An object moves with velocity vector (t,t?,-t) starting at (1,2,3) when t=0. Find the function r giving its location.
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- VECTOR DIFFERENTIATION: If R = e^(−t) i + ln(t^2+ 1) j - tant k. Find: (a) dR/dt, (b) d^2R/dt^2,(c) |dR/dt| ; (d) |d^2R/dt^2| at t = 0Gravitational potential The potential function for the gravitational force field due to a mass M at the origin acting on a mass m is φ = GMm/ | r | , where r = ⟨x, y, z⟩ is the position vector of the mass m, and G is the gravitational constant.a. Compute the gravitational force field F = -∇φ .b. Show that the field is irrotational; that is, show that ∇ x F = 0.Interpreting directional derivatives Consider the functionƒ(x, y) = 3x2 - 2y2.a. Compute ∇ƒ(x, y) and ∇ƒ(2, 3).b. Let u = ⟨cos θ, sin θ⟩ be a unit vector. At (2, 3), for what values of θ (measured relative to the positive x-axis), with 0 ≤ θ < 2π, does the directional derivative have its maximum and minimum values? What are those values?
- A. Find the directional derivative of the function at the given point in the direction of the vector v? f(x, y, z) = x2y+y2z P(1, 1, 1), v = (2, -1, 2) B. In which direction does the function f change the most rapidly at point P(1, 1, 1)? C. What is the rate of change of f at point P(1, 1, 1)?Determine the directional derivative of the function f (x, y, z) = x2 + xy + y + z2 in point P = (1; 2; 1) and in the direction of a vector orthogonal to the surface 2x2 - 3y2 - z + 1 = 0 in the point A = (2,1,6)You are walking on the surface described by f(x, y) = 3x^2 + 4xy + 6y^2. You arrive at the location P(0, 1, 6) and decide from that point on you want to walk in the direction of steepest ascent. Give a unit vector that gives the direction of steepest ascent from P, and give the rate of change of the ascent, with appropriate units (assuming distance on the surface z = f(x, y) is measured in meters).
- Evaluate along the curve y=x2 from (-1,1) to (2,4). First find the vector valued function r(t) defining the curve.A charged particle begins at rest at the origin. Suddenly, a force causes the particleto accelerate according to the vector function a(t) = ⟨ sin(t) , 6t , 2cos(t)⟩Find functions for the velocity, speed and position of the particle at time tThe temperature T, at a point (x, y) in the xy-plane is given by T(x, y) = 4x^2y^3degrees Celsius. Find a unit vector in the direction in which the temperaturedecreases most rapidly at (2, 3) and find this minimum rate of decrease intemperature at (2, 3).
- Find a vector parametrization of the curve x=−2z^2 in the xz-plane. Use t as the parameter in your answerCalculate the directional derivative in the direction of v at the given point. Remember to normalize the direction vector. f (x, y) = ln(x^2 + y^2), v = 3i − 2j, P = (1, 0)The flow lines (or streamlines) of a vector field are the paths followed by a particle whose velocity field is the given vector field. Thus the vectors in a vector field are tangent to the flow lines. (a) Use a sketch of the vector field F(x, y) = xi − yj to draw some flow lines. From your sketches, can you guess the equations of the flow lines? (b) If parametric equations of a flow line are x = x(t), y = y(t), explain why these functions satisfy the differential equations dx/dt = x and dy/dt = −y. (c) Solve the differential equations to find an equation of the flow line that passes through the point (x, y) = (−1, −1).