Find the average rate of change of [tex]f(x)=5 x^2-2 x+6[/tex] from [tex]x=2[/tex] to [tex]x=4[/tex].
Answer (2)
Calculate f ( 2 ) which equals 22.
Calculate f ( 4 ) which equals 78.
Calculate the average rate of change using the formula 4 − 2 f ( 4 ) − f ( 2 ) = 2 78 − 22 .
The average rate of change is 28 .
Explanation
Problem Setup We are asked to find the average rate of change of the function f ( x ) = 5 x 2 − 2 x + 6 from x = 2 to x = 4 . The average rate of change is given by the formula: 4 − 2 f ( 4 ) − f ( 2 ) So, we need to calculate f ( 2 ) and f ( 4 ) .
Calculating f(2) and f(4) First, let's calculate f ( 2 ) :
f ( 2 ) = 5 ( 2 ) 2 − 2 ( 2 ) + 6 = 5 ( 4 ) − 4 + 6 = 20 − 4 + 6 = 22 Now, let's calculate f ( 4 ) :
f ( 4 ) = 5 ( 4 ) 2 − 2 ( 4 ) + 6 = 5 ( 16 ) − 8 + 6 = 80 − 8 + 6 = 78
Calculating Average Rate of Change Now we can plug these values into the formula for the average rate of change: 4 − 2 f ( 4 ) − f ( 2 ) = 4 − 2 78 − 22 = 2 56 = 28
Final Answer Therefore, the average rate of change of the function f ( x ) from x = 2 to x = 4 is 28.
Examples
Understanding the average rate of change is crucial in many real-world applications. For instance, imagine you're tracking the growth of a company's revenue over a period. If the revenue was $22,000 in 2020 and grew to $78,000 in 2022, the average rate of change would tell you the average yearly increase in revenue. In this case, it's $28,000 per year, indicating the company's growth trend. This concept is also used in physics to calculate average velocity or in economics to analyze changes in market prices.
The average rate of change of the function f ( x ) = 5 x 2 − 2 x + 6 from x = 2 to x = 4 is 28 . This indicates an average increase of 28 units for every unit increase in x over this interval.
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