Find the average rate of change of [tex]f(x)=-3 x^2-5 x[/tex] from [tex]x=4[/tex] to [tex]x=7[/tex]. Simplify your answer as much as possible.
Answer (2)
The average rate of change of the function f ( x ) = − 3 x 2 − 5 x from x = 4 to x = 7 is − 38 . This is calculated using the changes in the function's values at those points. The process involves evaluating the function at both endpoints and applying the average rate of change formula.
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Calculate f ( 4 ) : f ( 4 ) = − 3 ( 4 ) 2 − 5 ( 4 ) = − 68 .
Calculate f ( 7 ) : f ( 7 ) = − 3 ( 7 ) 2 − 5 ( 7 ) = − 182 .
Calculate the average rate of change: 7 − 4 f ( 7 ) − f ( 4 ) = 3 − 182 − ( − 68 ) = 3 − 114 .
Simplify to find the average rate of change: − 38 .
Explanation
Problem Setup We are given the function f ( x ) = − 3 x 2 − 5 x and asked to find the average rate of change from x = 4 to x = 7 . The average rate of change of a function f ( x ) between two points x = a and x = b is given by the formula b − a f ( b ) − f ( a ) . In this case, a = 4 and b = 7 .
Calculate f(4) First, we need to calculate f ( 4 ) . Substituting x = 4 into the function, we get: f ( 4 ) = − 3 ( 4 ) 2 − 5 ( 4 ) = − 3 ( 16 ) − 20 = − 48 − 20 = − 68
Calculate f(7) Next, we need to calculate f ( 7 ) . Substituting x = 7 into the function, we get: f ( 7 ) = − 3 ( 7 ) 2 − 5 ( 7 ) = − 3 ( 49 ) − 35 = − 147 − 35 = − 182
Calculate Average Rate of Change Now, we can calculate the average rate of change using the formula: 7 − 4 f ( 7 ) − f ( 4 ) = 7 − 4 − 182 − ( − 68 ) = 3 − 182 + 68 = 3 − 114 = − 38
Final Answer Therefore, the average rate of change of f ( x ) = − 3 x 2 − 5 x from x = 4 to x = 7 is − 38 .
Examples
The average rate of change is a fundamental concept in calculus with numerous real-world applications. For instance, consider a car's distance traveled over time. If the car travels a certain distance between two points in time, the average rate of change represents the average speed of the car during that interval. Similarly, in economics, it can represent the average change in cost or revenue over a specific period, providing valuable insights for business decisions. Understanding average rate of change helps in analyzing trends and making predictions based on data.
