Home > Calculus > Average Rate of Change Over an Interval

How do you find the average rate of change of #f(x)= -1/x# from [1,2]?

Answer 1

#(f(2)-f(1))/(2-1) = 1/2#

The average rate of change of function #f# on interval #[a,b]# is

#(f(b) - f(a))/(b-a)#

It is the ratio of the changes, it may also be written #(Deltaf)/(Deltax)# and it may be thought of as the slope of the line through the endpoints of the graph of #f# on the interval.

Algebraically it is (one version of) the difference quotient. (The quotient of the differences in #f# values and #x# values..)

In this instance

#(f(2)-f(1))/(2-1) = ((-1/2)-(-1))/(2-1) = 1/2#

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Answer 2

Scarlett Allison

To find the average rate of change of ( f(x) = -\frac{1}{x} ) from ( x = 1 ) to ( x = 2 ), use the formula:

[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} ]

where ( a = 1 ) and ( b = 2 ).

[ f(1) = -1/1 = -1 ] [ f(2) = -1/2 ]

Plug these values into the formula:

[ \text{Average Rate of Change} = \frac{f(2) - f(1)}{2 - 1} ] [ = \frac{-1/2 - (-1)}{1} ] [ = \frac{-1/2 + 1}{1} ] [ = \frac{1 - 1/2}{1} ] [ = \frac{1/2}{1} ] [ = \frac{1}{2} ]

So, the average rate of change of ( f(x) = -\frac{1}{x} ) from ( x = 1 ) to ( x = 2 ) is ( \frac{1}{2} ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.