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Homework Help: Derivatives, continuity

  1. Oct 13, 2011 #1
    1. The problem statement, all variables and given/known data

    2 problems.

    Find an example of a function f such that :
    the line y=2 is a horizontal asymptote of the curve y=f(x)
    the curve intersects the line y=2 at the infinitive number of points

    The position of an object moving along x axis is given at time t by:
    s(t)= 4t-4 if 2<t<8
    and = -68 +t(20-t) if 8<=t<=10

    Determine the acceleration and the velocity at any time t. Is the velocity continuous? Is acceleration continuous?

    2. Relevant equations

    3. The attempt at a solution

    For the first one, I'm really puzzled how y=2 can be horizontal asymptote, AND that the curve intersects it at infinitive points.

    I'm guessing that the function is y=2cos(1/x), as , y=2 is an asymptote even if my guess is right, how exactly should I solve that?

    For the second one, I found derivatives of s and s'

    s'= 4 if 2<t<8
    s'= 20-2t if 8<=t<=10
    s''=0 if 2<t<8 and
    s"= -2 if 8<=t<=10

    what then? how do i check if they are continuous?

    I know the definition of continuous function and what exactly is horizontal asymptote, but I have no idea how to solve these two problems here :(.

    Last edited: Oct 13, 2011
  2. jcsd
  3. Oct 13, 2011 #2
    1. I guess your example is correct. You should use whatever definition of asymptote you have and see if it applies to your example. Just another, maybe slightly simpler example: 2+sin(x)/x

    2. Just check if the functions "fit together" at t=8. For the velocity, you have 4 and 20-2*8=4, so the velocity is continuous. For the acceleration, you have 0 and -2, so the acceleration is non-continuous. In this case, continuity just means that you can draw the graph of the function without having to lift the pen at some point, so it's very intuitive. Check http://en.wikipedia.org/wiki/Piecewise
  4. Oct 13, 2011 #3
    1. Yeah I figured my example was correct, does my function pass the fact that

    "the curve intersects the line y=2 at the infinitive number of points" though?

    I just have no idea if it does or not.

    2. I kinda figured that actually later on

    Thank you for your answer!
    Last edited: Oct 13, 2011
  5. Oct 14, 2011 #4
    Sorry for bumping, I still am not really sure if I did #1 right, both
    y=2cos(1/x) and y=2+sinx/x do seem to have y=2 as horizontal asymptotes, but I am not really sure how the curves intersects the line y=2 at the infinitive number of points, how can that be if the line y=2 is an asymptote?
  6. Oct 14, 2011 #5
    Ummm, I just noticed that the curve in your example does not intersect, but is rather tangent to y=2 infinitely many times. So, to be on the safe side, you should use my example.
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