How do you find the average rate of change of #f(x)=2x^2+1 # on [x,x+h]?
1 Answer
Explanation:
The average rate of change of a continuous function,
# (f(b)-f(a))/(b-a) #
So the average rate of change of the function
# Aroc = ( f(x+h)-f(x) ) / ( (x+h)-(x) )#
# " " = ( f(x+h)-f(x) ) / ( h ) \ \ \ \ \ ..... [1]#
# " " = ( 2(x+h)^2+1-(2x^2+1) ) / ( h )#
# " " = ( 2(x^2+2xh+h^2)+1 -2x^2-1 ) / ( h )#
# " " = ( 2x^2+4xh+2h^2 -2x^2 ) / ( h )#
# " " = ( 4xh+2h^2 ) / ( h )#
# " " = 4x+2h#
Which is the required answer.
Additional Notes:
Note that this question is steered towards deriving the derivative
# f'(x) = lim_(h rarr 0) (f(x+h)-f(x))/h #
This is the function we had in [1], so as we take the limit as
# f'(x) = lim_(h rarr 0) 4x+2h #
# " " = 4x#