simplify the difference quotients

Question

f(x+h)-f(x)/h and f(x)-f(a)/x-a   for the following function by rationalizing the numerator.

 

f(x)=sqr rt of x^2-8

Answer 1

f(x + h) - f(x)

____________  =

         h

 

 

 

√((x + h)² - 8) - √(x² - 8)

_______________________ =

                   h

 

 

 

√(x² + 2xh + h² - 8) - √(x² - 8)

___________________________  =

                     h

 

 

Multiply the expression by          √(x² + 2xh + h² - 8) + √(x² - 8) 

                                              ______________________________  
                                                √(x² + 2xh + h² - 8) + √(x² - 8)

 

 

 

x² + 2xh + h² - 8 - (x² - 8)

________________________________ 

h[√(x² + 2xh + h² - 8) + √(x² - 8)]

 

 

x² and the constant term 8 on top cancel each other out.

 

 

                h(2x + h)

__________________________________

h[√(x² + 2xh + h² - 8) + √(x² - 8)]

 

 

 

h cancels on top and bottom.  Final answer is

 

 

                    2x + h

_______________________________

√(x² + 2xh + h² - 8) + √(x² - 8)