Julia Pitre jdp8657

Week 1 1.4.2

url: http://http.tamu.edu/~jdp8657/homework.html

Problem: Construct the tangent line at t = p /3 to the circular curve in the text (x = cos t, y= sin t). What is the relationship between this line and the one in the previous exercise?

 

  1. The previous exercise consists of a helical curve :

    2. vector g(t) = (x, y, z)

    where x = cos t, y = sin t, z = t

  2. The general form for a tangent line is :

    3. vector x = vector g(to) + (t - to) * vector g’(to)

  3. The derivative of vector g(t) is :

    4. vector g’(t) = (-sint t, cos t, 1)

  4. At t= p /3, the equation for the tangent line of the helical curve is :

    5. vector x = (cos (p /3), sin (p /3), p /3) + (t- (p /3) ) * (-sin (p /3), cos (p /3), 1)

  5. When we find the equation for the circular curve, where x = cos t and y = sin t, we get:

    6.  

    vector x = (cos (p /3), sin (p /3) ) + (t- (p /3) ) * ( -sin (p /3), cos (p /3) )

     

    6. If we viewed both of these figures on the x-y axis, we would get the same circular figure. However, the z component is what makes the second figure a helical curve instead of a circle. The z component draws the circle out along the z axis. The relationship between the tangent lines is that the line for the circle can be viewed as the two dimensional representation of the tangent line for the helical curve.