• chutieuthichbeer

    khoảng 1 10 năm trước

    1. Sketch and also describe the curve given by

    a) R(t)=(t,0,t^3) [-1,3]---> -1<_ t <_ 3

    b) R(t)=[8cos(t)sin(2t), 8sin(t)sin(2t), 8cos(2t)]

    2. Find a vector function that defines the curve of intersection of the cylinder x^2 + y^2 = 1 and the plane y+z=2

    4. Show that for any differentiable vector function R(t) of constant length the vector dR/dt is perpendicular to R at any point t

    5. Find the length of the following curves. Graph the curves.

    a) r(t)=[t^2, sin(t) - tcos(t), cos(t) + tsin(t)] [0, Pi]----> 0<_ t <_ Pi

    b) r(t)=[cos(t), sin(3t), sin(t)] (nagetive infinity < t < positive infinity)

    6. Determine the vectors T, N, and B and also the curvature k at any point for the following curves
    a) r(t)=[sin(t), cos(t), sin(t)

    b) r(t)= [t^2, (t^3)2/3, t]

    7. For the curve y=lnx find the point where its curvature is a maximum. What happens to the curvature as x---> positive infinity ?

    8. Find a point on the curve r(t)=(t^3, 3t, t^4) where the normal plane is parallel to the plane 6x + 6y - 8z = 1.

    9. Find the velocity and speed, and the tangential and normal components of the acceleration if the position of an object is given by
    a) r(t)= i + (t^3 - 2)j + 2tk

    b) r(t)=[1; (1+t)/t; (1-t)^2/t]

    10. Find T, N, B, the curvature and the torsion for the curve .
    r(t)= [1sin(t)/square root 2; cos(t); 1sin(t)/square root 2]

    11. The force F(t)=(t^2, t-1, 1) Newton is applied to an object of mass 2 kg. At time t=0 the object is at the origin and has a velocity v=(1, -1, 3) . Find its position at time t.

  • mltr

    khoảng 1 10 năm trước
  • 1/ Hum có gì khó ... let t=-1,0,1,2,3 ... the other problem t =pi/4, pi/2 ... so on .. connect the dots ... if not sure how the curve go, check value of mid-point (b/w dots)

    2/ Find R(t)

    4/ dR/dt is a tangen at t ... R(t) x dR/dt = -1 (check formula again .. lâu rùi hủm nhớ rõ lém)

    5/ Find length = use line integral

    6/ hum nhớ

    7/ at x=1 .. after that, you can see on the graph .. google for lnx graph

    Will look at other problems later

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