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Let f(x,y)=x^3y^4/12. Then ∇f = , and Duf(−3,5) in the direction of the
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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.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.Let the velocity vector be v(t)=⟨5t^4,−3sin(t),4exp(2t)⟩ and the initial position vector be r(0)=⟨−1,6,−1⟩. Compute the position vector r(t).
- Calculate the directional derivative of g(x, y, z) = z² - xy + 4y2 in the direction v = (1,-4, 3) at the point P = (3, 1, -9). Remember to use a unit vector in directional derivative computation. (Use symbolic notation and fractions where needed.) Dvg(3, 1,-9) =A stone is thrown from a rooftop at time t=0 seconds. Its position at time t is given by r(t)=6ti−4tj+(9.8−4.9t^2)k. The origin is at the base of the building, which is standing on flat ground. Distance is measured in meters. The vector i points east, j points north, and k points up.(a) How high is the rooftop? meters.(b) When does the stone hit the ground? seconds.(c) Where does the stone hit the ground? (in meters)(d) How fast is the stone moving when it hits the ground? (in meters per second)Evaluate the directional derivative Duf (2, 3) if ƒ (x,y) and T2 - y u is the unit vector in the same direction as (-3, 4).
- Give a vector parametric equation for the line through the point (4,-3) that is perpendicular to the line <1−4t,3t−3>Find the derivative of the vector functionr(t)=ta×(b+tc) wherea=⟨−4,−3,5⟩ b=⟨−1,2,5⟩ and c=⟨−3,−2,−3⟩find the vector equation for the line through the point A = (5, -2, 1) at t = 0. and B = (-3, 1, -2) at t = 1.
- Find a vector in the plane parallel to the line ax + by = c.Find the cartesian equation of the straight line tangent to the plane curve given by the vector equation r(t) = (t- 6) i + + 3t - 2) į. at the point P(-5,4) on the curve. Express your answer in the form y = ax + b , where a and b are integers. equarion of the tangent line is given by y = 9x + 49Consider the following points: A(1, 1, −2), B(−1, 2, 1) and C(1, 0, 1). Determine theparametric equations of either the unknown line or the unknown plane in R^3a. the line passing through the point B that is perpendicular to the vector→AB ×→AC
