Question

FInd the average rate of change of f(x)=3x^2-7 between (a, f(a)) and (a+h, f(a+h)). Simplify your answer completely.

Answer 1

The function is f(x) = 3x^2 - 7

Substitute x = a in f(x)

f(a)  =  3a^2 - 7

Substitute x = a + h in f(x)

f(a + h)  =  3(a + h)^2 - 7

 

             =  3(a^2 + h^2 + 2ah) - 7

 

(a + h)  =  3a^2 + 3h^2 + 6ah - 7

 

The rate of change is 

 

[f(a + h) - f(a) ] / (a + h - a )  =  [ (3a^2 + 3h^2 + 6ah - 7 ) -  ( 3a^2 - 7 ) ] / h

 

                                                    =  [ 3a^2 + 3h^2 + 6ah - 7  -   3a^2 + 7  ] / h

 

                                                    =  [ 3h^2 + 6ah ] / h

 

                                                    =  h (3h + 6a) / h

 

[f(a + h) - f(a) ] / (a + h - a )  =  3h + 6a

 

Answer :

 

[f(a + h) - f(a) ] / (a + h - a )  =  3h + 6a