Interpreting the Sign of the First Derivative (Increasing and Decreasing Functions)

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Derivatives #12: Intro to Graphing Derivatives

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Key Questions

  • A function is monotonically increasing if it is always increasing.

    From a graph point of view, this means that moving to the right in the graph, the function will only increase.

    In more mathemtical notation:

    #(AA x,y in RR )(x>y)=>(f(x)>f(y))#

  • If the first derivative at a given point #c#, denoted #f'(c)#, is positive, then the function #f(x)# is increasing at that point.

    Remember that the derivative represents the rate of change of the function #f(x)#. If the first derivative is positive then the rate at which the function increases with #x# is also positive, meaning that as we make #x# larger, we make #f(x)# larger too.

    This is most obvious in the case of a simple function like #f(x) = 2x#. We know that #f'(x) = 2# at every point on the real line: the rate of change is positive. Hence, we expect #f(x)# to increase with #x#.

    Here are 2 more examples:

    enter image source here

    The red function is #f(x)# from E.S's example, the blue function is #g(x)=x^2# and the green function is #h(x)=x^3#. The derivatives for the new functions are: #g'(x)=2x# and #h'(x)=3x^2#.

    You can see the the red line is always increasing.

    The blue curve is decreasing when #x<0# and increasing when #x>0#. This matches the derivative #g'(x)=2x#; we can see that #g'# is negative when #x<0# and positive when #x>0#.

    The green curve is increasing everywhere except #x=0#. It's hard to tell from looking at the graph because it's pretty flat. However, examining the derivative #h'(x)=3x^2#, we can see that #x^2# is always positive except at #x=0#.

    So you can determine intervals of increase by looking at the graph although it may be difficult at some points. Or you can more accurately determine the intervals algebraically using the first derivative.

  • It can be done by looking at the sign of the first derivative.

    Remember: if the first derivative of a function is positive, then the function itself is increasing.

    For #f(x) = sin(x)# the first derivative equals

    #f'(x) =cos(x)#

    So the question becomes: when is #cos(x)>0#?

    The solution is pretty straight forward if you know your trigonometry:

    #-pi/2 < x < pi/2 mod 2pi#

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