An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (1)
Calculate the change in function value between x = 0 and x = 8 for each function.
Quadratic function: f ( 8 ) − f ( 0 ) = 130 − 2 = 128 .
Linear function 2 x + 2 : f ( 8 ) − f ( 0 ) = 18 − 2 = 16 .
Linear function 3 x + 2 : f ( 8 ) − f ( 0 ) = 26 − 2 = 24 .
Exponential function: f ( 8 ) − f ( 0 ) = 258 − 3 = 255 .
The exponential function increases at the fastest rate: f ( x ) = 2 x + 2 .
Explanation
Understanding the Problem We are given four functions: a quadratic function f ( x ) = 2 x 2 + 2 , two linear functions f ( x ) = 2 x + 2 and f ( x ) = 3 x + 2 , and an exponential function f ( x ) = 2 x + 2 . We want to determine which function increases at the fastest rate between x = 0 and x = 8 . To do this, we will calculate the change in the function value, f ( 8 ) − f ( 0 ) , for each function.
Quadratic Function Analysis For the quadratic function f ( x ) = 2 x 2 + 2 , we have f ( 0 ) = 2 ( 0 ) 2 + 2 = 2 and f ( 8 ) = 2 ( 8 ) 2 + 2 = 2 ( 64 ) + 2 = 128 + 2 = 130 . The change is f ( 8 ) − f ( 0 ) = 130 − 2 = 128 .
First Linear Function Analysis For the first linear function f ( x ) = 2 x + 2 , we have f ( 0 ) = 2 ( 0 ) + 2 = 2 and f ( 8 ) = 2 ( 8 ) + 2 = 16 + 2 = 18 . The change is f ( 8 ) − f ( 0 ) = 18 − 2 = 16 .
Second Linear Function Analysis For the second linear function f ( x ) = 3 x + 2 , we have f ( 0 ) = 3 ( 0 ) + 2 = 2 and f ( 8 ) = 3 ( 8 ) + 2 = 24 + 2 = 26 . The change is f ( 8 ) − f ( 0 ) = 26 − 2 = 24 .
Exponential Function Analysis For the exponential function f ( x ) = 2 x + 2 , we have f ( 0 ) = 2 0 + 2 = 1 + 2 = 3 . We calculated f ( 8 ) = 2 8 + 2 = 256 + 2 = 258 . The change is f ( 8 ) − f ( 0 ) = 258 − 3 = 255 .
Comparing the Rates of Increase Comparing the changes in f ( x ) for each function, we have:
Quadratic function: 128
First linear function: 16
Second linear function: 24
Exponential function: 255
The exponential function has the largest change (255) between x = 0 and x = 8 . Therefore, the exponential function increases at the fastest rate.
Final Answer The exponential function f ( x ) = 2 x + 2 increases at the fastest rate between x = 0 and x = 8 .
Examples
Understanding the rate of increase of different functions is crucial in various real-world applications. For instance, in finance, comparing linear growth (simple interest) versus exponential growth (compound interest) helps investors understand how their investments can grow over time. Similarly, in biology, population growth can be modeled using exponential functions, while resource consumption might be modeled using linear functions. Comparing these rates helps predict potential ecological imbalances. In computer science, algorithm efficiency can be analyzed by comparing the growth rates of different algorithms as the input size increases.
