Match the vector-valued functions (a)–(f) with the space curves (1)–(vi) in Figure 10. (a) r(t) = (t + 15, e0.081 cos t, e0.081 sin t) %3D 25t (b) r(t) = (cos 1, sin 1, sin 12t) (c) r(t) %3D %3D
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Could you explain (a) (b) (c) please?
Exaplain in words please.
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- Find the parametric equation for a line that is tangent (intersection) of the two fields below: −2x + 3y + 7z = −2 and x + 2y - 3z = −5Find a vector equation for the tangent line to the curve of intersection of the cylinders x2+y2=25 and y2+z2=20 at the point (3, 4, 2).Suppose that z is an implicit function of x and y in a neighborhood of the point P = (0, −3, 1) of the surface S of equation: exz + yz + 2 = 0 An equation for the tangent line to the surface S at the point P, in the direction of the vector w = (3, −2), corresponds to:
- Sketch the vector-valued function r(t) = ⟨cos(t), 1, sin(t)⟩, indicating the orientation of the curve with an arrow.Determine the parametric equation for a line that is tangent (intersection)of the two fields below:−2x + 3y + 7z = −2 and x + 2y - 3z = −5A particle moves along a circular path over a horizontal xy coordinate system, at constant speed. At time t1 = 4.90 s, it is at point (4.60 m, 5.00 m) with velocity (2.00 m/s)ĵ and acceleration in the positive x direction. At time t2 = 13.6 s, it has velocity (–2.00 m/s)î and acceleration in the positive y direction. What are the x and y coordinates of the center of the circular path? Assume at both times that the particle is on the same orbit.
- give the position vectors of particles moving alongvarious curves in the xy-plane. In each case, find the particle’s velocityand acceleration vectors at the stated times, and sketch them asvectors on the curve. Motion on the parabola y = x2 + 1r(t) = ti + (t2 + 1)j; t = -1, 0, and 1Suppose a > 0the vector function that describes the motion of a particle for t ≥ 0. Find the tangential and normal components of accelerationConsider the parametric curve segment (t, t2), t ∈ [0, 1]. What is the firstorder derivative of the curve at t = 0? Show that exactly the same curve segment can be re-parameterized so that the first-order derivative at t = 0 is different.
- ) Compute the directional derivative of f(x, y) = x2y at the point (1, 1) in adirection normal to the ellipse x2 + 2y2 = 1 at the point (1, 0).Show that the vector-valued function r(t) = ti + 2t cos tj + 2t sin tk lies on the cone 4x2 = y2 + z2 . Sketch the curve.Find an equation of the tangent plane to the surface z=−1x2+3y2−3x+1y+1 at the point (2, 2, 5).