If f is defined by #f(x)= x^3 +2x^2 + x#, how do you find the value of x when the average rate of change of f on the interval #x=-1# to #x =2# is equal to the instantaneous rate of change on the interval from #x=-1# to #x=2#? The answer is x=0.786?

Answer 1

Ok, this is a good one!
Consider that the instantaneous rate of change is the derivative, #f'(x)#, while the average is the difference between two values (evaluated in #x=a and x=b#) of your function in an interval divided by the interva (#b-a#)l!!!

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Answer 2

To find the value of ( x ) when the average rate of change of ( f ) on the interval ( x = -1 ) to ( x = 2 ) is equal to the instantaneous rate of change on the same interval, you can follow these steps:

  1. Find the average rate of change of ( f ) on the interval ( x = -1 ) to ( x = 2 ).
  2. Find the derivative of ( f(x) ) with respect to ( x ).
  3. Evaluate the derivative at ( x = 0.786 ).
  4. Verify if the average rate of change and the instantaneous rate of change are equal at ( x = 0.786 ).

If the given answer is x = 0.786, then you would verify by calculating the average rate of change and the instantaneous rate of change at ( x = 0.786 ) and confirming if they are indeed equal.

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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.

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.

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.

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.

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