the triangle with vertices (1,0,0), (0, 2, 0), (0, 1, 1).Find the Directional derivative of the function f(x,y, z) = Vx² + y² + z² from point (1,2, –2) in the direction of vector v = (-6,6 ,–3). %3D