Q4:

            Compute and simplify the difference quotient  for ,

and then use it to find the average rate of change of  between  and

                             a) ,     b) ,     c)

 

 

 

A4:

            If  then  

 

But this difference quotient is just the average rate of change from  to ,

and so the average rate of change of  from  to 2.5 is 3.5, to 2.1 is 3.1, and to 2.01 is 3.01.

 

We could also compute the average rates directly as follows:

For case a) we have  ,

and we could do cases  b) and c) similarly.

 

But doing the simplification first is clearly much easier  especially if there are a lot of cases to consider, or if we are interested in seeing a pattern such as what happens at

 

What does happen at ?

(The applet at http://www.langara.bc.ca/~acooper/mathlets/SecTan/applet.htm might help you to see what’s going on)