Results 1 to 6 of 6

Math Help - Why do positive gradients for tangents show a curve is increasing?

  1. #1
    Newbie
    Joined
    Jul 2010
    Posts
    18

    Why do positive gradients for tangents show a curve is increasing?

    Imagine you have a curve and you find the derivative of that curve.

    It is a known fact that if the derivative is positive in certain areas then the curve is increasing in those areas. But why is this?

    The derivative shows the gradient of a tangent at a certain point. So if you find the derivative to be positive for a certain region (say x=0 to x=2), you know that all the gradients of tangents are positive in that area. But people seem to assume the curve is increasing in this area as well why is this?

    Sorry if i have worded the question badly it is hard to put into words, but i was just solving questions where is asked me to find where the curve was increasing and i started to think why positive gradients-of-tangents proved that the curve was increasing

    thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Unknown008's Avatar
    Joined
    May 2010
    From
    Mauritius
    Posts
    1,260
    A curve increases when the y value of the curve increases as the x value of the curve increases.

    The gradient at a point is the gradient of the tangent at that point, and this gradient can be found using:

    \frac{y_2 - y_1}{x_2 - x_1}

    It is positive when y2 > y1 and x2 > x1, or when y1 > y2 and x1 > x2.

    The moment you have the y and x coordinates of a point greater than the y and x coordinates of another point, you have an increasing 'curve' (it can be a line).

    Does this answer your question?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Jul 2010
    Posts
    18
    Maybe a better question to ask is why is the gradient of the tangent (at a point) the same as the gradient at that point on the curve?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor Unknown008's Avatar
    Joined
    May 2010
    From
    Mauritius
    Posts
    1,260
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,393
    Thanks
    1327
    Quote Originally Posted by deltasalt View Post
    Maybe a better question to ask is why is the gradient of the tangent (at a point) the same as the gradient at that point on the curve?
    Perhaps because "gradient of the tangent" is the definition of the "gradient of a curve"?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Newbie
    Joined
    Jul 2010
    Posts
    18
    Thanks for the help I gave it some thought and understand it better now. I think I was just starting to confuse myself
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. show that there exists an increasing function
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: October 12th 2011, 03:53 AM
  2. Replies: 3
    Last Post: February 15th 2011, 02:54 PM
  3. Show that a sequence is increasing
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: March 31st 2010, 08:31 PM
  4. Tangents to a curve
    Posted in the Calculus Forum
    Replies: 2
    Last Post: May 16th 2009, 09:58 AM
  5. Show this curve has 2 tangents at (0,0)
    Posted in the Calculus Forum
    Replies: 2
    Last Post: March 23rd 2009, 01:36 AM

Search Tags


/mathhelpforum @mathhelpforum