Particle MotionA particle moves in the xy-plane along the curve represented by the
(a) Use a graphing utility to graph r. Describe the curve.
(b) Find the minimum and maximum values of
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Calculus (MindTap Course List)
- 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?arrow_forwardAnalyzing motion Consider the position vector of the following moving objects.a. Find the normal and tangential components of the acceleration.b. Graph the trajectory and sketch the normal and tangential components of the acceleration at two points on the trajectory. Show that their sum gives the total acceleration. r(t) = 2 cos t i + 2 sin t j, for 0 ≤ t ≤ 2πarrow_forwardAngular speed Consider the rotational velocity fieldv = ⟨ -2y, 2z, 0⟩ .a. If a paddle wheel is placed in the xy-plane with its axis normalto this plane, what is its angular speed?b. If a paddle wheel is placed in the xz-plane with its axis normalto this plane, what is its angular speed?c. If a paddle wheel is placed in the yz-plane with its axis normalto this plane, what is its angular speed?arrow_forward
- Gravitational 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.arrow_forwardTorsion of a helix Compute the torsion of the helixr(t) = ⟨a cos t, a sin t, bt⟩, for t ≥ 0, a > 0, and b > 0.arrow_forwardIntegrals of Line and Work A cyclist rides up a mountain along the path shown in the figure. She makes one complete revolution around the mountain in reaching the top, while her vertical rate of climb is constant. Throughout the trip, she exerts a force described by the vector field F(x,y,z) = z2i + 3y2j + 2xk What is the work done by the cyclist in travelling from A to B?arrow_forward
- Harmonic functions A scalar-valued function φ is harmonicon a region D if ∇2φ = ∇ ⋅ ∇φ = 0 at all points of D. Show that the potential function φ(x, y, z) = | r | -p is harmonicprovided p = 0 or p = 1, where r = ⟨x, y, z⟩ . To what vectorfields do these potentials correspond?arrow_forwardNonuniform straight-line motion Consider the motion of an object given by the position function r(t) = ƒ(t)⟨a, b, c⟩ + ⟨x0, y0, z0⟩, for t ≥ 0,where a, b, c, x0, y0, and z0 are constants, and ƒ is a differentiable scalar function, for t ≥ 0.a. Explain why r describes motion along a line.b. Find the velocity function. In general, is the velocity constant in magnitude or direction along the path?arrow_forwardFlux of the radial field Consider the radial vector field F = ⟨ƒ, g, h⟩ = ⟨x, y, z⟩. Is the upward flux of the field greater across the hemisphere x2 + y2 + z2 = 1, for z ≥ 0, or across the paraboloid z = 1 - x2 - y2, for z ≥ 0?Note that the two surfaces have the same base in the xy-plane and the same high point (0, 0, 1). Use the explicit description for the hemisphere and a parametric description for the paraboloid.arrow_forward
- Surface integrals of vector fields Find the flux of the following vector field across the given surface with the specified orientation. You may use either an explicit or a parametric description of the surface. F = ⟨e-y, 2z, xy⟩ across the curved sides of the surfaceS = {(x, y, z): z = cos y, | y | ≤ π, 0 ≤ x ≤ 4}; normal vectorspoint upward.arrow_forwardSurface integrals of vector fields Find the flux of the following vector field across the given surface with the specified orientation. You may use either an explicit or a parametric description of the surface. F = r/ | r | 3 across the sphere of radius a centered at the origin,where r = ⟨x, y, z⟩ ; normal vectors point outward.arrow_forwardA 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 tarrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageElementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning