Over which interval does [tex]$f$[/tex] have a positive average rate of change?
[tex]$f(x)=x^2+10$[/tex]
A. [-4,-1]
Answer (1)
Calculate the average rate of change using the formula b − a f ( b ) − f ( a ) .
For the interval [ − 4 , − 1 ] , the average rate of change is − 1 − ( − 4 ) f ( − 1 ) − f ( − 4 ) = − 5 , which is negative.
For the interval [ 1 , 4 ] , the average rate of change is 4 − 1 f ( 4 ) − f ( 1 ) = 5 , which is positive.
For the interval [ − 5 , − 2 ] , the average rate of change is − 2 − ( − 5 ) f ( − 2 ) − f ( − 5 ) = − 7 , which is negative.
For the interval [ − 1 , 2 ] , the average rate of change is 2 − ( − 1 ) f ( 2 ) − f ( − 1 ) = 1 , which is positive.
Therefore, the function f has a positive average rate of change over the intervals [ 1 , 4 ] and [ − 1 , 2 ] .
Explanation
Understanding Average Rate of Change We are given the function f ( x ) = x 2 + 10 and asked to find the interval over which f has a positive average rate of change. The average rate of change of a function f over an interval [ a , b ] is given by the formula: b − a f ( b ) − f ( a ) We need to find an interval [ a , b ] such that this expression is positive.
Testing Interval [-4, -1] Let's analyze the given option (A) [ − 4 , − 1 ] . Here, a = − 4 and b = − 1 . We calculate f ( a ) and f ( b ) :
f ( a ) = f ( − 4 ) = ( − 4 ) 2 + 10 = 16 + 10 = 26 f ( b ) = f ( − 1 ) = ( − 1 ) 2 + 10 = 1 + 10 = 11 Now, we compute the average rate of change: b − a f ( b ) − f ( a ) = − 1 − ( − 4 ) 11 − 26 = 3 − 15 = − 5 Since − 5 < 0 , the average rate of change over the interval [ − 4 , − 1 ] is negative.
Testing Interval [1, 4] Now let's test the interval [ 1 , 4 ] . Here, a = 1 and b = 4 . We calculate f ( a ) and f ( b ) :
f ( a ) = f ( 1 ) = ( 1 ) 2 + 10 = 1 + 10 = 11 f ( b ) = f ( 4 ) = ( 4 ) 2 + 10 = 16 + 10 = 26 Now, we compute the average rate of change: b − a f ( b ) − f ( a ) = 4 − 1 26 − 11 = 3 15 = 5 Since 0"> 5 > 0 , the average rate of change over the interval [ 1 , 4 ] is positive.
Testing Interval [-5, -2] Now let's test the interval [ − 5 , − 2 ] . Here, a = − 5 and b = − 2 . We calculate f ( a ) and f ( b ) :
f ( a ) = f ( − 5 ) = ( − 5 ) 2 + 10 = 25 + 10 = 35 f ( b ) = f ( − 2 ) = ( − 2 ) 2 + 10 = 4 + 10 = 14 Now, we compute the average rate of change: b − a f ( b ) − f ( a ) = − 2 − ( − 5 ) 14 − 35 = 3 − 21 = − 7 Since − 7 < 0 , the average rate of change over the interval [ − 5 , − 2 ] is negative.
Testing Interval [-1, 2] Now let's test the interval [ − 1 , 2 ] . Here, a = − 1 and b = 2 . We calculate f ( a ) and f ( b ) :
f ( a ) = f ( − 1 ) = ( − 1 ) 2 + 10 = 1 + 10 = 11 f ( b ) = f ( 2 ) = ( 2 ) 2 + 10 = 4 + 10 = 14 Now, we compute the average rate of change: b − a f ( b ) − f ( a ) = 2 − ( − 1 ) 14 − 11 = 3 3 = 1 Since 0"> 1 > 0 , the average rate of change over the interval [ − 1 , 2 ] is positive.
Conclusion From the calculations above, we found that the average rate of change is positive for the interval [ 1 , 4 ] and [ − 1 , 2 ] .
Examples
Understanding the average rate of change is crucial in many real-world applications. For instance, consider a car accelerating. The average rate of change of its velocity over a time interval tells us the average acceleration during that period. Similarly, in economics, the average rate of change of a company's revenue over a quarter can indicate its growth trend. This concept helps in making informed decisions, whether it's predicting future performance or optimizing current strategies.
