Find the average rate of change of [tex]f(x)=4 x+5[/tex] from [tex]x=4[/tex] to [tex]x=9[/tex]. Simplify your answer as much as possible.
Answer (2)
The average rate of change of the function f ( x ) = 4 x + 5 from x = 4 to x = 9 is 4 .
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Calculate f ( 4 ) by substituting x = 4 into the function: f ( 4 ) = 4 ( 4 ) + 5 = 21 .
Calculate f ( 9 ) by substituting x = 9 into the function: f ( 9 ) = 4 ( 9 ) + 5 = 41 .
Determine the average rate of change using the formula 9 − 4 f ( 9 ) − f ( 4 ) = 9 − 4 41 − 21 = 5 20 .
Simplify the expression to find the average rate of change: 5 20 = 4 .
Explanation
Understanding the Problem We are asked to find the average rate of change of the function f ( x ) = 4 x + 5 from x = 4 to x = 9 . The average rate of change is a measure of how much the function's output changes per unit change in its input over a specific interval.
Formula for Average Rate of Change The average rate of change of a function f ( x ) over the interval [ a , b ] is given by the formula: b − a f ( b ) − f ( a ) In this case, a = 4 and b = 9 .
Calculating f(4) First, we need to find the value of the function at x = 4 :
f ( 4 ) = 4 ( 4 ) + 5 = 16 + 5 = 21
Calculating f(9) Next, we find the value of the function at x = 9 :
f ( 9 ) = 4 ( 9 ) + 5 = 36 + 5 = 41
Finding the Difference in Function Values Now, we can calculate the difference in the function values: f ( 9 ) − f ( 4 ) = 41 − 21 = 20
Finding the Difference in x Values And the difference in the x values: 9 − 4 = 5
Calculating the Average Rate of Change Finally, we can calculate the average rate of change: 9 − 4 f ( 9 ) − f ( 4 ) = 5 20 = 4
Final Answer The average rate of change of f ( x ) = 4 x + 5 from x = 4 to x = 9 is 4 .
Examples
Imagine you're tracking the distance a cyclist covers over time. The average rate of change helps you determine the cyclist's average speed between two points in time. For example, if the cyclist's distance is given by f ( x ) = 4 x + 5 , where x is time in hours and f ( x ) is distance in kilometers, the average speed between 4 and 9 hours is 4 km/hour. This concept is crucial for understanding motion and making predictions in various fields like physics and sports analytics.
