Find the average rate of change for the given function.
[tex]f(x)=x^2+6 x \text { between } x=0 \text { and } x=4[/tex]
The average rate of change is $\square$ . (Simplify your answer.)
Answer (1)
Calculate f ( 0 ) : f ( 0 ) = 0 2 + 6 ( 0 ) = 0 .
Calculate f ( 4 ) : f ( 4 ) = 4 2 + 6 ( 4 ) = 16 + 24 = 40 .
Calculate the average rate of change: 4 − 0 f ( 4 ) − f ( 0 ) = 4 − 0 40 − 0 = 4 40 = 10 .
The average rate of change is 10 .
Explanation
Understanding the Problem We are asked to find the average rate of change of the function f ( x ) = x 2 + 6 x between x = 0 and x = 4 . The average rate of change of a function between two points is the change in the function value divided by the change in the x value. In other words, it's the slope of the secant line connecting the two points on the graph of the function.
Calculating f(0) First, we need to find the value of the function at x = 0 . We have
f ( 0 ) = ( 0 ) 2 + 6 ( 0 ) = 0 + 0 = 0
So, f ( 0 ) = 0 .
Calculating f(4) Next, we need to find the value of the function at x = 4 . We have
f ( 4 ) = ( 4 ) 2 + 6 ( 4 ) = 16 + 24 = 40
So, f ( 4 ) = 40 .
Calculating the Average Rate of Change Now, we can calculate the average rate of change using the formula:
Average rate of change = 4 − 0 f ( 4 ) − f ( 0 ) = 4 − 0 40 − 0 = 4 40 = 10
So, the average rate of change is 10.
Final Answer Therefore, the average rate of change of the function f ( x ) = x 2 + 6 x between x = 0 and x = 4 is 10 .
Examples
Imagine you're tracking the distance a cyclist travels over time. The function f ( x ) = x 2 + 6 x represents the distance in meters the cyclist has traveled after x seconds. Finding the average rate of change between x = 0 and x = 4 seconds tells you the cyclist's average speed during that time interval. This concept is useful for understanding how quantities change over specific intervals, whether it's speed, temperature, or population growth.
