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- Find the area. y = 3 cos(4x), y = 3 − 3 cos(4x), 0 ≤ x ≤ π/4Suppose that the position of one particle at time t is given by x1 = 3 sin(t), y1 = 2 cos(t), 0 ≤ t ≤ 2? and the position of a second particle is given by x2 = −3 + cos(t), y2 = 1 + sin(t), 0 ≤ t ≤ 2?. (a) Graph the paths of both particles. How many points of intersection are there? __ points of intersection (b) Are any of these points of intersection collision points? That is, are the particles ever at the same place at the same time? If so, find the collision points. (Enter your answers as a comma-separated list of ordered pairs of the form (x, y). If there are no collision points, enter DNE.) (x, y) =__ (c) Describe what happens if the path of the second particle is given by x2 = 3 + cos(t), y2 = 1 + sin(t), 0 ≤ t ≤ 2?. The circle is centered at (x, y) = __ There are _ intersection point(s), and there are _ collision point(s).A force of F(x)=x2−cos(3x)+2, x is in meters, acts on an object. What is the work required to move the object from x=2 to x=5 ?
- The prolate cycloid given by x = 2t − π sin t and y = 2 − π cos t crosses itself at the point (0, 2), as shown in Figure 10.32. Find the equations of both tangent lines at this pointA particle moves in the xy-plane in such a way that its path is defined by x = et cos t and y = et sin 2t. Find the speed of the particle when t = /2.A rectangle is to be inscribed under the arch of the curve y = 4 cos (0.5x) from x = -π to x = π. What are the dimensions of the rectangle with largest area, and what is the largest area?
- Qifi: Is the differential 'eq'uati::m’ (co§:xv-:|- In jz) dqé - (% + eJ’) dy = 0 exact?Find a parametrization of the right branch (x > 0) of the hyperbola(x/a)2−*(y/b)2= 1 using cosh t and sinh t. How can you parametrize the branch x < 0?When a particle is located a distance meters from the origin,a force of cos ( pie x / 3 )newtons acts on it. How much work is done in moving the particle from x = 1 to x = 2 ?Interpret your answer by considering the work done from x = 1 to x = 1.5 to and from x =1.5 to x = 2?