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How do you find the average rate of change of #f(x)=2x^2+2# from -1 to 1?

Answer 1

Average Rate of Change: #color(green)(0)#

Since #f(-1)=f(1)# the amount of change is #0#

#rarr # average rate of change is #0# (sometimes the rate is negative and sometimes it's positive, but in the end the average is 0).

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Answer 2

Aurora Alvarado

To find the average rate of change of ( f(x) = 2x^2 + 2 ) from ( x = -1 ) to ( x = 1 ), you first need to calculate the values of ( f(x) ) at the endpoints of the interval. Then, you find the difference in the function values and divide it by the difference in the input values.

Calculate ( f(-1) ):
( f(-1) = 2(-1)^2 + 2 = 2(1) + 2 = 2 + 2 = 4 )
Calculate ( f(1) ):
( f(1) = 2(1)^2 + 2 = 2(1) + 2 = 2 + 2 = 4 )
Find the difference in function values:
( f(1) - f(-1) = 4 - 4 = 0 )
Find the difference in input values:
( 1 - (-1) = 1 + 1 = 2 )
Calculate the average rate of change:
( \text{Average Rate of Change} = \frac{f(1) - f(-1)}{1 - (-1)} = \frac{0}{2} = 0 )

Therefore, the average rate of change of ( f(x) = 2x^2 + 2 ) from ( x = -1 ) to ( x = 1 ) is 0.

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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.