Evaluate the unoriented line integral +y² + z²) ds along the parametric curve a : [0, 1] → R³ with a(t) = (t, cos(3t), sin(3t)). (2a² + y? + z²) ds =
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- Write parametric equations for a cycloid traced by a point P on a circle of radius a as the circle rolls along the x -axis given that P is at a maximum when x=0.Find a set of parametric equations for the tangent line to the curve of intersection of the surfaces z = x2 + y2 , z = 4 − y at the given point (2, −1, 5).Find parametric equations for the line tangent to the curve of intersection of the surfaces tan-1(x? + y? + z²) + e(2 Inx+3In y) – xvz and 15 sec-1(2v2x + 2v2y + v2z) at the point (1, 1, 1).
- a) Find the equation of tangent line of curve that is given by parametric equations x = 3cost + sint y = e2t at the point (3, 1). b) Find the extrema points of f(x, y) = x³ + 6x² + 3y? – 12xy + 9xFind a set of parametric equations for the tangent line to the curve of intersection of the surfaces x2 + y2 + z2 = 14, x − y − z = 0, at the given point (3, 1, 2).³Find the second derivative for the following set of parametric equations. a) x = √t, y = 2t + 4 b) x = sec(t), y = tan(t)
- Find the equation of the tangent line of the parametric equations x = 2 – 3 cos 0, y = 3+ 2 sin 0 at the points (–1,3) and (2,5). .Sketch the plane curve represented by the given parametric equations x = 4 cos t + 2, y = 4cos t - 1. Then use interval notation to give relation’s domain and range.Let F(x, y, z) = (6xyz+4x) i+(3a²z+ze-")j+(3x²y-e¬³) k, and let C be the portion of the parametric curve given by x = 6 cos t, y = 6 sin t, z = 3+3 sin(11t/2) that starts at (6,0, 3) and ends at (-6,0,0) (see diagram to the right). Calculate %D F. dr. Show your work, and circle your final answer. (-6, 0, 6), (6, 0, 3) x.
- Find an equation, in slope-intercept form, of the line tangent to the curve with parametric equations x = t² + 4t and y = Int at the point (5,0). y =Find a parametric equation for the path of a particle that moves halfway around the circle (x^2)+(y-1)^2 = 9 counterclockwise, if the particle starts at (0,4). Let x and y be in terms of t for t between 0 and pi.Find parametric equations for the semicircle x2 + y2 = a2, y> 0, using as parameter the slope t = dy/dx of the tangent to the curve at (x, y).