Find the average rate of change of the function over the given intervals.
[tex]f(x)=10 x^3+10[/tex];
a) [tex][2,4][/tex],
b) [tex][-5,5][/tex]
a) The average rate of change of the function [tex]f(x)=10 x^3+10[/tex] over the interval [tex][2,4][/tex] is ____.
(Simplify your answer.)
Answer (2)
Calculate f ( 2 ) and f ( 4 ) to find the change in the function's value over the interval [ 2 , 4 ] .
Apply the average rate of change formula: 4 − 2 f ( 4 ) − f ( 2 ) = 2 650 − 90 = 280 .
Calculate f ( − 5 ) and f ( 5 ) to find the change in the function's value over the interval [ − 5 , 5 ] .
Apply the average rate of change formula: 5 − ( − 5 ) f ( 5 ) − f ( − 5 ) = 10 1260 − ( − 1240 ) = 250 .
Explanation
Problem Analysis We are given the function f ( x ) = 10 x 3 + 10 and asked to find the average rate of change over the intervals [ 2 , 4 ] and [ − 5 , 5 ] . The average rate of change of a function f ( x ) over an interval [ a , b ] is given by the formula b − a f ( b ) − f ( a ) .
Calculating Average Rate of Change for [2,4] a) For the interval [ 2 , 4 ] , we need to calculate f ( 2 ) and f ( 4 ) .
f ( 2 ) = 10 ( 2 ) 3 + 10 = 10 ( 8 ) + 10 = 80 + 10 = 90
f ( 4 ) = 10 ( 4 ) 3 + 10 = 10 ( 64 ) + 10 = 640 + 10 = 650
Now, we can calculate the average rate of change over the interval [ 2 , 4 ] :
4 − 2 f ( 4 ) − f ( 2 ) = 4 − 2 650 − 90 = 2 560 = 280
Calculating Average Rate of Change for [-5,5] b) For the interval [ − 5 , 5 ] , we need to calculate f ( − 5 ) and f ( 5 ) .
f ( − 5 ) = 10 ( − 5 ) 3 + 10 = 10 ( − 125 ) + 10 = − 1250 + 10 = − 1240
f ( 5 ) = 10 ( 5 ) 3 + 10 = 10 ( 125 ) + 10 = 1250 + 10 = 1260
Now, we can calculate the average rate of change over the interval [ − 5 , 5 ] :
5 − ( − 5 ) f ( 5 ) − f ( − 5 ) = 5 − ( − 5 ) 1260 − ( − 1240 ) = 5 + 5 1260 + 1240 = 10 2500 = 250
Final Answer a) The average rate of change of the function f ( x ) = 10 x 3 + 10 over the interval [ 2 , 4 ] is 280 .
b) The average rate of change of the function f ( x ) = 10 x 3 + 10 over the interval [ − 5 , 5 ] is 250 .
Examples
The average rate of change is a fundamental concept in calculus with many real-world applications. For instance, consider a rocket launch where f ( t ) represents the rocket's altitude at time t . The average rate of change over an interval [ a , b ] would tell us the rocket's average speed during that time period. Similarly, in economics, if f ( x ) represents the cost of producing x units of a product, the average rate of change over an interval [ a , b ] gives the average cost per unit produced as production increases from a to b . This concept is also used in physics to calculate average velocity and acceleration, and in finance to determine the average growth rate of an investment.
The average rate of change of the function f ( x ) = 10 x 3 + 10 over the interval [ 2 , 4 ] is 280 and over the interval [ − 5 , 5 ] is 250.
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