For the graph of y=cosx

Question

a) For what values of x does the instantaneous rate of change appear to equal 0?

b) For what values of x does the instantaneous rate of change appear to reach a maximum value? A minimum value?

 

Answer 1

Hi Amber,

 

When a question asks about the instantaneous rate of change at a particular moment, it is asking what the slope is at that particular moment.  When we are looking for an instantaneous rate of change equal to 0, we are looking for when is the slope of the curve equal to 0.  We have slopes of curves equal to 0 when the function's curve goes from increasing to decreasing OR decreasing to increasing.  So you are looking for where does the graph of cosine go from increasing to decreasing or decreasing to increasing.  Look at a graph of cosine.  Notice that when x = 0, you have one of these moments when the curve goes from increasing to decreasing.  Do you see another place this happens?  Look at x = 2pi.  Notice this happens again.  Now look at when x = pi.  Notice that curve goes from decreasing to increasing.  Therefore, we have found three places on the function y = cos x where the curve changes from increasing to decreasing or decreasing to increasing.  Therefore, at each of these three places (These places are x = 0, pi, and 2pi) are places where the slope of the curve is 0.  Therefore, at each of these three places we have an instantaneous rate of change of 0.

 

Please let me know if this helps or if I can help further.

 

Thank you,

Tyler