Can someone help me find the average rate of change? f(x)=2x^2+1

Question

f(x)=2x^2+1 between the points (x, f(x)) and (x + h, f(x + h)).

Answer 1

The average rate of change is defined by

 

[f(b) - f(a)] / (b - a)

 

where a and b represent the intervals in the x-coordinate.  So applying the coordinates into the formula, the rate of change between the given points is

 

[f(x + h) - f(x)] / (x + h - x) =

 

[f(x + h) - f(x)] / h

 

 

The average rate of change is now the difference quotient.  Now we apply the given function into this.

 

[2(x + h)² + 1 - (2x² + 1)] / h =

 

[2(x² + 2xh + h²) + 1 - (2x² + 1)] / h =

 

[2x² + 4xh + 2h² + 1 - 2x² - 1] / h =

 

[4xh + 2h²] / h

 

 

Factor out a 2h.

 

2h(2x + h) / h

 

 

Cancel out an h.

 

2(2x + h) =

 

4x + 2h