Which expression can be used to determine the average rate of change in [tex]f(x)[/tex] over the interval [tex]$[2,9]$[/tex]?
Answer (2)
The average rate of change of a function f ( x ) over an interval [ a , b ] is given by b − a f ( b ) − f ( a ) .
For the interval [ 2 , 9 ] , the average rate of change is 9 − 2 f ( 9 ) − f ( 2 ) .
Simplifying the expression gives 7 f ( 9 ) − f ( 2 ) .
The expression f ( 9 − 2 ) = f ( 7 ) does not represent the average rate of change.
The correct expression is 7 f ( 9 ) − f ( 2 ) .
Explanation
Understanding the Problem The problem asks us to identify the expression that calculates the average rate of change of a function f ( x ) over the interval [ 2 , 9 ] . We are given a candidate expression f ( 9 − 2 ) and need to determine if it correctly represents the average rate of change.
Recalling the Formula The average rate of change of a function f ( x ) over an interval [ a , b ] is given by the formula: b − a f ( b ) − f ( a ) This formula calculates the change in the function's value divided by the change in the input variable over the interval.
Applying the Formula In our case, the interval is [ 2 , 9 ] , so a = 2 and b = 9 . Plugging these values into the formula, we get: 9 − 2 f ( 9 ) − f ( 2 )
Simplifying the Expression Simplifying the denominator, we have: 7 f ( 9 ) − f ( 2 ) This expression represents the average rate of change of f ( x ) over the interval [ 2 , 9 ] .
Evaluating the Given Expression The given expression is f ( 9 − 2 ) = f ( 7 ) . This expression simply evaluates the function at x = 7 and does not represent the average rate of change. The average rate of change requires evaluating the function at both endpoints of the interval and calculating the difference in function values divided by the difference in input values.
Final Answer Therefore, the expression that determines the average rate of change in f ( x ) over the interval [ 2 , 9 ] is 7 f ( 9 ) − f ( 2 ) .
Examples
Imagine you're tracking the growth of a plant over a week. If the plant was 2 inches tall on Monday and 9 inches tall on Sunday, the average rate of change in height per day is calculated using the same principle as the average rate of change of a function. It helps you understand how quickly the plant is growing on average each day, even if the growth isn't constant.
The expression for the average rate of change of the function f ( x ) over the interval [ 2 , 9 ] is 7 f ( 9 ) − f ( 2 ) . This formula calculates the difference in function values at the endpoints divided by the difference in input values. Other options like f ( 7 ) do not yield the average rate of change.
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