Particle Motion A 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: Early Transcendental Functions
- Torsion 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_forwardHarmonic 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_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_forward
- Nonuniform 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_forwardInterpreting 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_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
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