Find the average rate of change of [tex]f(x)=-2 x^3+3 x^2[/tex] from [tex]x=-1[/tex] to [tex]x=3[/tex]. Simplify your answer as much as possible.
Answer (2)
The average rate of change of the function f ( x ) = − 2 x 3 + 3 x 2 from x = − 1 to x = 3 is − 8 . This means the function decreases by 8 units for each unit increase in x over this interval.
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First, find the values of the function at the given points: f ( 3 ) and f ( − 1 ) .
Calculate f ( 3 ) = − 2 ( 3 ) 3 + 3 ( 3 ) 2 = − 27 and f ( − 1 ) = − 2 ( − 1 ) 3 + 3 ( − 1 ) 2 = 5 .
Apply the average rate of change formula: 3 − ( − 1 ) f ( 3 ) − f ( − 1 ) = 4 − 27 − 5 .
Simplify to find the average rate of change: 4 − 32 = − 8 .
Explanation
Problem Setup We are asked to find the average rate of change of the function f ( x ) = − 2 x 3 + 3 x 2 from x = − 1 to x = 3 . The average rate of change is given by the formula: 3 − ( − 1 ) f ( 3 ) − f ( − 1 )
Evaluating the function First, we need to evaluate f ( 3 ) and f ( − 1 ) .
f ( 3 ) = − 2 ( 3 ) 3 + 3 ( 3 ) 2 = − 2 ( 27 ) + 3 ( 9 ) = − 54 + 27 = − 27 f ( − 1 ) = − 2 ( − 1 ) 3 + 3 ( − 1 ) 2 = − 2 ( − 1 ) + 3 ( 1 ) = 2 + 3 = 5
Calculating the average rate of change Now, we can plug these values into the formula for the average rate of change: 3 − ( − 1 ) f ( 3 ) − f ( − 1 ) = 3 − ( − 1 ) − 27 − 5 = 4 − 32 = − 8
Final Answer The average rate of change of f ( x ) from x = − 1 to x = 3 is − 8 .
Examples
The average rate of change is a fundamental concept in calculus with numerous real-world applications. For instance, consider a company tracking its revenue over a period. If the revenue was R ( t ) = − 2 t 3 + 3 t 2 (in thousands of dollars) where t is time in years, the average rate of change from year t = − 1 to t = 3 would tell us how the revenue changed on average per year during that period. In this case, the average rate of change is -8, meaning the company's revenue decreased by an average of $8,000 per year between year -1 and year 3. This information is crucial for making informed business decisions.
