Find the average rate of change of $g(x)=x^2+5 x+8$ from $x=4$ to $x=9$.
The average rate of change is $\square$.
Answer (1)
Calculate g ( 4 ) and g ( 9 ) :
g ( 4 ) = 4 2 + 5 ( 4 ) + 8 = 44 g ( 9 ) = 9 2 + 5 ( 9 ) + 8 = 134
Apply the average rate of change formula: 9 − 4 g ( 9 ) − g ( 4 ) = 5 134 − 44
Simplify the expression: 5 90 = 18
The average rate of change is: 18
Explanation
Understanding Average Rate of Change We are asked to find the average rate of change of the function g ( x ) = x 2 + 5 x + 8 from x = 4 to x = 9 . The average rate of change of a function g ( x ) over an interval [ a , b ] is given by the formula: b − a g ( b ) − g ( a ) In this case, a = 4 and b = 9 .
Calculating g(4) and g(9) First, we need to calculate g ( 4 ) and g ( 9 ) :
g ( 4 ) = ( 4 ) 2 + 5 ( 4 ) + 8 = 16 + 20 + 8 = 44 g ( 9 ) = ( 9 ) 2 + 5 ( 9 ) + 8 = 81 + 45 + 8 = 134
Calculating Average Rate of Change Now, we can plug these values into the formula for the average rate of change: 9 − 4 g ( 9 ) − g ( 4 ) = 9 − 4 134 − 44 = 5 90 = 18 So, the average rate of change of g ( x ) from x = 4 to x = 9 is 18.
Final Answer The average rate of change of the function g ( x ) = x 2 + 5 x + 8 from x = 4 to x = 9 is 18 .
Examples
The average rate of change is a fundamental concept in calculus with numerous real-world applications. For instance, consider a car's distance from its starting point modeled by the function g ( x ) = x 2 + 5 x + 8 , where x represents time in hours. Finding the average rate of change between x = 4 and x = 9 hours tells us the average speed of the car during that 5-hour interval. This concept is also used in economics to determine the average change in cost or revenue over a specific period, helping businesses make informed decisions about production and pricing strategies.
