Place the three functions in order from the fastest decreasing average rate of change to the slowest decreasing average rate of change on the interval $[0,3]$.
[tex]f(x)=16(\frac{1}{2})^x[/tex]
[tex]g(x)[/tex]
| x | g ( x ) |
| -1 | 81 |
| 0 | 27 |
| 1 | 9 |
| 2 | 3 |
| 3 | 1 |
[tex]h(x)[/tex]
[ ] [tex]f(x)[/tex]
[ ] [tex]h(x)[/tex]
[ ] [tex]g(x)[/tex]
Answer (1)
Calculate the average rate of change for f ( x ) on [ 0 , 3 ] : 3 − 0 f ( 3 ) − f ( 0 ) = 3 2 − 16 = 3 − 14 .
Calculate the average rate of change for g ( x ) on [ 0 , 3 ] : 3 − 0 g ( 3 ) − g ( 0 ) = 3 1 − 27 = 3 − 26 .
Compare the average rates of change: 3 − 26 < 3 − 14 , so g ( x ) decreases faster than f ( x ) .
Since we have no information about h ( x ) , the order from fastest to slowest decreasing rate is g ( x ) , f ( x ) , h ( x ) . The final answer is g ( x ) , f ( x ) , h ( x ) .
Explanation
Understanding the Problem We are given two functions, f ( x ) = 16 ( 2 1 ) x and g ( x ) defined by a table of values. We are asked to order the three functions f ( x ) , g ( x ) , and h ( x ) from the fastest decreasing average rate of change to the slowest decreasing average rate of change on the interval [ 0 , 3 ] . We don't have any information about h ( x ) , so we can't include it in the ordering.
Calculating Average Rate of Change for f(x) First, we need to calculate the average rate of change for f ( x ) on the interval [ 0 , 3 ] . The average rate of change is given by the formula: 3 − 0 f ( 3 ) − f ( 0 ) We calculate f ( 0 ) and f ( 3 ) :
f ( 0 ) = 16 ( 2 1 ) 0 = 16 ( 1 ) = 16 f ( 3 ) = 16 ( 2 1 ) 3 = 16 ( 8 1 ) = 2 So the average rate of change for f ( x ) is: 3 − 0 2 − 16 = 3 − 14 ≈ − 4.67
Calculating Average Rate of Change for g(x) Next, we calculate the average rate of change for g ( x ) on the interval [ 0 , 3 ] . The average rate of change is given by the formula: 3 − 0 g ( 3 ) − g ( 0 ) From the table, we have g ( 0 ) = 27 and g ( 3 ) = 1 . So the average rate of change for g ( x ) is: 3 − 0 1 − 27 = 3 − 26 ≈ − 8.67
Comparing the Rates of Change Since we don't have any information about h ( x ) , we cannot determine its average rate of change. Therefore, we can only compare f ( x ) and g ( x ) .
Comparing the average rates of change, we have 3 − 26 < 3 − 14 . This means that g ( x ) has a faster decreasing average rate of change than f ( x ) .
Final Answer Therefore, the order from the fastest decreasing average rate of change to the slowest decreasing average rate of change is g ( x ) , f ( x ) , and h ( x ) since we don't know anything about h ( x ) .
Examples
Understanding rates of change is crucial in many real-world scenarios. For instance, consider the depreciation of two cars, where f ( x ) represents the value of one car and g ( x ) represents the value of another car over time. By calculating and comparing their average rates of change, you can determine which car loses value faster. This helps in making informed decisions about investments, maintenance, and resale strategies. Similarly, in business, comparing the rates of change of different marketing strategies can help allocate resources effectively.
