See the circular helix with equation D(t)= (-15 cost,15 sin t,8t). 1. Look for the value of t such that D(t)= (0,−15,12π).
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- Consider the parametric equations x = t2 - 1 and y = t2 + 2t. (a) Find (dy)/(dx) and (d2y)/(dx2). (b) Set up, but do not evaluate, an integral representing the arc length over the interval 2 ≤ t ≤ 4.A slime mold grows in a spiral pattern with polar radius r=θ3r=θ3. If the mold grows by continuing the spiral endpoint with constant angular velocity, θ=ωtθ=ωt, what will be the cartesian location of the endpoint att=10t=10?Given the parametric equation x=(4/5)t5/2 and y= (1/4)t4-t find the arc length of the curve on the interval 0 less than or equal to t less than or equal to 1
- Consider the parametric equation x = t2 - 1 and y = t2 + 2t. (a) find (dy)/(dx) and (d2y)/(dx2). (b) Set up, but do not evaluate, and integral representing the arc length over the interval 2 ≤ t ≤ 4.The ammonia molecule can be treated as a symmetric rigid rotator, with its 3 hydrogen atoms lying in the xy plane and the nitrogen atom lying above the plane on the z axis, as shown in the attached image. If we call the moment of inertia about the z acis I3, and the moments about the pair of axes perpendicular to the z axis I1, the rotational energy of the molecule can be written as Eq 1 (see provided image). Say that at time t = 0, the wave function of the ammonia molecule is given as Eq 2 (see provided image), where the Yl, m1(θ, Φ) are the sphereical harmonics. What are the energy eigenvalues for this symmetric rigid rotator? Find (θ, Φ, t) , the wave function at time t.Find the arc length of the curve given in parametric form by c(t) = (t^3, t^2) for 0 ≤ t ≤ 1.
- The curve C is parametrized byr(t) = < 3t2, 5 sin(t2), 5 cos(t2) >, 0 ≤ t ≤ 2. (a) Compute the arc-length function s(t). (b) If the curve C is reparametrized using the arc-length parameter s, determine the interval c ≤ s ≤ d where this parameter varies. What is the value of c? (c) In part (b), what is the value of d?Find the arc length of the curve on the given interval. Parametric Equations x = √t, y = 3t − 1 Interval 0 ≤ t ≤ 1Consider the curve C parametrized byx = −1 + 6 sin t and y =(1/2)− 6 cos t for −π ≤ t ≤ 7π. What is the radius, and the center (x,y), how many times is it transversed and in what direction?
- Consider the following. x = sin(6t), y = −cos(6t), z = 24t; (0, 1, 4?) (a) Find the equation of the normal plane of the curve at the given point. (b)Find the equation of the osculating plane of the curve at the given point.Consider the following. x = sin(6t), y = −cos(6t), z = 24t, (0, 1, 4?) Find the equation of the normal plane of the curve at the given point. Find the equation of the osculating plane of the curve at the given point.Determine a > 0 so that ∫Cx^2 + y^2 + (z/a)ds = π + (π^2)/2, where C is the part of the circular helix parameterized by r (t) = (cos (at), sin (at), at), 0≤t≤π.