Find the average rate of change of [tex]g(x)=3 x^2-4 x[/tex] from [tex]x=3[/tex] to [tex]x=8[/tex]. Simplify your answer as much as possible.
Answer (2)
The average rate of change of the function g ( x ) = 3 x 2 − 4 x from x = 3 to x = 8 is 29 . This is calculated by finding the values of the function at those points and applying the average rate of change formula. The final answer is 29 .
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Calculate g ( 3 ) = 3 ( 3 2 ) − 4 ( 3 ) = 15 .
Calculate g ( 8 ) = 3 ( 8 2 ) − 4 ( 8 ) = 160 .
Calculate the average rate of change using the formula 8 − 3 g ( 8 ) − g ( 3 ) = 5 160 − 15 .
Simplify the result to find the average rate of change: 29 .
Explanation
Problem Analysis We are asked to find the average rate of change of the function g ( x ) = 3 x 2 − 4 x from x = 3 to x = 8 . The average rate of change is the change in the function value divided by the change in x .
Calculate g(3) First, we need to find the value of the function at x = 3 . We have
g ( 3 ) = 3 ( 3 ) 2 − 4 ( 3 ) = 3 ( 9 ) − 12 = 27 − 12 = 15 .
Calculate g(8) Next, we need to find the value of the function at x = 8 . We have
g ( 8 ) = 3 ( 8 ) 2 − 4 ( 8 ) = 3 ( 64 ) − 32 = 192 − 32 = 160 .
Calculate Average Rate of Change Now, we can calculate the average rate of change using the formula:
8 − 3 g ( 8 ) − g ( 3 ) = 8 − 3 160 − 15 = 5 145 = 29 .
Final Answer The average rate of change of g ( x ) from x = 3 to x = 8 is 29 .
Examples
The average rate of change is a fundamental concept in calculus and has many real-world applications. For example, if g ( x ) represents the distance traveled by a car after x hours, then the average rate of change from x = 3 to x = 8 represents the average speed of the car between the 3rd and 8th hour of the trip. Similarly, in economics, if g ( x ) represents the profit of a company after x years, then the average rate of change from x = 3 to x = 8 represents the average annual profit growth between the 3rd and 8th year.
