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- A particle at (1, 0, 0) starts moving in space in such a way that its position vector at any time t ≥ 0 is R~ (t) = (cost + tsin t)ˆi + (sin t − t cost)ˆj + t²ˆk, t ≥ 0. (a) Find parametric equations for the line tangent to the trajectory of the particle at the point where t = π/2. (b) Calculate the acceleration of the particle at time t = π/2. (c) Calculate the total distance traveled by the particle in the time interval 0 ≤ t ≤ π/2(3z-4)/(z(2z+1))dz for a curve: |z|=1 by cauchy integral formulaA space curve Let w = x2e2y cos 3z. Find the value of dw/ dt at the point (1, ln 2, 0) on the curve x = cos t, y = ln (t + 2), z = t.
- 3Express the square of the arc length differential on the surface z=xy. What does it take to find the shortest curve (geodesic) on this surface that reaches from point (1,1,1) to point(2,3,6)?A thin plate of constant density d = 1 occupies the region enclosed by the curve y = 36/(2x + 3) and the line x = 3 in the first quadrant. Find the moment of the plate about the y-axis.or this problem, consider a particle traveling within the force field F = < -y,x,1/2 > along the parametrized curve r(t) = < t cos(t),t sin(t),1/2t > from the point (0,0,0) to the point (2pi,0,pi) Explain why the work done moving the particle along the path in this force field is positive. Compute the work done on a particle traveling along the given parametrized curve within the force field.
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- Let C be the boundary of the region in the first quadrant bounded by the x-axis, a quarter-circle with radius 9, and the y-axis, oriented counterclockwise starting from the origin. Label the edges of the boundary as C1,C2,C3 starting from the bottom edge going counterclockwise. Give each edge a constant speed parametrization with domain 0≤t≤1 How can I solve the blanks in the attached picture? Thank youStarting from the point (3,−4,−2) reparametrize the curve r(t)=(3+1t)i+(−4−3t)j+(−2−3t)k in terms of arclength. HINT. Your result should be the position of the "particle", which moves along the curve, after traveling distance s from the initial point.The temperature of a solid is given by the function B(x, y,2) =x*y: +4x, where x, y, z are space coordinates with respect to the centre of the solid. Find the directional derivative of at P(1.-2.1) along a =2i-2j+k. What are the magnitude and direction of the fastest decrease of the temperature from the point P? Also find the divergence of the temperature gradient at P.